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The Sharp Erd\H{o}s-Tur\'an Inequality

Erd\H{o}s and Tur\'an proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erd\H{o}s-Tur\'an inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erd\H{o}s-Tur\'an inequality is obtained based on a thorough understanding of these equilibrium distributions.

1. Introduction 1.1. Main Results and Consequences. Erdős and Turán prove a beautiful result on the distribution of roots for polynomials f (z) ∈ C[z] in their 1950 work [ET50], that is, roughly speaking, if |f (z)| attains small value on the unit circle |z| = 1, then firstly the magnitudes of all roots of f (z) are close to 1, secondly the angular distribution of the roots is close to equidistribution on T = R/Z. See [Sou19, Figure 1] for a pictorial illustration. The statement on the magnitude of roots is a consequence of Jensen's formula in complex analysis, see e.g. [Sou19, Theorem 1] for a compact treatment. The statement on the angular distribution requires much more work.
The main goal of this paper is to give a sharp inequality characterizing this phenomenon of equidistribution in angles. Let f (z) = n k=0 a k z k be a degree n polynomial with complex coefficients and a 0 = 0. We denote its roots by r j e 2πiθj for 1 ≤ j ≤ n with θ j ∈ T = R/Z. For α ≤ β < α + 1, we write N f (α, β) to be the number of roots with θ j ∈ [α, β] when considered as a subset in T. The theorem relates two quantities of f (z). We first define the discrepancy of a polynomial f to be measuring how large f is on the unit circle up to a normalizing factor. Our main theorem is the following, Moreover, this inequality is sharp.
Theorem 1.1 gives a sharp improvement upon the original inequality of Erdős and Turán, where the constant √ 2 is replaced with 16. By a family of polynomials constructed in [AM96], it is clear that H[f ] is the optimal scaling. Therefore the last unknown component is the constant, which is solved by Theorem 1.1.
Historically, one of the motivations for Erdős and Turán to study this type of inequality is from number theory. The question of bounding the number of real solutions of a polynomial is of great interest. One perspective for the Erdős-Turán inequality is to give an error term for the angular distribution when the uniform distribution is considered as the main term in n (1.4) The dependence of f in the error term can be bounded by √ 2 · log 1/2 ( f L ∞ (|z|=1) / |a 0 a n |) by Theorem 1.1. Bloch and Pólya [BP32] first investigated how to bound the number of real zeros of a polynomial f (z) by the size of its coefficients. Schmidt [Sch32] and Schur [Sch33] refined the result by giving a bound in terms of a certain height just involving the degree n and sum of coefficients. By applying the inequality to small intervals near the x-axis, Erdős and Turán recover the result of Schmidt and Schur as an immediate consequence. Moreover due to a different choice of height function, the upper bound of Erdős and Turán gives a slight improvement up to the constant. We denote N θ (f ) (respectively N + (f ) and N − (f )) to be the number of roots with angle θ (respectively positive and negative) of f (z) ∈ C[z]. As a by-product of Theorem 1.1, we also obtain sharp estimates for the number of real roots with a given sign in terms of H[f ].

Theorem 1.2 (Sharp Estimates for Signed Real Roots).
For any polynomial f (z) ∈ C[z] with f (0) = 0 and any θ ∈ T, we have the sharp inequality (1.5) In particular, we have the sharp estimate for the number of positive real roots of f (z) and the number of negative real roots of f (z).
Although the sharp estimate for the total number of real roots (including both positive and negative) does not follow directly from Theorem 1.1, we will prove in a separate forthcoming note the sharp estimate for the number of real roots in terms of H[f ] from a similar approach we prove Theorem 1.1.
This inequality also has close connections to the theory of complex analysis and harmonic functions. In [Gan54], Ganelius first made the connection of this inequality to harmonic functions. Let u and v be a pair of conjugate harmonic functions inside the unit circle. Ganelius proved a theorem bounding the variation of v by the maximal value of u and ∂v/∂θ in the unit disk. He then applies it to deduce the inequality of Erdős and Turán with a better constant. Mignotte [Mig92] slightly refined the result by replacing the upper bound of u with the integral of u + (the positive part of u) on the unit circle. We manage to show that Ganelius' conjugate function problem turns out to be equivalent to the Erdős and Turán inequality, thus Theorem 1.1, in turn, implies a sharp estimate on harmonic functions. To our knowledge, such an equivalence has been neither noticed nor utilized before.

Theorem 1.3 (Sharp Estimates for Harmonic Functions
). Let f (z) = u(z)+iv(z) be an analytic function in |z| < 1 with f (0) = 0. Suppose u ≤ H and ∂v/∂θ ≤ K in |z| < 1 where H, K > 0. Then we prove the following sharp estimate for |z 1 |, |z 2 | < 1. (1.7) 1.2. Questions and History. In an earlier work [ET40], Erdős and Turán studied the relation between the discrepancy of a distribution {z i } ⊆ [−1, 1] and the values of f (z) = j (z − z j ) on [−1, 1]. This might be the origin of the inequality we are considering in this paper. Actually we can consider Theorem 1.1 as an analogue by replacing the interval [−1, 1] with the unit circle. Indeed it has been observed by [Sch33] that in order to prove Theorem 1.1 it suffices to consider polynomials with unimodular roots. Therefore the question at the core is fundamentally the following: given f (z) = j (z −z j ) with z j = e 2πiθj , what is the minimal value of max |z|=1 |f (z)|? With such a simple and clean form of this question, the inequality of Erdős and Turán inevitably has connections to many questions in different areas. Besides the application to bounding the number of real roots of polynomials, in [ET50] this inequality is also applied to recover the result of Jentzsch [Jen16] and Szegő [Sze34] in complex analysis that every point of the circle of convergence for a power series is a limit point of zeros of its partial sums. It is noted by [ET40] that such an equidistribution theorem has a potential theoretic characterization that dates back to Hilbert in 1897. We can imagine that the roots of a polynomial are negatively charged particles and they repel each other with a force, and equidistribution corresponds to a state where the energy is minimized. Indeed, one of our main tools in this paper will be to apply recent development in modern potential theory and energy minimization to review this old question. Potential theoretic approach has also appeared in studies of closely related questions, for example in Bilu's result on equidistribution of roots for f (z) ∈ Z[x] [Bil97], see [Gra07] for a nice exposition. This result also has applications towards small points on abelian varieties [Zha95,Zha98]. In [CGPS00], this inequality is applied to study distribution of roots of Fekete polynomials, which has applications on the distribution of L-function values.
After the fundamental works of Erdős and Turán [ET40,Erd42,ET50] in the 1940s, questions on equidistribution and discrepancy have been generalized in various directions. We refer interested audiences to [AB02] for a collection of results on generalizations. We give some examples, far from being exhaustive, in the following. Instead of using this notion of height H[f ], one can bound the discrepancy D[f ] using other quantities. Mignotte in [Mig92] gives a bound of D[f ] in terms of h[f ] := 1 n T (log |f (z)| √ |a0an| ) + dθ ≤ H[f ], which is equivalent to take L 1 (T) norm of log |f (z)|/ |a 0 a n | whereas H[f ] is close to a L ∞ norm of log |f (z)|/ |a 0 a n |. See also [Sou19,CDF + 21] for inequalities involving h[f ]. In [Bil97], Bilu uses the Mahler measure for an integral polynomial M [f ] = exp( T log |f (z)| dθ) to bound the discrepancy. This is also generalized to higher dimensions. Hüesing [Hue01] has used energy to bound discrepancy for signed measures. Instead of working over T, [AB97a, AB97b,AB99] studies the discrepancy of µ on a general quasiconformal curve and bound the discrepancy in terms of the value log |f (z)| on the curve. On the segment [−1, 1], [Bla92,Tot93] studies the equidistribution distribution of simple roots of a polynomial. Results in higher dimensions are also studied in [Sjö72,G00]. More recently, in [Ste21], Steinerberger studies the bound for Wasserstein distance from the uniform distribution, which generalizes the notion of discrepancy on T. We also mention works on roots distribution of polynomials related to Erdős-Turán inequality. For more refined discussion on the modulus of roots for polynomials, see [Erd08]. Erdős-Turán inequality is also applied to study roots of polynomials with small coefficients [OP93], Fekete polynomials [CGPS00], and random polynomials [HN08,PS14].
Despite the huge amount of generalization, the original form of Erdős and Turán inequality hasn't been improved much ever since the early 1950s. As commented in [AB02], the constant remains the only factor that is not sharp in this inequality. Early work of Ganelius [Gan54] improved this constant to 2π/k ≈ 2.5619 where k is the Catalan constant. In a very recent work [Sou19], Soundararajan uses h[f ] to bound D[f ] by an elegant Fourier argument. As a consequence it sharpens the constant to 8/π ≈ 2.5464. In the AIM workshop in 2021, a group involving the second author sharpens the tools of Soundararajan. In [CDF + 21], the group improves the constant to 4/ √ π ≈ 2.2567 by giving a complete solution of a Fourier optimization problem. This seems to be the best one can extract from this Fourier argument. We remark that optimizing the inequality with respect to h[f ] will be a different problem, as the optimal polynomial will probably be different. It is observed by Mignotte [Mig92], an unpublished note of Soundararajan and the recent AIM work [CDF +  Indeed, given f (z) = a n j (z − r j e 2πiθj ) andf (z) = j (z − e 2πiθj ), one can observe that |f (z)| |a 0 a n | ≤ |f (z)|, for |z| = 1, (1.8) whereas the discrepancy remains the same, see for example [Sou19, Section 3]. Therefore to study the optimal constant, it suffices to focus on f with all roots on the unit circle T. After normalization to monic polynomials, we can assume |a 0 | = a n = 1, so H[f ] is simply the maximal value of log(|f (z)| 1/n ) on unit circle T.
To demonstrate the phenomenon quantified by this inequality, we look at two contrasting polynomials (z − 1) n and z n − 1 with all roots on the unit circle. The polynomial f (z) = (z − 1) n is the most discrepant polynomial with degree n since z = 1 is a root of multiplicity n and the angular distribution is supported on a single point θ = 0, which is far away from being equidistributed. Meanwhile |f (z)| attains the maximal value 2 n at z = −1, which is exponentially large in the degree, and D[f ] is achieved at α = β = 0 with the value D[f ] = 1. Theorem 1.1 predicts 1 ≤ √ 2 · log 2 ≈ 1.18. The other polynomial f (z) = z n − 1 has all roots that are n-th roots of unity, and is the most evenly distributed polynomial with degree n. The discrepancy is D[f ] = 1/n and max |z|=1 |f (z)| is as small as 2. In this case Theorem 1.1 says 1/n ≤ √ 2 log 2/n. This also implies that H/D 2 cannot be bounded from above.
1.3.2. From Discrete to Continuous. Although this question is originally stated as one about roots distribution of polynomials, it can be considered as a question just about distributions of n points on the unit circle, thanks to Schur's observation and the convenient translation between polynomials and their roots. Since the degree of f can be arbitrarily large, the extreme case will clearly take place when n is large enough. This leads to one of the main ideas in proving Theorem 1.1, that is, to solve the discrete problem using continuous methods. Instead of considering discrete measures supported on n points, we will enlarge the class of distributions of consideration to all probability measures M on T.
Under the language of probability measures, we can describe the root distribution of f (z) by an empirical measure of n points where δ θj is the Dirac function at θ j .
where I ⊂ T is any closed interval on the unit circle and We then extend the question to an optimization problem of H[ρ]/D[ρ] 2 for general ρ ∈ M. Notice that we can also conveniently approximate a probability distribution by empirical measures with large n, see Section 6 for the details. Therefore it suffices for us to prove the following.
Theorem 1.4. For all probability measure ρ ∈ M on T, we have the sharp inequality (1.11) 1.3.3. Potential Theory. As an optimization problem, we see that the optimizer is not among the trivially extremal ones, in view of the two previous examples. So one of the main challenges is to construct distributions with the optimal value. The main tool for us to characterize the shape of the optimal distribution is potential theory and energy minimization. We start by giving a potential theoretic description of Theorem 1.4. The reader can also find basic potential theoretic facts in [ST13]. As we noted before H[ρ] = − ess inf(W * ρ), where W (x) = − log |2 sin(πx)|. We can view W as a pairwise interaction potential among particles. Its negative sign stands for the repulsion when two particles are close. For a probability distribution ρ ∈ M, we denote V [ρ] := W * ρ to be the potential field generated by ρ with the logarithmic potential W , then the total potential energy of ρ is (1.12) Since W (x) is mean zero over T, we see V [1 T ] = W * 1 T ≡ 0 and it is easy to show that 1 T is the unique probability measure with E[1 T ] ≤ 0. Such a probability measure ρ that minimizes the total energy is called the energy minimizer or equilibrium. On the other hand, it was shown in [Gra20] that D[ρ] := 2 · d ∞ (ρ, 1 T ), which is the Wasserstein-infinity distance between ρ and the uniform distribution 1 T . We mention that Wasserstein distances are the most natural distance in studies like optimal transport.
Then (1.11) as (1.13) Now we can interpret Theorem 1.4 as saying if the potential generated by ρ is close to that of the equilibrium, then ρ is close to the equilibrium under the d ∞ metric. In a forthcoming work, the authors will exactly use this formulation and give an application of Erdős-Turán type inequality towards stability of energy minimizers.
1.3.4. Energy Minimization. In this part, we characterize the property for the optimal distributions. We first restrict this optimization problem of H/D 2 from ρ ∈ M to ρ ∈ M D≥d , where D[ρ] ≥ d. Then the optimal value for H/D 2 can be approximated as d approaches 0, (1.14) Then in Section 2, our main goal is to translate the problem of optimizing H[ρ]/D[ρ] 2 with fixed d into an energy minimization problem with certain fixed external potentials U . Given an external potential U , we define for arbitrary ρ ∈ M m , which is the set of measures over T with total mass m. We study the properties of energy minimizers in Section 2.3. The theory of energy minimization guarantees that for fixed m and U , if W and U are both nice, then there exists a unique ρ ∈ M with minimal E U . Moreover, such an energy minimizer ρ always satisfies the condition We say that a measure ρ satisfying 1.16 is sediment with respect to U . Intuitively this condition is saying that all of the mass of ρ is resting at the bottom of the total potential V U [ρ] generated by ρ and U . It is shown that such a sediment distribution ρ is unique. Moreover among all ρ ∈ M m , the unique sediment ρ sees the highest bottom, i.e. has the largest ess inf V U [ρ]. See Proposition 2.8 for detailed treatment.
On the other hand, we develop a transport plan called microscopic diffusion in Section 2.2 to study the minimizer of H/D 2 in M D≥d . It shows that if a certain ρ ∈ M D≥d has (W * ρ)(x 0 ) > ess inf(W * ρ) for some x 0 ∈ supp ρ, then we can always construct a local diffusion such that W * ρ increases for x away from x 0 . Therefore as long as the local operation stays away from the endpoints of I witnessing D, we will be able to decrease H/D 2 . The key of this argument lies in the convexity of our potential function W (x) = − log |2 sin(πx)|. What it implies is an ultimate surprise and stunning connection, the minimizer of H/D 2 in M D≥d is the unique sediment distribution with respect to the external potential U generated by Dirac mass(es) at endpoints of I. This leads to the characterization of minimizer(s) for H/D 2 in Theorem 2.1. For this part, our method will also work in general for H/D α for any α ≥ 0 and for a more general class of interaction potentials W .
1.3.5. Construction of Minimizers. Next, our goal is to construct explicit distributions satisfying the characterization in Theorem 2.1. For each Dirac mass(es) configuration ρ d (position M and mass m), there exists a unique sediment distribution with respect to U = W * ρ d . Therefore it suffices to construct sediment distributions for each M and m in order to find a pool of candidates for the minimizer. In Section 3, we start with constructing a larger family of distributions, called stationary distribution with respect to U , which satisfies (1.17) Intuitively these are the distributions where all mass of ρ are stationary and has total force 0. The construction heavily depends on the fact that the kernel of the Hilbert transform is derivative of 6 logarithm. Due to this elegant connection between logarithmic potential and Hilbert transform, we are able to construct these distributions ρ as the real part (g) of some nice analytic functions g(z), while (g) = − 1 π (W * (g)) = − 1 π V U [ρ] is the derivative of the generated potential. By constructing analytic functions g(z) which are either totally real or totally imaginary, we thus obtain ρ where (W * (g)) = V U [ρ] is 0 on supp ρ. In order to evaluate the minimal value of H/D 2 conveniently, we also give the analogous construction for a similar class of distributions over R, using the Hilbert transform over R.
From this construction, we obtain a family of distributions over T parametrized by M , m and another parameter m 1 = (−M,M ) ρ(x) dx. By imposing the condition that ρ is sediment with respect to its Dirac masses configuration, we then get rid of the parameter and obtain a family just parametrized by M and m. We can discuss distributions over R similarly, and obtain a class of distributions called admissible distribution, also parametrized by two parameters.
We expect this construction to be useful for various types of optimization problems. In order to reduce the complexity of the problem over T, we make a comparison between sediment distributions over T and admissible distributions over R. A natural operation to get a distribution over T from a distribution µ over R is to take the periodization ρ • , see (5.1),(5.2). Notice that both families are parametrized by their Dirac mass(es) configurations. Therefore we can associate each sediment distribution ρ with a periodization ρ • of a unique admissible distribution µ over R. Then in Section 5, we firstly relate two functionalsH andD for distributions over R (see Section 4 for the definitions ofH andD for distributions over R ) to H and D over T, and prove the first comparison theorem where ρ • and ρ are both distributions over T and share the Dirac masses configuration, see Theorem 5.7. This concludes the chain of comparisons and proves the inequality in Theorem 1.4.

Notations and Preliminaries.
We will denote the torus T = R/Z, and use a real number x to indicate a point in T. Similarly, we will use an interval [a, b] ⊆ R with length less than 1 to indicate an interval I in T. For an interval I in T, we use I c to denote its complement. By measures over T, we always mean Borel measures on T. We denote M m to be the set of all measures ρ with total mass T ρ = m, thus M = M 1 is the set of all probability measures on T. For a point x ∈ T, we will denote a Dirac function located at x by δ x . A sequence {ρ n } of probability measures weakly converges to ρ if for all f ∈ C(T), we have lim n→∞ T f dρ n = We will always use W to denote a potential function that describes the pairwise interaction between particles. In some occasions, there is also an external potential which we denote by U . For ρ ∈ M, we denote V [ρ] := W * ρ to be the potential generated by ρ and V U [ρ] := W * ρ + U to be the total potential.
For ρ ∈ M, for W a potential function, several important functionals we use are where ess inf(f ) = sup{a ∈ R : |{x ∈ T : f (x) < a}| = 0} is the essential infimum of f . We will suppress the dependence of W in H W [ρ] when there is no confusion. We denote for some positive real number 0 < d ≤ 1. For us, the Fourier transform (respectively Fourier coefficients) of u is defined bŷ respectively when u is a function over R and T.
As an auxillary mollifier, we define ψ as It is even and supported on [−a, a]. For a ≤ 1/2, ψ a can be interpreted as a function on T via identifying T with [−1/2, 1/2), and thenψ a (k) =ψ(ak) for k ∈ Z whereψ is the Fourier transform of ψ on R.
1.5. Organization of Contents. In Section 2, we characterize minimizer(s) of H/D 2 in the class of probability measures M D≥d for any d > 0. We prove our main result Theorem 2.1 by relating minimizer(s) of H/D 2 and minimizer(s) of potential energy with logarithmic interaction and external potentials. Such a connection is established by showing that both minimizers are sediment distributions in Section 2.2 and Section 2.3 respectively. In Section 3, we construct sediment distributions, see in Propositions 3.4, 3.5 and 3.6. We achieve so by constructing a larger class of distributions by using the property of Hilbert transform. Meanwhile we also construct analogue distributions over R in Section 3.1 and Section 3.4.
In Section 4, we formulate an optimization problem over R analogous to minimizing H/D 2 over T, and then solve the minimal value for distributions constructed in Section 3.4.
In Section 5, we solve the optimization problem over T by relating it to the optimization problem over R via two comparison arguments, Theorem 5.2 in Section 5.1 and Theorem 5.7 in Section 5.3.
Finally, we conclude the proofs of Theorems 1.1 and 1.4 in Section 6.1, Theorem 1.2 in Section 6.2, and Theorem 1.3 in Section 7. Emanuel Carneiro, Mithun Das, Alexandra Florea, Angel V. Kumchev, Amita Malik, Micah B. Milinovich and Caroline Turnage-Butterbaugh for helpful conversations. The authors would like to thank José A. Carrillo and Dino J. Laurenzini for suggestions on earlier drafts.

Minimizer in M D≥d
In this section, we will study minimizer(s) of G among probability measures over T with D[ρ] ≥ d. In particular, for fixed 0 < d < 1, we are interested in what the value inf M D≥d G is, and whether it can be achieved and whether it is unique, and if it can be achieved or approximated how the minimizer(s) look like.
We will focus on the last two questions above in this section, and save the discussion on the minimal value in Section 4 and Section 5. Instead of just working with the functional sin(πx)|, in this section we will work in general with functional in the form where α ≥ 0 is an arbitrary real number and W : T → (−∞, ∞] is any potential function satisfying the following assumptions: • (H5): there exists a constant C 2 > 0 such that for all 0 < r < 1/2 and x ∈ T, It is straightforward to verify that W (x) = − log |2 sin(πx)| satisfies (H1) -(H5). We compute that W = 2π/(1 − cos(2πx)) = π/ sin 2 πx ≥ π. It follows from Jensen's formula that W has mean value 0. For (H5), it suffices to notice that −c log |x| ≤ W (x)−2 inf W ≤ −C log |x| for some positive constants c and C for x ∈ [−1/2, 1/2] and log |x| satisfies the condition via a computation.
Our main theorem for this section is the following: for some 0 < m ≤ 1/2, and ρ 1 ∈ M 1−2m is supported on two (possibly empty) closed intervals J I c and K I. (iii) Let ρ be a minimizer as in 1. Then ρ is a sediment distribution with respect to W , i.e., (2.4) (iv) Further assume that W satisfies (H5). Let ρ be a minimizer. Then ρ is the unique probability measure satisfying (2.4) in the class of probability measures with the same Dirac mass configuration.
Theorem 2.1 shows the existence of the minimizer(s) for G α within M D≥d and describes the shape of minimizer(s) to a great extent. Moreover, it shows that minimizer(s) have a characterizing property (2.4) and is the unique measure to satisfy this property with a fixed Dirac mass configuration. This translates the problem of constructing minimizers of G α to a problem of constructing sediment distributions, which we will do in Section 3.
We organize this section as the following. We first prove Theorem 2.1(i)in Section 2.1, see Corollary 2.3. In Section 2.2, we prove (ii) and (iii) in Theorem 2.6 and Corollary 2.5. Finally we prove (iv) in Section 2.3 in Corollary 2.9.
2.1. Properties of D and H. In this subsection, we will give some basic results about the functionals D and H in Lemma 2.2. Then we will show that it implies Theorem 2.1(i) as a corollary.
(i) There exists a closed interval I such that Before we prove this lemma, we first apply it to show Theorem 2.1 (i). Proof. Lemma 2.2(ii) shows that if ρ n ρ and ρ n ∈ M D≥d , then ρ ∈ M D≥d . If {ρ n } is a minimizing sequence of G = H/D α in the class M D≥d , then we can take a weakly convergent subsequence, still denoted as {ρ n }, and ρ n ρ ∈ M D≥d . Combined with Lemma 2.2(iv), this shows the existence of a minimizer ρ ∈ M D≥d .
We further show that ρ can be made even. Without loss of generality, we assume that where each I n is a closed interval on T. We may represent I n by [a n , b n ] for some a n ∈ [0, 1), b n ∈ [a n , a n + 1). By compactness of T, one can form a subsequence, still denoted as {I n = [a n , b n ]}, such that lim n→∞ a n = a and lim n→∞ b n = b. By dominated convergence theorem Proof of (ii): ] be a closed interval witnessing D[ρ] and ρ n ρ ∈ M. We first prove lim inf n→∞ D[ρ n ] ≥ D [ρ]. For > 0 with b − a < 1 − 2 we define the continuous function (2.8) Then for any n and > 0, by definition of D and φ we have (2.9) Therefore for any > 0, by weak convergence of the sequence . Let I n = [a n , b n ] be the intervals witnessing D[ρ n ]. We may take a subsequence of {ρ n }, still denoted as {ρ n }, such that lim n→∞ D[ρ n ] = lim sup n→∞ D[ρ n ], and lim n→∞ a n = a 0 , lim n→∞ b n = b 0 exist. For any > 0, letφ (x) be given as in (2.8) with a, b replaced by a 0 − , b 0 + respectively, then for sufficiently large n, we have (2.11) and since ρ n ρ. Therefore for any > 0 and I = [a 0 − , b 0 + ], we have Assume on the contrary that (2.14) is false. Since W * ρ ∈ L 1 , almost every x ∈ T is a Lebesgue point for W * ρ. Without loss of generality, assume x = 0 is a Lebesgue point and there exists > 0 and a subsequence {ρ n k } such that By definition of Lebesgue points, where ψ a is as in (1.23). Therefore, one can choose a such that and W * ψ a is a continuous function. Therefore, by the weak convergence of {ρ n k }, we have Notice that (W * ρ n k )(x)ψ a (x) dx ≥ ess inf(W * ρ n k ). Therefore we get a contradiction with (2.17). Finally, (iv) on lower semi-continuity of G α follows directly from (ii) and (iii).
where m 1 , m 2 > 0 are determined by the moment conditions Then for λ > 0 sufficiently small, ρ λ is a probability measure with Proof. We start by showing the sign of a finite difference using the convexity of W . For any then the difference of potential from splitting the Dirac mass at u is x−u y dz dy > 0 and C 1 > 0 is as given in (H4).
Here m 1 and m 2 are uniquely determined because the coefficient matrix in (2.21) is invertible. Now for any which is strictly larger than ess inf V U [ρ] by taking λ > 0 small enough. Therefore we obtain the conclusion. Figure 1. Microscopic Diffusion: horizontal arrows indicate the diffusion of mass near x 0 , vertical arrows indicate the change of the generated potential W * ρ, which is increasing away from x 0 and decreasing near x 0 .

Microscopic diffusion operation basically implies that if ρ is a minimizer of
as long as x ∈ supp ρ\{±M }. Suppose not, by Proposition 8.1, we can always find a small neighborhood (x 0 − , x 0 + ) for some x 0 ∈ supp ρ\{±M } to apply microscopic diffusion and strictly decrease H. Since the diffusion operation is local and away from ±M , it won't decrease D, we then get a contradiction with ρ being a minimizer. This proves Theorem 2.1(iii).
This in turn imposes a really strong restriction on the shape of supp ρ for a minimizer ρ. Suppose (x 1 , x 2 ) ⊂ (supp ρ) c and x 1 , x 2 ∈ supp ρ and x i are not ±M . Then by Corollary 2.5 is right continuous at x 1 and left continous at x 2 by Proposition 8.1. Therefore by convexity V [ρ](x) < ess inf V [ρ] for x ∈ (x 1 , x 2 ), which is a contradiction. Therefore there exists no (x 1 , x 2 ) ⊂ (supp ρ) c with x i ∈ supp(ρ)\{±M }. This helps us to further characterize the shape of the minimizer in the following theorem.
for some 0 < m ≤ 1/2, and ρ 1 ∈ M 1−2m is supported and non-zero on two (possibly empty) closed intervals J I c and K I.
Before we give the proof of Theorem 2.6, we first give a lemma on W .
Now we are ready to prove Theorem 2.6.
Proof of Theorem 2.6. Firstly, since I is maximizing I (ρ − 1), its endpoints ±M ∈ supp ρ. By the reasoning above Theorem 2.6, there is no open interval ( We will show that J =Ī c and K = I.
Finally, since J Ī c and K I and I witnesses D[ρ], it follows that ρ must have Dirac masses at ±M .

Energy Minimization.
Our main goal in this section is to prove the uniqueness in Theorem 2.1(iv). We will do so by using the idea of energy minimization in potential theory.
To set up the question, we will consider an external potential U ∈ L 1 (T) of the form where ρ + and ρ − are nonnegative continuous functions, ρ d is the sum of finitely many positive Dirac masses and W satisfies (H1)-(H4). As a consequence, U is bounded from below and at every point x ∈ T\ supp ρ d we have U (x) < ∞ and continuous. Define the total energy functional for ρ ∈ M m with m ≥ 0. The dependence of E U on U will be omitted when it is clear from the context. Here the two terms in (2.33) physically represent the pairwise interaction energy and potential energy respectively. For any ρ ∈ M m , both terms in E U [ρ] take value in (−∞, ∞] since U and W (implied by (H1) and (H3)) are bounded from below, therefore finiteness of E U [ρ] implies finiteness of both terms.
Then we integrate the inequality d 2 dt 2 E[ρ t ] ≥ 0 from t to 1 and get d dt . Suppose the statement is not true, then by lower (2.45) satisfying the (1.16). Since ρ 1 is the energy minimizer of E U , it is clear that ρ 1 does not contain any Dirac mass. Therefore the Dirac mass configuration of ρ is completely determined by m and M in U .
. Furthermore ρ is the unique probability measure satisfying (1.16) in the class of probability measures with the same Dirac mass configuration.
For the purpose of our discussion in Section 3, we also give a refined version of Proposition 2.8 in the case that U is the sum of two Dirac masses of the same size at −M and M , where we restrict ρ to a smaller class of measures with the prescribed total mass on the two intervals (−M, M ) and (M, 1 − M ).
which is the unique ρ satisfying There exists a unique (m 1 , m 2 ) with either of the following holds: and in this case ρ ∞,m1,m2 = ρ ∞ is the minimizer of E U in M 1−2m .
Since the proof is very similar to Proposition 2.8, we will only give a sketch of the proof.
Proof. Existence of energy minimizer: Let ρ n ρ ∈ M m1,m2 minimizing E U . Then ρ is a minimizer of E by lower semicontinuity of E with respect to weak convergence.
Characterizing property: If ρ ∞,m1,m2 does not satisfy the first line in (2.47), a similar transport plan with (2.38) where S is now a subset of (−M, M ) will construct ρ β with smaller E U . Similarly for the second line of (2.47).

Construction of Distributions
Our main goal for this section is to construct a class of measures that are candidates for the unique minimizer of G. In Section 2, we prove that the unique minimizer of G α in M D≥d must be a certain Dirac mass configuration in the format ρ d = m(δ M + δ −M ) together with the unique sediment distribution with respect to U = W * ρ d in M 1−2m under the potential W . In this section, we will first give the constructions of a class of (signed) measures which we loosely refer to as stationary distribution. These measures satisfy the condition that We demonstrate the inclusion of all mentioned classes as following: Although our final goal and the above inclusions are all about measures over T, we take a detour to discuss signed measures over R where the minimal value of another functionalG, defined in (4.2) as an analogue of G, is relatively easier to determine. We will discuss the minimal value of G in Section 5 and its analogueG over R in Section 4. For the current section, we will focus on the construction of interesting measures. From now on, we will take W (x) = − log |2 sin(πx)| for x ∈ T andW (x) = − log |x| for x ∈ R. The construction heavily deploys the facts that we are using logarithmic potential, which is closely related to the kernel of Hilbert transform, on both R and T. We will first give the construction over R since it is relatively cleaner, and save the construction over T to after.
3.1. Stationary Distributions over R. In this section, we firstly construct a family of signed measures over R. To simplify the notation, given a sequence of real numbers we denote S ⊂ R to be the union of intervals and denote the product Similarly for T M and T R .
Lemma 3.1. Given a sequence of real numbers If further more where a k is given by (3.5) and φ(z) = √ z is the analytic function on C\iR ≤0 . It is clear that is continuous in C y≥0 since the pole of g c (z) at M k is exactly cancelled by that of g d (z) due to the value of a k . Now we analyze g on the real line. If x ∈ S then g c (x) = µ c (x) is real and positive. If is purely imaginary. Therefore for x ∈ R\{M k | 1 ≤ k ≤ n} we have (3.10) The function (g)(x) can be extended to R continuously since g is continuous on C y≥0 . The function (g)(x) is continuous at any x ∈ R and (g) ∈ L 2 (R), since as |x| → ∞ (3.11) Therefore Hilbert transform H[ (g)] exists and it is a standard property that where the second equality follows sinceW (x) = − 1 x . We finished proving (3.6) by adding the term −W /π * µ d to both sides of (3.12).

18
Finally, we prove R µ = 0. We take a contour integral of g along for a large r and small , with counterclockwise direction. Since g is analytic in a neighborhood of the domain enclosed by γ = γ r, , we have γ g(z) dz = 0. Therefore its real part is { e iθ +P :0≤θ≤π} g(z) dz = 0.
(3.14) The first term in (3.14) converges to R µ(x) dx − n j=1 a j as r → ∞. For the second term, the contribution from −1 + g c (z) vanishes as r → ∞, since on upper half plane | − 1 + g c (re iθ )| ≤ C/r 2 under (3.7). The contribution from g d (z) for each M k -term is (3.15) The third term vanishes as → 0 since g(z) is continuous in a neighborhood of L k and R k . Therefore by letting r → ∞ and → 0, we obtain R µ(x) dx = 0.
Remark 3.2. If (3.7) holds then |µ(x)| ≤ C x 2 for large |x|, thusW * µ is well-defined on R except at M k . We can imagine that µ d generates an external potential U =W * µ d and µ c − 1 is a signed measure on R. Since (W * µ) = p.v.W * µ = 0 in S = supp µ c , the total potential V U [µ c − 1] =W * µ is a constant on each connected component of supp µ c . This is the reason why we consider µ as stationary distributions.
Then we apply the above lemma to obtain the following type I, II, III measures on R, denoted as µ I , µ II , µ III , which are stationary in the sense of Remark 3.2. Firstly by taking the sequence −1/π < 0 < 1/π, we obtain (3.16) Next for R > 1, by taking −R < −1 < 0 = 0 < 1 < R we obtain (3.20) See Figure 3 for examples of constructions above.
3.2. Stationary Distributions over T. We now construct a family of stationary measures over T. It will contain the unique minimizer of E U with U being generated by the Dirac mass configuration ρ d = m(δ M + δ −M ) in Proposition 2.8 for all m and M , therefore includes the unique minimizer of G α in Theorem 2.1 (after subtracting ρ d ).
The construction is similar to that of Lemma 3.1. For this construction, we will always represent points in T as [−1/2, 1/2). We will use the conformal mapping ω(z) = i 1−z 1+z that maps the unit disc to the upper half plane. Notice that ω(e 2πix ) = tan πx for any −1/2 < x < 1/2. We also similarly simplify the notation before the construction. Given a sequence of real numbers − 1 2 < l 1 < m 1 < r 1 ≤ l 2 < m 2 < r 2 ≤ · · · ≤ l n < m n < r n < 1 2 , we denote and Similarly for T m (x) and T r (x).
Lemma 3.3. Given a sequence of points on T denoted by − 1 2 < l 1 < m 1 < r 1 ≤ l 2 < m 2 < r 2 ≤ · · · ≤ l n < m n < r n < 1 2 . (3.23) We define ρ = ρ c + ρ d on T be given by Then there exists a constant C 1 such that Proof. Let L j = w(e 2πilj ) = tan πl j , M j = tan πm j , R j = tan πr j . Then similar to the function g in (3.8), we define g = g c + g d − C g for |z| ≤ 1 with where φ is as appeared in (3.8), a j is given in (3.25), and C g = C 0 + iC 1 is a complex constant such that g(0) = 0. Similarly with Lemma 3.1, here g is analytic in the unit disc and continuous on |z| = 1. At z = −1 we have lim z→−1, |z|<1 g(z) = 1 − n j=1 a j i ω(−e −2πim j ) . Then, similar to the proof of Lemma 3.1, we have which are functions in L 2 (T). The kernel for the Hilbert transform on T is exactly (3.30) By construction g(0) = 0, we get The above lemma is useful for our application to the energy minimizers only when C 1 = 0. Using ω(0) = i and g d (0) = − i a i , this condition is Then we may apply the mean value principle to the analytic function g to get i.e. . (3.35) Then we will construct some energy minimizers. Using the sequence − 1 π sin −1 2m < 0 < 1 π sin −1 2m with n = 1 in Lemma 3.3, we obtain the following: The measure ρ I,c is the unique minimizer of E U in M 1−2m for U = W * ρ I,d .
See Figure 2 (a) for an example of this construction.
Proof. We verify (3.33) and invoke (3.35) by computing (3.37) where the corresponding measure ρ in Lemma 3.3 is In order to construct a probability measure, we divide ρ by its total mass and obtain ρ I , which is the sum of ρ I,c and ρ I,d given in (3.36). By Lemma 3.3 we obtain the formula for (W * ρ I ) . It is positive in [− 1 π sin −1 2m, 0) and negative in (0, 1 π sin −1 2m] and zero otherwise, indicating that W * ρ I is smallest in supp ρ I,c . Therefore ρ I,c ∈ M 1−2m satisfies the characterizing condition (2.34) for U = ρ I,d * W with M = 0, and by Proposition 2.8, it is also the unique minimizer of E U and maximizer of ess inf V U in M 1−2m .
Similarly we can obtain the unique minimizer when M = 0 by applying Lemma 3.3. Using the sequence −R < −M < −L ≤ L < M < R with n = 2, we obtain the following: Proof. Similarly with Proposition 3.5, we verify (3.33) and invoke (3.35) by computing where the corresponding ρ = ρ c + ρ d in Lemma 3.3 is We are going to show in this section that we can further narrow down candidates from stationary distributions to sediment distributions. While doing so, we will eliminate the parameter m 1 and reduce the number of parameters from three to two.  Lemma 3.7. Let U be a C 1 function on (0, X) with X ∈ (0, ∞] and U < 0, and µ 1 , µ 2 be signed measures on (0, X) with (0,X) µ 1 dx = (0,X) µ 2 dx. If (µ 1 (y) − µ 2 (y)) dy = 0, (3.48) And strict inequality holds as long as µ 1 = µ 2 .
Proof of Proposition 3.6. The uniqueness of m 1 follows directly from the uniqueness of minimizer of E U in M 1−2m in Corollary 2.10.
3.4. Admissible Distributions over R. In this section, we will give a statement for distributions over R that serves as the analogue of Proposition 3.6 in Section 3.3. We will not only prove similar qualitative results like existence and uniqueness in Proposition 3.6, but also provide quantitative analysis for this family of distributions over R.
Recall the three types of distributions defined in (3.16), (3.17) and (3.19). The distributions µ II and µ III form a family parametrized by two parameters R and L where µ II corresponds to cases where L = 0. We will first make the observation in Proposition 3.8 that (W * µ)(∞) = 0 for all three types of µ. Then by imposing the condition (W * µ)(x) = ess inf(W * µ), for x ∈ supp µ c , (3.55) we further cut down by one parameter and obtain a family of stationary measures µ over R just parametrized by R in Proposition 3.11. We call this family of signed measures over R together with µ I the admissible distributions. Figure 2 (c) and (d) illustrate the difference between admissible distributions and non-admissible distributions. Although technically speaking they are not measures and do not satisfy the sediment condition, they serve a similar role on R with that of sediment measures (together with the Dirac masses) over T. Such connection will be made clearer in Section 5. for large |x|. We can assume x > 0 sinceW * µ is even. For |y| ≤ √ x/2, we have ln |1 − y (3.58) Next for √ x/2 < |y| ≤ x/2, similarly we have ln |1 − y x | |y| x and |µ(y)| 1/y 2 , thus (3.59) Finally for |y| > x/2 using a change of variable y = xy 1 we get (3.60) Therefore (3.56) is proved.
In the following proposition, by imposing (3.55), we will further get rid of the parameter L and obtain a one-parameter family of distributions parametrized by R. We first explain how the condition (3.55) is equivalent to a condition on (W * µ)(0). For Type II, we recall in We now give the formula for (W * µ)(0) for µ constructed in (3.17) and (3.19) by integrating against (W * µ) . AlthoughW * µ is not continuous at x = 1, we still have where we plug in µ = (−1 + µ c ) + µ d by Lemma 3.1 for the second equality. The third equality follows from Remark 3.9. Then by the explicit expression of (W * µ) in (3.18) and (3.20) we obtain (3.62) We denote the last integral by Φ(L, R).
In order to study (W * µ)(0) for Type II and III distributions over R, we first prove some useful properties of this function Φ(L, R). This verifies the condition (3.46) with µ i = µ L,Ri using the mean zero property of µ III (x). One can also check the condition (3.47) with U =W since µ i are L ∞ functions near 0, have the mean-zero property, and decay at least as 1/|x| 2 at infinity. Therefore, noticing that µ i are even, we can apply Lemma 3.7 (with X = ∞) to get Φ(L, R 2 ) > Φ(L, R 1 ) by the decreasing property ofW (x) in |x|. Similarly one can prove the monotonicity in L.
Proof of (ii): Since Φ(0, R) is strictly increasing in R, it suffices to show that Notice that Φ(0, R) can be explicitly evaluated as which indicates the sign of limit as R → 1 + and R → ∞. We denote the unique root in (1, ∞) of √ R 2 − 1 ln(R + √ R 2 − 1) − R = 0 by R c . It is approximately 1.8102. Proof of (iii): Note that it follows Φ(0, R c ) = 0 and the monotonicity of Φ that if Φ(L, R) = 0 for some L ≥ 0, then R ≤ R c . Now since Φ(L, R) is increasing in both L and R, it suffices to show for any R c ≥ R > 1 that Φ(2 − R, R) > 0, Φ( 2 − R 2 , R) < 0. (3.66) To see Φ( √ 2 − R 2 , R) < 0, we use a change of variable u = x 2 and then write 2 (3.67) Similarly to see Φ(2 − R, R) > 0, we write 1 π Φ(2 − R, R) as (3.68) The last integrand is clearly positive since Now we are ready to state the following.
In this case µ III (x) < 0 for |x| = 1. This function L(R) is decreasing in R.
Proof of Proposition 3.11. Proof of (i): Lemma 3.10(i) and (ii) implies that Φ(0, R) = (W * µ II )(0) ≥ 0 if and only if R ≥ R c . To see that µ II (x) < 0 for |x| = 1, we notice that for x ≥ R, Proof of (ii): By Lemma 3.10, the function Φ(L, R) is increasing in both L and R. Therefore for each 1 < R < R c we have Φ(0, R) < Φ(0, R c ) = 0. On the other hand, by (3.62), Therefore, there exists a unique L = L(R) such that Φ(L, R) = 0. The decreasing property of L(R) then follows from the monotonicity of Φ(L, R) in R and L. To see µ III (x) < 0 for |x| = 1 when 1 < R < R c and L = L(R), it suffices to note (3.74) The last inequality follows from Lemma 3.10(iii).
It follows from Proposition 3.11 that for each R > 1, we get a unique stationary distribution satisfying (3.55). We now collect these distributions and the Type I distribution and their scaling functions into a family.
Definition 3.12 (Admissible Distributions). Let µ be a signed measure on R. Then we say µ is admissible if it is one of the following: for 1 < R < R c and some λ > 0. We call λ the scaling factor of µ.
Aside from the Type I distribution, this family is parametrized by two parameters λ and R. By the notation in Lemma 3.1, we have µ = −1 + µ c + µ d . By abuse of notation, we will also denote the corresponding term for a general admissible µ with a scaling factor λ by µ c and µ d .

An Optimization Problem over R
In this section, our goal is to determine the minimal value of an analogous goal functional G for admissible distributions over R. Recall that we have constructed and defined admissible distributions in Section 3.4 and Definition 3.12. Now we define for admissible µ that . (4.2) The relation betweenH,D and the functionals H, D on T will be revealed in Section 5 by Theorem 5.2 and Theorem 5.7. Then we state the main result of this chapter as an optimization problem.
where we use the mean-zero property of µ, shown in Lemma 3.1, in the fourth inequality. Then we discussD. FirstlyD[µ I ( · λ )] = λ = λD[µ I ] by definition. If µ = µ II,R or µ III,L(R),R , thenD It follows from Lemma 4.2 that it suffices to consider µ I , µ II and µ III to determine minimal value ofG. We first compute this value for µ I directly.
Before we studyG[µ III ], we first give a lemma on the monotonicity ofH andD. Proof. Let µ = µ III,L(R),R . We first considerH. Similar with (4.5), we computẽ where the third equality uses (3.18), and the second last equality follows from (3.71) for L = L(R). ThenH[µ III,L(R),R ] is strictly increasing in R since L(R) is strictly decreasing in R. By definition ofD and the mean-zero property of µ,D being increasing in R is equivalent to (1,∞) µ III,L(R),R dx being decreasing in R. For i = 1, 2 and 1 < R i < R c , we denote L i := L(R i ) and µ i := µ III,Li,Ri . Suppose the contrary, then there exists 1 < R 1 < R 2 < R c such that We now compare the numerator of µ i in (3.19), which is . (4.10) By the assumption (4.9) and the fact that µ 1 (R 2 ) > µ 2 (R 2 ) = 0, we see that µ 1 − µ 2 must be decreasing in x and negative when x is large enough. Therefore there exists some x 0 ∈ (R 2 , ∞) such that We claim that In fact, this can be seen by separating into the following cases. If 0 < X ≤ 1, we use the first inequality in (4.11); if 1 < X ≤ x 0 , we firstly notice that (4.9) implies that (0,1] µ 1 (x) dx > (0,1] µ 2 (x) dx by the mean-zero property of µ, then we use the first inequality in (4.11) for (1, X); if X > x 0 , then the second inequality in (4.11) implies that [X,∞) µ 1 (x) dx < [X,∞) µ 2 (x) dx, then we apply the mean-zero property of µ. Now we apply Lemma 3.7 with X = ∞ and get which contradicts the property (W * µ)(0) = 0 for µ of Type III, proved in Proposition 3.11. Now we consider µ III,L(R),R with R c ≥ R ≥ 1.1. We take a discretization R 0 < R 1 < · · · < R n = R c of R with n = 19 and denote µ i := µ III,L(Ri),Ri andH i :=H[µ i ] andD i :=D[µ i ]. We then numerically verify in Table 4 We thus prove the following.
Proposition 4.6. For 1.1 ≤ R ≤ R c and µ = µ III,L(R),R , we haveG[µ] > 1/2. 4.3. Small R. In this section, we give an estimate ofG[µ] for µ = µ III,L(R),R when R is close to 1. In Section 4.2 we have shown Theorem 4.1 for R ≥ 1.1, therefore we focus on 1 < R ≤ 1.1. We start with giving a lower bound forH and an upper bound forD in terms of R and L = L(R). They serve as good approximations ofH andD when R is close to 1.
Lemma 4.7. For any 1 < R < R c and µ = µ III,L(R),R , we havẽ Proof. The lower bound forH follows from estimating the integral in (4.8). By a change of variable x = √ y, we havẽ (4.17) 2 Approximation of integrals are computed in Matlab with error no more than 10 −4 . As a result, the error of is no more than 10 −3 for any k.
Proof. Using Lemma 4.7 and R 2 + L 2 > 2 and R + L < 2 from Lemma 3.10, we obtain that (4.25) We notice that the last expression is equal to 1 when L = 1, therefore it is larger than 1/2 when L is close to 1, equivalently when R is close to 1. In fact, this can be quantified by observing that the last expression is increasing in L. When L = 0.85, it is approximately 0.58, therefore it must be greater than 1/2 for any 0.85 ≤ L < 1. Since R ≥ √ 2 − L 2 , we have verifiedG[µ] > 1/2 for any 1 < R ≤ √ 2 − 0.85 2 , and in particular, it is true for 1 < R ≤ 1.1.

From R to T
In this section, we will compare the values ofG[µ] with G[ρ], where µ is admissible over R and ρ (without its Dirac masses) is sediment over T. We have constructed these two family of distributions in Section 3.4 and 3.3 respectively. In Section 4, we have determinedG[µ] ≥ 1/2 for all admissible µ. In order to establish a comparison, we first introduce a way to associate a signed measure ρ • ∈ M with each admissible distribution µ over R, via periodization, in Section 5.1. We then relate the quantitiesH[µ] andD[µ] with H[ρ • ] and D[ρ • ] in Theorem 5.2 and Section 5.2. In particular, we show that G[ρ • ] ≥G[µ]. Finally we prove a comparison principle using the convexity of logarithmic potential, showing that for each minimizer ρ ∈ M of G in M D≥d there exists a unique admissible µ whose associated ρ • satisfying G[ρ] ≥ G[ρ • ] > 1/2. 5.1. Periodization. In this section, we bridge the admissible distributions over R to distributions over T via a periodization. For each signed measure µ with R |µ(x)| dx < ∞ over R, we define its periodization This definition does not depend on the choice of representatives for x in T. Now for each admissible µ, we associate a signed measure over T If µ is in (3.16), (3.17) and (3.19) up to a scaling factor λ , then µ = −1 + µ c + µ d using the notation in Lemma 3.1, we can break down ρ For simplicity, we will drop the associated µ in these notations when there is no confusion. We now characterize properties of ρ •,c and ρ •,d for µ.
Next we establish the key connection between the functionalsH andD over R and the functionals H and D over T. Note that ρ • is not necessarily a measure but only a signed measure over T. Therefore we extend the functionals H by defining for arbitrary signed measures ρ over T.

A Comparison Principle via Convexity.
In this section, our main goal is to establish the following comparison principle between energy minimizers associated with different external potentials. The proof of this principle essentially takes advantage of the convexity of log potential. where U * ∈ C 1 (T) is even. Let m 1 , m 2 ≥ 0, and ρ and ρ be the energy minimizer for U and U in M m1,m2 (c.f. (2.46)) respectively. If and ρ and ρ are Hölder continuous functions, then We first give some preparations.
Lemma 5.4. Let u be Hölder continuous on T with exponent β. Then its Hilbert transform H[u] is defined everywhere over T and is Hölder continuous with exponent β 1 for any 0 < β 1 < β.
Proof. Recall that the kernel for Hilbert transform over T is exactly W , and it is an odd function. Therefore Here the last integrand is integrable because |W (y)| |y| −1 near y = 0 and |u(x − y) − u(x)| ≤ C|y| β . Therefore H[u](x) is defined everywhere.
To show the Hölder continuity, we take x 1 < x 2 and denote = x 2 − x 1 > 0. Then for any β 1 < β. This finishes the proof.
Let m 0 > 0, and ρ ∈ M m0 be even. Define its cumulative function as is strictly increasing and piece-wise continuous. We will write m(x) and X(m) in short when there is no confusion.
Lemma 5.5. Let m 0 > 0 and ρ ∈ M m0 be even and Hölder continuous. Let m(x) be the cumulative function for ρ and X(m) be its inverse function. Then we have the change-of-variable formula Proof. Since ρ is Hölder continuous, both W * ρ and W * ρ are well-defined everywhere by Lemma 5.4, and X(m) is guaranteed to be strictly increasing and piece-wise continuous. By symmetry of ρ and a substitution y = X(m), we get (5.27) We can consider W * ρ similarly, and it suffices to prove Without loss of generality, we can assume x ∈ supp ρ and ρ(x) > 0, since otherwise p.v. can be removed. Denote M 0 = m(x). Then by previous analysis, we obtain that W (x − y)ρ(y) dy → 0, as → 0. (5.30) By the Hölder continuity of ρ, we get ρ(x ± ) = ρ(x) + O( α ). Therefore when is small enough, |X(M 0 ± ) − X(M 0 )| . The above limit then easily follows.
Now we are ready to give the following comparison principle, which is the key to proving Proposition 5.3. We will compare two measures ρ and ρ . For the notation, we will write m for m[ρ ] and X for X[ρ ]. Similarly for ρ .
Lemma 5.6. Let ρ and ρ be in M m0 that are Hölder continuous and even. If for some x 0 ∈ (0, 1/2), then there exists 0 < x < x ≤ 1/2 and x ∈ supp ρ and x ∈ supp ρ such that Proof. For convenience, we will define X (m) to be the smallest x ∈ [0, 1/2] such that m (x) = m, and X (m) to be the largest x ∈ [0, 1/2] such that m (x) = m. Then X is lower-semicontinuous and X is upper-semicontinuous. By the assumption (5.31), we see that where the integrand is x −X (m) x −X (m) x −X (m) x −X (m) (5.36) We can again show that z = 0 is not in the domain of any of these integrals: trivially true when m < m s and use X (m) < X (m) for m > m s . Now it is clear that the first and third integral are positive, and the second and fourth integral can combine as x −X (m) x −X (m) In the last integral, we have z and x both in [0, 1/2]. Then it follows from the fact that W (x) = π/ sin 2 πx is even and decreasing in (0, 1/2] that the integrand is always positive. We thus finish proving (W * ρ )(x ) ≥ (W * ρ )(x ). Now we are ready to prove the main proposition in this section.
Proof of Proposition 5.3. Recall the notation that for external potential U , we obtain ρ ∈ M m0 as the energy minimizer as proved in Proposition 2.8, and the generated potential is V := U + W * ρ . By Proposition 2.8, we have V (x) = 0 on supp(ρ ). We denote the cumulative function m for ρ . It follows from the assumption that m (M ) = m 1 /2. Similarly everything holds also for U . We claim that Suppose not, then we apply Lemma 5.6 to get x and x satisfying (5.32). Since V = W * ρ + U = 0 on supp(ρ ) and similarly for ρ , (5.32) then implies On the other hand, Here the first term is positive since U > 0 by the expression of U and M / ∈ [x , x ] since m (M ) = m (M ) = m 1 /2. The second term is positive since −U * (x ) > 0 by the assumption. Therefore we get a contradiction and prove the claim (5.38).
Finally, applying Lemma 3.7 with X = 1 2 , we get (W * ρ )(0) > (W * ρ )(0), and the other conclusion (W * ρ )( 1 2 ) < (W * ρ )( 1 2 ) can be obtained similarly. 37 We will formulate energy minimization problems for ρ • , by specifying external potentials U and U and their corresponding minimizer ρ and ρ . For small > 0 and r ≥ 0, define a cutoff function φ r, (x) = max 0, 1 − 1 dist x, {y ∈ [−1/2, 1/2) : |y| > x J + r} , (5.48) where dist denotes the distance function on T. Combined with (5.47), for each , there exists a unique r = r( ) ∈ (0, ) such that We thus write φ in short for φ r( ), . Then we define and let ρ be the unique minimizer in M m1,m2 for E U . It is clear that is C 1 since ρ •,c is Hölder continuous. We will verify (5.19) in Lemma 5.8. Then by Lemma 5.3 to get By the construction of U and W (x) < 0 for x ∈ (0, 1/2), we have Here the second inequality is a consequence of ρ being an energy minimizer in M 1−2m , together with the fact that 1/2 ∈ supp ρ . There ρ + m(δ M + δ −M ) must be of Type II in Proposition 3.5. Notice that we can assume 0 ∈ supp ρ since otherwise supp ρ ∩ We now verify (5.19) for the constructed potential U and U in Theorem 5.7.
In particular, Therefore, considering the possibility of 2x 0 − x lying in the second or third piece of (5.60), we have . Therefore the last quantity above is nonnegative if is small enough, which proves (5.62). Then we define a function G by otherwise. (5.65) Here R G ∈ (0, R • ) is determined by T G dx = 0, which is possible since the positive parts of G are below F , by (5.62). Then F − G is supported on {|x| ≤ x 0 } with the same sign properties as F , and we may apply (5.61) to show that (W * (F − G))(x 0 ) > 0, and in fact, bounded from below uniformly in and x 0 . Since G is symmetric around x 0 in a small neighborhood of x 0 , this neighborhood makes no contribution to (W * G)(x 0 ), and other parts only contribute O( ) to (W * G)(x 0 ) since they are away from x 0 with total mass O( ). Therefore we get the positivity of (W * F )(x 0 ) for all x J + r − ≤ x 0 ≤ x J + r if is small enough.

From Continuum to Discrete
In this chapter we finalize the proof of Theorem 1.4 on the minimization problem of G for measures. Then we show that it implies Theorem 1.1 via a construction of polynomials whose roots distribution approximates the minimizing distribution over T. Finally based on our proof of Theorem 1.1, we prove Theorem 1.2 on giving a sharp bound of the number of signed real roots for an arbitrary complex polynomial.
6.1. Proof of Main Theorem. In this section, we collect all results we prove before and deduce the main theorem.
Proof of Theorem 1.1 and Theorem 1.4. Recall that we first extend the functional D and H from the set of discrete probability measures We now compute the value of G[ρ] for sediment ρ with M = 0. In Proposition 3.4 we construct ρ = ρ I,m (x) ∈ M parametrized by m. By expression of ρ, we have D[ρ] = 2m is taken at the origin since ρ(x) < 1 for x = 0. On the other hand, it follows from the construction that W * ρ is stationary. By Lemma 3.3, by letting C 1 = 0, we obtain Due to the sign of W * ρ I (x), we see that H[ρ] = −(W * ρ)(1/2). We can compute the value using mean-zero property of W * ρ where the last equality follows from integration by parts and the fact that (W * ρ I )(1/2) = (W * ρ I )( sin −1 (2m) π ). It then follows from (6.3) and (6.2) that 2my 1 − (2my) 2 dy > 2m 2 .
(6.4) Therefore we prove the inequality in Theorem 1.4. One can also evaluate the limit lim m→0+ (W * ρ I )( 1 2 )/m 2 = 2, thus lim Therefore we also show that √ 2 sharp in Theorem 1.4, thus finish proving Theorem 1.4. Since M emp ⊂ M, we also prove the inequality in Theorem 1.1.
It now suffices to prove that Theorem (1.3) is sharp. We will do so by constructing ρ in where m j,1 , m j,2 ≥ 0 are determined by the moment conditions (j+1)/n j/n m j,1 δ(y − j n ) + m j,2 δ(y − j + 1 n ) − ρ I,c (y) y k dy = 0, k = 0, 1. (6.8) The weak convergence of {ρ n } to ρ is clear. Similar to the proof of (2.27) in Lemma 2.4, we may show that using the convexity of W . Therefore applying (6.9) to those j with x / ∈ [j/n, (j + 1)/n), we have where j x -th interval contains x. For large n, we have (W * δ)(x) > 0 for |x| < 1/n since W is positive near 0. We can also bound the other term using ρ I,c ≤ 1 and W = − log |2 sin πx|, | W * (ρ I,c χ [jx/n,(jx+1)/n] ) (x)| ≤ sup a∈T a+1/n a |W | dx log n n . (6.11) Therefore we obtain . Therefore for each > 0, we can find ρ = ρ I,m such that G[ρ] < 1/2 + . For this ρ, we can construct {ρ n } as above so that G[ρ n ] < 1/2 + 2 for n large enough. For a ρ emp = N j=1 c j δ(x − x j ), N ∈ N ∈ M emp , we can view G[ρ emp ] as a function of c = (c 1 , . . . , c N ), and indeed a continuous function in terms of c. Therefore, by replacing each c j by a nearby rational number while keeping N j=1 c j = 1, one can find a ρ n ∈ M emp-rat with G[ρ n ] < 1 2 + 3 . Since is arbitrary, we finish proving the sharpness of the constant √ 2 in Theorem 1.1. (ρ − 1) dx. (6.14) It is easy to see that if ρ f = 1 n j θ j is the empirical measure from a degree n polynomial f (z), then R[ρ f ] = N + (f )/n. (6.16) Therefore it suffices to prove the inequality is sharp. Notice that in the proof of Theorem 1.1, we have constructed ρ I,m such that G[ρ I,m ] < 1/2 + for any > 0. By the expression of ρ I,m we see that D[ρ I,m ] = R[ρ I,m ], therefore we also have for these ρ I,m that H[ρ I,m ]/R[ρ I,m ] 2 < 1/2 + . We then construct ρ n ρ I,m in the same way to approximate ρ = ρ I,m . Since lim n→∞ D[ρ n ] = 2m = R[ρ n ], we also have lim n→∞ H[ρ n ]/R[ρ n ] 2 = 1/2. Therefore we can choose ρ n such that H[ρ n ]/R[ρ n ] 2 < 1/2 + 2 . Finally the construction for ρ n is the same since R is also continuous in c when ρ emp = N j=1 c j δ(x − x j ). Therefore we can find ρ n with H[ρ n ]/R[ρ n ] 2 < 1/2 + 3 . Since is arbitrary, we finish proving the sharpness of constant √ 2. The upper bound for N θ (f ) is exactly the same since N θ (f (z)) = N + (f (z · e −2πθi )) and

Formulation in Harmonic Functions
In his 1952 work [Gan54], Ganelius formulates a question in harmonic functions and uses it to improve the constant in the original Erdős-Turán inequality proved by [ET50]. This approach of harmonic functions has been further developed by Mignotte in [Mig92]. In this section, our goal is to show that our sharp version of Erdős-Turán inequality in turn implies a sharp upper bound for harmonic functions in Ganelius' formulation.
Proof of Theorem 1.3. In order to prove inequality (1.7), we first note that it suffices to prove (1.7) for |z| = 1 in the case where f (z) is analytic in |z| < 1 + and u ≤ H and ∂v ∂θ (z) ≤ K. Indeed, if |z i | < 1 − for i = 1, 2, we can consider g(z) = f (z(1 − )) instead. We still have the bound u g ≤ H and ∂vg ∂θ (z) ≤ K(1 − ) ≤ K. Now since f (z) is analytic on |z| < 1 + , u and v are both harmonic functions on |z| < 1 + . The (1.7) for z i and f then follows from that for z i /(1 − ) and g. By maximal value principle, sup |z1|,|z2|≤1 |ṽ(z 1 ) −ṽ(z 2 )| is achieved at 3 Notice that θ is taken to be in [0, 1] in this current formulation whereas in [Gan54] and [Mig92] θ is taken to be in [0, 2π]. This results in a change of √ 2π in the constant C.

Appendix: Continuity of Potential
We list several results on the continuity of the generated potential V = W * ρ. Recall that a function f is lower semicontinous at x = a ∈ T if lim inf (i) The generated potential V is lower semicontinuous and is C 2 on T\ supp ρ.
(ii) There exists a constant C > 0 such that V (x) ≥ C at every x / ∈ supp ρ. (iii) Let (x 1 , x 2 ) ⊆ (supp ρ) c be an interval with endpoints x 1 , x 2 ∈ supp ρ. Then V is right continuous at x 1 and left continuous at x 2 .
Proof. Proof of (i): The lower semicontinuity of V for general W can be found in [BCLR13, Lemma 2] and for W = − ln |2 sin(πx)| is automatic. For the sake of completeness, we include the proof here. The continuity of W away from 0, together with (H3), implies that W is bounded from below, say, by −C 1 . By including the possibility of ∞ value, we have lim x→x0 W (x) = W (x 0 ) for any x 0 ∈ T. Therefore, for any x ∈ T and sequence {x n } with lim n→∞ x n = x, V (x) + C 1 = T (W (x − y) + C 1 )ρ(y) dy = T lim n→∞ (W (x n − y) + C 1 )ρ(y) dy ≤ lim inf n→∞ T (W (x n − y) + C 1 )ρ(y) dy = lim inf n→∞ V (x n ) + C 1 . (8. 2) The inequality uses Fatou's lemma on the nonnegative functions W (x n − y) + C 1 . The lower semicontinuity of V then follows.