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Szego's theorem on Parreau-Widom sets

In this paper, we generalize Szego's theorem for orthogonal polynomials on the real line to infinite gap sets of Parreau-Widom type. This notion includes Cantor sets of positive measure. The Szego condition involves the equilibrium measure which is shown to be absolutely continuous. Our approach builds on a canonical factorization of the M-function and the covering space formalism of Sodin-Yuditskii.


Introduction
Let dν = w(θ) dθ 2π + dν s be a finite positive measure on the unit circle ∂D, with dν s singular to dθ. A classical result of Szegő [39] reads where P is the set of polynomials vanishing at zero. So the infimum on the left-hand side is > 0 if and only if the integral on the right-hand side, also known as the Szegő integral, is convergent (i.e., > −∞). Strictly speaking, Szegő only considered absolutely continuous measures but one can allow for a singular part too. There are many equivalent forms of Szegő's theorem, see, e.g., [34,Chap. 2], and it is bound up with asymptotics of Toeplitz determinants. From the point of view of orthogonal polynomials, perhaps the most suitable formulation is due to Verblunsky [41]. It replaces the left-hand side in (1.1) by where {α n } ∞ n=0 are the recurrence coefficients of the associated monic orthogonal polynomials. Without any problems, one can carry over the result of Szegő to measures supported on an interval of the real line. Let dµ = f (t)dt + dµ s be a probability measure supported on [−2, 2] and let {a n , b n } ∞ n=1 be the recurrence coefficients of the associated orthonormal polynomials. Then It is possible to allow for point masses of dµ s outside [−2, 2] as long as the mass points {x k } satisfy the condition see [23,36] for details. If infinite in number, the x k 's thus have to accumulate sufficiently fast at the endpoints ±2.
The aim of this paper is to establish a version of Szegő's theorem on sets in R much more general than an interval, namely what we shall call Parreau-Widom sets. The notion of such sets will be introduced in Sect. 2 and we shall only give a brief description here. Among all regular compact subsets of R, the Parreau-Widom condition (2.6) singles out those for which the values of the Green's function at critical points are summable. In particular, it allows for a great number of sets with infinitely many components.
Our main result, Thm. 2 (in Sect. 5), goes beyond the recent monograph [35] of Simon as to generalizing the Szegő-Shohat-Nevai theorem, using the language of [35]. The situation of 'finite gap sets' was studied in [5,6], inspired by Widom's famous paper [42] and the landmark paper [24] of Peherstorfer-Yuditskii. For a finite gap set e, the product in (1.2) has to be replaced by (1.3) a 1 · · · a n Cap(e) n , where Cap(e) is the logarithmic capacity of e. Moreover, the Szegő condition takes the form where dµ e is the equilibrium measure of e. For comparison, dθ/2π is the equilibrium measure of ∂D and dµ [a,b] is a scaled arcsine distribution. While [42] uses multiple-valued functions, [24] makes heavy use of the covering space formalism introduced by Sodin-Yuditskii [37]. The present paper also builds on [37], as do [5,6], and we give some background on uniformization theory in Sect. 2. The core of [24,25] is to establish Szegő asymptotics of orthogonal polynomials on so-called homogeneous sets in R. This is done under the Szegő condition and a Blaschke condition (similar to (1.5)) by comparing the solutions of two extremal problems. As a by-product, the implication '⇐' of (1.2) (or rather (1.6) below) is obtained. In contrast, the main tool of the present paper is a step-by-step sum rule obtained from a canonical factorization of the M -function. This technique was developed by Killip-Simon [17] and is applied repeatedly in the monograph [35]. We shall establish the desired factorization in Sect. 3 and arrive at the step-by-step sum rule in (3.32)- (3.33). In Sect. 4, we show that the equilibrium measure of a Parreau-Widom set is absolutely continuous, provided the set has positive Lebesgue measure. Hence the Szegő integral is a relative entropy (up to some constant), and known properties of relative entropy combined with uniform upper bounds on certain eigenvalue sums allow us to iterate the step-by-step sum rule and pass to the limit. With all preparations in place, the proof of Thm. 2 is merely half a page.
Every now and then we shall use the language of Jacobi matrices rather than the one of measures. As is well known, there is a one-one correspondence between compactly supported (nontrivial) probability measures on R and bounded Jacobi matrices. Given dµ, the associated Jacobi matrix is given by are the recurrence coefficients of the orthonormal polynomials P n (x, dµ). When supp(dµ) is compact, these coefficients are bounded. The spectrum of J, viewed as an operator on ℓ 2 (N), coincide with supp(dµ) and we shall often refer to dµ as the spectral measure of J. In the language of Jacobi matrices (and with g the Green's function for C \ ), the main result of the paper reads: n=1 be a Jacobi matrix with spectral measure dµ = f (t)dt + dµ s and let ⊂ R be a Parreau-Widom set of | | > 0. Assume that σ ess (J) = and denote by {x k } the eigenvalues of J outside , if any. On condition that Moreover, if one and hence both of the equivalent conditions in (1.6) hold true, then 0 < lim inf n→∞ a 1 · · · a n Cap( ) n ≤ lim sup n→∞ a 1 · · · a n Cap( ) n < ∞.
While the Szegő integral relates to relative entropy and allows for defining an outer function (see Sect. 3), the product a 1 · · · a n is the reciprocal of the leading coefficient in P n (x, dµ). So if is rescaled to have capacity 1, the leading coefficients in the orthonormal polynomials are bounded above and below.
With (1.1) as a starting point, Hayashi [14] set out to generalize Szegő's theorem to Riemann surfaces. Naturally, such a generalization may not be as clean and explicit as (1.2) or (1.6). Interestingly, the results of [14] simplify when the Riemann surface R is of Parreau-Widom type. Let dω τ be the harmonic measure on ∂R for a point τ ∈ R. When h ≥ 0 belongs to L 1 (∂R, dω τ ), Hayashi proved that where H ∞ τ (R) is the set of bounded analytic functions on R vanishing at τ , and C > 0 is some constant depending on τ . Provided that ∂R ⊂ R, and if the infimum on the left-hand side can be related to a 1 · · · a n / Cap(∂R) n , this inequality may prove the implication '⇒' of (1.6) when dµ s = 0. In order to be able to include a singular part of the measure, [14] needs an extra assumption on R which is equivalent to the Direct Cauchy Theorem. We shall not discuss this issue here but refer the reader to [13] and the recent paper [44].
Acknowledgements: The author would like to thank Alexandru Aleman, Katsuhiro Matsuzaki, Barry Simon, Mikhail Sodin, Peter Yuditskii, and Maxim Zinchenko for useful discussions and comments while preparing the paper.

Parreau-Widom sets and uniformization theory
In this section we start by introducing the notion of a Parreau-Widom set on the real line. This notion covers a large class of compact subsets of R and allows for a set to have infinitely many 'gaps', yet no isolated points. The precise definition will be given below. It relies on potential theory and key roles will be played by the Green's function and the equilibrium measure. We refer the reader to [8,10,18,21,38,40] for background and more details on potential theory.
In the second part of the section, we give a brief account on uniformization theory. The universal covering map will be brought into play and we relate the Green's function to Blaschke products of the underlying Fuchsian group. See, e.g., [1,20] or [35,Chap. 9] for further details.

Parreau-Widom sets
Let ⊂ R be a compact set. We shall always assume that the logarithmic capacity of , denoted Cap( ), is positive so that the domain Ω = C \ has a Green's function. For fixed y ∈ Ω, we let g Ω ( · , y) be the Green's function for Ω with pole at y. Recall that this function is positive and harmonic on Ω \ {y}. Moreover, (2.1) g Ω ( · , y) + log | · −y| is harmonic at y. The special case y = ∞ will be of particular interest to us and we write g instead of g Ω ( · , ∞). Not only is g( · ) − log | · | harmonic at ∞, but we also have the expansion where γ( ) = − log Cap( ) is the so-called Robin's constant for .
There is a unique probability measure on of minimal logarithmic energy. This measure is called the equilibrium measure of and will be denoted dµ . It minimizes the integral among all probability measures dµ on and the minimal energy is given by I(dµ ) = γ( ). The logarithmic potential of dµ brings us back to the Green's function through the relation Besides having positive logarithmic capacity, we will assume that each point of is a regular point for Ω, that is, This in particular means that has no isolated points as such points are irregular for Ω. In short we say that is regular when (2.4) holds. Equivalent to this is the Green's function being continuous on all of C. It will often be useful to write in the form where ∪ j is a countable union of disjoint open subintervals of [α, β] and α < α i = β j < β for all i, j. We shall refer to (α j , β j ) as a 'gap' in and to (2.5) as the infinite (or finite) gap representation of . While the Green's function vanishes on and in particular at the endpoints α i and β j , it is strictly concave on each of the gaps. Since g cannot be constant on any interval in R \ , this follows from (2.3) and the fact that log is concave. Hence there is a unique point c j ∈ (α j , β j ) at which g attains its maximum on (α j , β j ). The c j 's are the critical points of g since g ′ (c j ) = 0 for each j and g ′ never vanishes outside [α, β].
Definition. Suppose ⊂ R is a compact set of Cap( ) > 0 and suppose is regular. We call a Parreau-Widom set if where g is the Green's function for C \ with pole at ∞ and {c j } are the critical points of g.

Remark.
Note that the definition is independent of which y ∈ Ω is taken as pole of the Green's function. If the values of g Ω ( · , y) at critical points are small enough to be summable for one y, the same applies to all y. The choice of y = ∞ is made for convenience.
The above terminology is inspired by the monograph [13] of Hasumi. In comparison, we say that ⊂ R is a Parreau-Widom set if the domain Ω = C \ is a Riemann surface of Parreau-Widom type in the language of [13,Chap. 5]. While originally introduced by Parreau in [22], Widom [43] showed that such surfaces have sufficiently many analytic functions. See, e.g., [13] for more details.
Clearly, any finite gap set (cf. [4][5][6]) is a Parreau-Widom set. But the notion goes way beyond. Jones-Marshall [16,Sect. 3] proved that it includes infinite gap sets which are homogeneous in the sense of Carleson [3]. By definition, this means there is an ε > 0 such that Carleson introduced this geometric condition to avoid the possibility of certain parts of to be very thin, compared to Lebesgue measure. See [7,45] for further results on homogeneous sets.
Example. Remove the middle 1/4 from the interval [0, 1] and continue to remove subintervals of length 1/4 n from the middle of each of the 2 n−1 remaining intervals. Let be the set of what is left in [0, 1]; this is a fat Cantor set of | | = 1/2. One can show that |(t − δ, t + δ) ∩ | ≥ δ/4 for all t ∈ and all δ < 1.

Uniformization theory
When has at least one gap, the domain Ω = C \ is not simply connected. So only in the trivial case of being an interval, Ω is conformally equivalent to the unit disk. For general Parreau-Widom sets, we shall employ uniformization theory as in the seminal paper [37] of Sodin-Yuditskii. There is a map x : D → Ω, which is onto but only locally one-to-one, and a Fuchsian group Γ of Möbius transformations on D so that This map is called the universal covering map and we fix it uniquely by requiring Note that Γ is isomorphic to the fundamental group π 1 (Ω) and hence a free group on as many generators as the number of gaps in . Since Cap( ) > 0, it follows from a theorem of Myrberg (see, e.g., [40, Chap. XI]) that Γ is of convergent type. This means and hence the Blaschke products defined by are convergent for z, w ∈ D. By convention, a factor in (2.10) reduces to z if γ(w) = 0. Note that B( · , w) is analytic on D with simple zeros at {γ(w)} γ∈Γ . The link back to potential theory is given by In particular, the Green's function can be written as where B is shorthand notation for B( · , 0). We point out that

A canonical factorization of the M-function
Let ⊂ R be a Parreau-Widom set and consider a Jacobi matrix J = {a n , b n } ∞ n=1 with σ ess (J) = . The spectrum of J thus contains and consists only of isolated eigenvalues outside . We denote these eigenvalues, if any, by {x k }. Should there be infinitely many of them, the x k 's accumulate nowhere but at some (or all) of the endpoints of , viz., α, β and α j , β j in the representation (2.5).
Let dµ = f (t)dt+dµ s be the spectral measure of J and introduce the m-function (or Stieltjes transform of dµ) by It is well known that m is a Nevanlinna-Pick function (i.e., m is analytic in C\R and Im m(x) ≷ 0 for Im x ≷ 0). Since dµ is compactly supported, we readily see that near ∞. In fact, one can write down the Laurent expansion of m µ around ∞ in terms of the moments of dµ. More importantly, the boundary values To be even more specific, we have The m-function remains analytic in the gaps of , and also below α and above β, except at the eigenvalues {x k } where it has simple poles. Moreover, m is real-valued and strictly increasing on any interval in R \ σ(J) as its derivative is > 0 there. So the poles and zeros of m interlace on each of the intervals in R \ and the same applies to m(x) − a for any a ∈ R. A major role in what follows will be played by the function Here x is the covering map defined in (2.8)-(2.9). Compared to m, the function M has the advantage of being meromorphic on D rather than C \ . It follows immediately from (2.9) and (3.2) that near z = 0. As a direct consequence of (2.8), M is automorphic with respect to Γ (i.e., M γ(z) = M (z) for every z ∈ D and all γ ∈ Γ). The poles of M are situated at the points p ∈ D for which To better keep track of this set, we introduce a fundamental set for Γ as follows.
Consider first the open set which is known as the Ford fundamental region. Geometrically, F is the unit disk with a number of orthocircles (and their interior) removed. More precisely, one has to remove two orthocircles for each generator γ j of Γ (or one for γ j and one for γ −1 j ) since the action of γ j can be described as inversion in some orthocircle C j in the upper half-plane following complex conjugation, and γ −1 j acts similarly with respect to the conjugate circle in the lower half-plane. Besides being symmetric in the real line, the set F has the important properties that 1) no two of its points are equivalent under Γ (i.e., if z ∈ F then γ(z) / ∈ F ), 2) γ∈Γ cl γ(F ) = D, where 'cl' refers to closure within D.
We can preserve these two properties and even get a disjoint union in 2) without taking closure by considering This is a fundamental set for Γ as it contains one and only one point of each Γ-orbit.
Returning to the poles of M , let p k be the unique point in F such that x(p k ) = x k . Then we can write the collection of poles as {γ(p k )} k,γ∈Γ . Note that p k either belongs to (−1, 1) \ {0} or lies in one of the C j 's, depending on whether x k is situated outside [α, β] or contained in (α j , β j ) for some j. The minus sign on the right-hand side in (3.5) ensures that Im M (z) ≷ 0 when z ∈ F and Im z ≷ 0.
We now aim at establishing an all-important result about M , namely that it is a function of bounded characteristic on D with no singular inner part (under a certain condition on the poles {x k }, also known as the Blaschke condition). This result can also be found in [37, Sect. 5] but we include a complete proof of the statement here, partly due to its importance and partly to make the present paper more self-contained. Proposition 1. Let and J be given as above. In addition, assume that the eigenvalues {x k } satisfy the condition where g is the Green's function for C \ with pole at ∞. Then the function M defined in (3.5) has bounded characteristic.
Remark. The condition (3.9) is equivalent to For by (2.12), it implies k,γ∈Γ |γ(p k )| = k |B(p k )| > 0, so that k B(z, p k ) converges to an analytic function on D with simple zeros at {γ(p k )} k,γ∈Γ . As none of these zeros belong to F \ {p k }, the product k |B(p k , z)| is > 0 there. Hence (3.10) follows from (2.11). Since every compact set K ⊂ C \ σ(J) is the image (under x) of a compact subset of F \ {p k }, we have a uniform bound of the form (3.11) k g Ω (x k , y) ≤ C for all y ∈ K.
Before the proof, let us briefly recall the notion of bounded characteristic (see, e.g., [15] or [21]). For a meromorphic function h, one defines the proximity function by m(r, h) = 2π 0 log + |h(re iθ )| dθ 2π and the counting function by where n(t, h) is the number of poles of h in |z| < t (counted with multiplicity). If these poles are denoted {p k }, the above integral can also be written as is called the characteristic function of h and if lim r↑1 T (r) < ∞, we say that h has bounded characteristic in |z| < 1.
When z = 0 is not a pole of h, the Cartan identity states that This formula, obtained by applying Jensen's formula to h(z) − e iθ and integrating over the unit circle, becomes very useful when the solutions to h(z) = e iθ are under control for all θ. In a similar way and by use of potential theory, Frostman [9] established the more general estimate Hence the task is reduced to dealing with values of the Green's function. Due to interlacing, we immediately see that (3.15) k: x k (a)∈(αj ,βj) So the a-points in gaps of do not present any problems. Outside [α, β] most of the a-points interlace with poles too. But for a > 0, there may be an a-point (x min (a), say) of m below the smallest eigenvalue of J and for a < 0, there may be an a-point above the largest eigenvalue. However that may be, we always have a pointwise (in a = 0) estimate of the form for some constant C := C(a) depending on a. For the choice of C = g(x min (1)), the estimate holds uniformly for a ≥ 1. Combining (3.15) and (3.16), we thus arrive at valid for a ∈ [1, 2], say. This completes the proof.
As a function of bounded characteristic, M has angular boundary values M (e iθ ) a.e. on the unit circle and admits a factorization of the form where π 1 , π 2 are Blaschke products corresponding to zeros and poles of M , and dρ is a singular measure on ∂D. Naturally, M (e iθ ) coincide a.e. with m(t + i0) for suitable t ∈ . As e iθ runs through ∂F := F ∩ ∂D, the corresponding values of t cover precisely twice and the same applies when e iθ traverses γ(F)∩∂D for arbitrary γ ∈ Γ. With reference to Pommerenke [28], see also [32], we have γ∈Γ |γ(F) ∩ ∂D| = 2π (since is a Parreau-Widom set) and whenever h • x is integrable on ∂D. Moreover, by preservation of the equilibrium measure under the covering map (see, e.g., [13,Chap. 3] or [8, Chap. 2]), for every h ∈ L 1 ( , dµ ).
The exponential of the integral in (3.17) that involves log |M (e iθ )| is called the outer part of M while exp of the second integral is referred to as the singular inner part of M . The following result will be crucial to us. Theorem 1. In the setting of Prop. 1, the function M has no singular inner part. In other words, it can be factorized as where z k and p k belong to the fundamental set F and are chosen in such a way that {x(z k )} and {x(p k )} are the zeros and poles of m in R \ .
Proof. We start by showing that M − ε has no singular inner part for ε > 0. The result will then follow taking ε ↓ 0. Given ε > 0, write m ε := m + ε in the form where ξ is defined a.e. on R by ξ(t) = 1 π Arg m ε (t + i0). The trick is to split the integral into two parts, namely i) and ii) R\ . We consider each of the two parts separately.
Since is compact, the behaviour of m is controlled by the function As a Stieltjes transform, ϕ is holomorphic in C \ and vanishes at ∞. Hence ϕ • x is analytic on D and in order to show that exp{ϕ • x} has no singular inner part, it suffices to prove that ϕ • x belongs to the Hardy space H 1 . For every f ∈ H 1 has a complex Poisson representation of the form A simple computation shows that So the imaginary part of ϕ • x is bounded and by M. Riesz' theorem on conjugate functions (see, e.g., [31,Chap. 17]), this implies that ϕ • x ∈ H p for all p < ∞.
ii)m(x) = exp Since m ε is real-valued (on R) away from σ(J), the function ξ only takes the values 0 and 1 in R \ (except at poles and zeros where it is not defined). More precisely, ξ = 1 on every interval of the form (x k , y k ), where x k is a pole and y k the following zero or some β j , whichever comes first. Furthermore, if lim t↓αi m ε (t) < 0 for some i, then ξ = 1 on the interval (α i , y), where y is the first zero after α i (or some β j ). Otherwise ξ = 0, and this in particular means that ξ vanishes below x − (= the minimum of α and the smallest pole) and above y + (= the maximum of β and the largest zero). Hence it suffices to consider the function Let P := {x k } denote the set of poles of m. In case these poles accumulate at all endpoints of , we can write ψ as where y k by definition is the first zero to the right of x k . The behaviour of m at ∞ ensures that 1) no zero can come before the smallest pole in (−∞, α), 2) there will always be a zero after the largest pole in (β, ∞).
The representation in (3.21) remains valid in general if we abuse notation and allow for certain x k 's and y k 's to coincide with suitable α i 's or β j 's. In all circumstances, the interval (x k , y k ) is either contained in a gap of or in R \ [α, β]. So each factor in (3.21) -and hence the full product -is positive on , except perhaps at certain endpoints where it vanishes or is not defined. Naturally, the x k 's can be ordered and we set Since k (y k −x k ) ≤ y + −x − , the finite product in (3.22) converges uniformly to ψ(x) on compact subsets of C \ P . The strategy for showing thatm • x has no singular inner part is as follows. We know that ψ • x admits a factorization of the form (3.17), involving a ratio of Blaschke products and a singular measure dρ. Move the Blaschke products to the left-hand side, take the logarithm and compare the real-parts. Because of (2.11), we get (3.23) log The goal is to show that the harmonic function on the left-hand side is the Poisson integral of its boundary values (i.e., the first integral on the right-hand side since g Ω ( · , y) vanishes on for all y ∈ Ω). This clearly implies dρ to be the trivial measure. As it seems hard to tell whether the harmonic function in question is the real part of an H p -function for suitable p > 1, we proceed by approximation. For 0 < δ 1 , δ 2 < 1, consider the function (3.24) φ n; δ1,δ2 (x) = log δ 1 + ψ n (x) 1 + δ 2 ψ n (x) + k≤n g Ω x, y k (δ 1 , n) − g Ω x, y k (1/δ 2 , n) , where {y k (δ, n)} are the zeros of δ + ψ n ( · ). It is clear that y k (δ, n) ∈ (x k , y k ) for all k, and y k (δ 1 , n) ր y k as δ 1 ↓ 0 while y k (1/δ 2 , n) ց x k as δ 2 ↓ 0. Since we see from (2.1) that φ n; δ1,δ2 is harmonic on Ω and continuous throughout C. Hence its maximum and minimum is assumed on (= the boundary of Ω). The Möbius transformation z → (δ 1 + z)/(1 + δ 2 z) maps R + onto the interval (δ 1 , 1/δ 2 ) and since g Ω · , y k (δ, n) vanishes on , we therefore have The uniform convergence of ψ n implies (by Hurwitz's theorem) that y k (δ, n) → y k (δ) as n → ∞, where {y k (δ)} are the zeros of δ + ψ( · ). We claim that φ n; δ1,δ2 converges locally uniformly on Ω to It suffices to consider compact sets K ⊂ Ω for which the intersection K ∩ (R \ ) is a closed interval, L = [a, b] say. There are only finitely many y k (δ)'s in this interval and if none of them are endpoints of L (i.e., = a or b), we have precisely the same number of y k (δ, n)'s in L for n large enough (again, by Hurwitz's theorem). Clearly, is bounded on K, uniformly in n, and the claim will follow by dominated convergence if we can find C > 0 such that k: y k (δ,n) / ∈L g Ω x, y k (δ, n) ≤ C for all x ∈ K and n sufficiently large. By concavity of the Green's function and with {c x,j } the critical points of g Ω ( · , x), it follows that (3.28) k: y k (δ,n) / ∈L g Ω x, y k (δ, n) ≤ g Ω (a−η, x)+g Ω (b+η, x)+ k: x k / ∈L g Ω x k , x + j: cj / ∈L g Ω c x,j , x for η > 0 small and n sufficiently large. We thus get the desired C on the lines of the remark after Prop. 1. The estimate (3.25) continues to hold in the limit n → ∞ so that φ δ1,δ2 • x is a bounded harmonic function on D. Hence it can be written as the Poisson integral of its boundary values, that is, All that remains is now to let δ 2 ↓ 0 and then δ 1 ↓ 0 in (3.29) and (3.26). Recalling that ψ ≥ 0 a.e. on , we get by monotone convergence that Since y k (δ) converges to y k as δ ↓ 0 and to x k as δ ↑ ∞, it follows from (3.26) that if we use dominated convergence as above. In conclusion, ψ • x has no singular inner part.
For ε > 0 small enough, the largest zero of m ε lies to the right of β and when ε ↓ 0, it converges to ∞. We therefore get the factor B(z) in (3.19).
Along the lines of [17,33] we shall now rewrite (3.19) to a nonlocal step-by-step sum rule and introduce first some notation. Let J n be the n times stripped Jacobi matrix (i.e., the matrix obtained from J by removing the first n rows and columns) and denote by dµ n = f n (t)dt + dµ n,s its spectral measure. In particular, J 1 = {a n+1 , b n+1 } ∞ n=1 and we let m 1 be the associated mfunction. More generally, m n denotes the m-function for J n (or Stieltjes transform of dµ n ) and M n is short for −m n • x. Furthermore, we use {x n,k } to denote the eigenvalues of J n (or poles of m n ) in R \ and write p n,k for the points in F for which x(p n,k ) = x n,k .
Related to coefficient stripping is the Stieltjes expansion 1 m 1 (z) which by iteration leads to a continued fraction representation of m. As a direct consequence of (3.30), we see that the zeros of m coincide with the poles of m 1 . Moreover, taking boundary values of the imaginary parts and recalling that m(t + i0) = 0 a.e., we get that Pulled back to ∂D, this means for a.e. θ, provided that Im M 1 (e iθ ) = 0 a.e. or, equivalently, Im M (e iθ ) = 0 a.e. on ∂D. When the set {θ : Im M (e iθ ) = 0} has full measure (i.e., f (t) > 0 for a.e. t ∈ ), we can therefore write (3.19) in the form This representation, relating M and M 1 , provides us with step-by-step sum rules. For our purpose it suffices to compare the constant terms. If we divide by B(z) in (3.31) and let z → 0, the left-hand side reduces by (3.6) and (2.13) to a 1 / Cap( ). According to (2.12), the logarithm of the Blaschke product (over k) on the right-hand side simplifies to k g(x k ) − g(x 1,k ) and the integral becomes log(f /f 1 )dµ , using (3.18). So we end up with Iteration now leads to provided that either log f or log f n is integrable with respect to dµ . The underlying assumption (3.9) automatically implies that k g(x n,k ) < ∞ for all n (cf. Prop. 3).

Preparatory results
In this section we present the last results needed to prove our main theorem. Throughout the section, denotes a Parreau-Widom set of positive Lebesgue measure. We start by showing that the equilibrium measure of is absolutely continuous with respect to Lebesgue measure. This enables us to relate the Szegő integral to relative entropies. Then we establish upper bounds for eigenvalue sums like the one in (3.9), but now for Jacobi matrices that are different from -and yet related to -the original J.

Absolute continuity of dµ
We shall prove the following result: The equilibrium measure dµ is absolutely continuous with respect to Lebesgue measure.
Needless to say, the statement can only be valid when | | > 0. Our proof relies on two lemmas, inspired by [19], [11]. While the first applies to any probability measure on R, the second is more specific to equilibrium measure. Lemma 1. Let dρ = w(t)dt + dρ s be a probability measure on R and let be its Stieltjes transform. If m ρ (t + i0)/(t + i) belongs to L 1 (R), then dρ is absolutely continuous (i.e., dρ s = 0).
Proof. Recall that the boundary values m ρ (t+i0) exist for a.e. t ∈ R and that w(t) = 1 π Im m ρ (t+ i0) a.e. on R. Our goal is thus to prove that By the assumption on m ρ (t + i0)/(t + i), the integral defines a holomorphic function in C \ R. We readily see that and since the right-hand side has the same boundary values as m ρ (a + ib) for b ↓ 0 (see, e.g., [30,Thm. 5.30] , and therefore f (x) must be constant and the goal is now easily achieved noting that Lemma 2. Let K ⊂ R be a compact set of positive Lebesgue measure and denote by m K the Stieltjes transform of the equilibrium measure dµ K . Then the boundary values m K (t + i0) are purely imaginary for a.e. t ∈ K.
Remark. In the language of [11,12,26,27,29], the lemma says that dµ K (or m K ) is reflectionless on K. The statement is consistent with the fact that the equilibrium potential is constant a.e. on K.
Proof. Write It is known that m Kn has the desired property for each n ∈ N (see, e.g., [35,Chap. 5]). By passing to a subsequence, if necessary, we can assume that dµ Kn w − → dµ K as n → ∞. Hence m Kn → m K , locally uniformly on C \ K.
The trick is now to consider the exponential representation of the m-function. We have where ξ n (t) = 1 π Arg m Kn (t + i0) a.e. on R. Obviously, ξ n (t) = 0 for t < α and ξ n (t) = 1 for t > β. So the right-hand side in (4.1) reduces to But more importantly, we have ξ n (t) = 1/2 for all t ∈ K n , n ∈ N. The uniform convergence of m Kn implies that γ α ξ n (t)dt converges uniformly for γ in [α, β] (see, e.g., [2,Sect. 2]). Equivalently, ξ n (t)dt w − → ξ(t)dt as measures on [α, β] (where ξ is short for ξ ∞ ). Since the Green's function for C \ K n converges locally uniformly to the Green's function for Ω := C \ K, the critical points converge too. Therefore, where {c j } are the critical points of g Ω . We conclude that ξ(t) = 1/2 for a.e. t ∈ K and the result follows.
Proof of Proposition. By Lemma 1, it suffices to show that m (t + i0)/(t + i) is integrable on R. We split the integral into 3 parts, namely i) , ii) ∪j (αj ,βj) , and iii) R\[α,β] . i) According to Lemma 2, m (t + i0) is purely imaginary a.e. on . When restricted to , |m (t + i0)| is therefore the absolutely continuous part of a finite measure. Hence, ii) By (2.3), we can relate m to the derivative of g in the gaps of and get the estimate Since is a Parreau-Widom set, the sum on the right-hand side is < ∞. iii) Recall from (3.2) that m decays like −1/t at ∞. Therefore, This completes the proof.
By Prop. 2, we can write the equilibrium measure of as dµ = f (t)dt. It follows from (3.18) and Lemma 2 that Hence the Szegő integral (i.e., log f dµ ) is closely related to a relative entropy, specifically Moreover, if either log f or log f n belongs to L 1 ( , dµ ) then we can write the integral in (3.33) as where S( · ) is short notation for the relative entropy S(dµ | · ).
Lemma 3. Let J be a bounded Jacobi matrix and suppose that σ ess (J) ∩ (a, b) = ∅ for some a < b.
If J has no eigenvalues in (a, b), then both P n JP n and (1 − P n )J(1 − P n ) have at most one eigenvalue between a and b.
Remark. Note that J (n) := P n JP n is the upper left n × n corner of J while (1 − P n )J(1 − P n ) is equal to J n , the n times stripped matrix.
Proof. By the spectral mapping theorem, J has no eigenvalues in (a, b) if and only if (J − a)(J − b) ≥ 0. Put differently, this means Given a projection P , write (P JP ) 2 = P J 2 P − P J(1 − P )JP in order to get where all operators are restricted to Ran(P ). When P = P n or P = 1 − P n , the perturbation P J(1 − P )JP has rank one and introduces at most one eigenvalue in (a, b).
In what follows, we specialize to the situation where J = {a n , b n } ∞ n=1 has essential spectrum equal to (i.e., the setting of Section 3). For arbitraryJ without eigenvalues, letJ (n) be the finite rank perturbation given by  Proof. For the first two series, the choice of works. This follows immediately from Lemma 3. To estimate the third, consider first J (n) ⊕J (i.e., set a n = 0 inJ (n) ). SinceJ has no eigenvalues, this direct sum has the same eigenvalues as J (n) . The rank two perturbation coming from a n > 0 can be written as 0 a n a n 0 = a n 2 1 1 1 1 + a n 2 −1 1 1 −1 .
Near the right end of a gap, the negative rank one perturbation −1 1 1 −1 interlaces the eigenvalues and the positive rank one perturbation ( 1 1 1 1 ) then moves the eigenvalues to the right. Near the left end of a gap, a similar argument applies. Hence, and the result follows.
In the next section, we takeJ to be the Jacobi matrix corresponding to the equilibrium measure of .

Szegő's theorem
We are now ready to prove the main result of the paper. Theorem 2. Let J = {a n , b n } ∞ n=1 be a Jacobi matrix with spectral measure dµ = f (t)dt + dµ s and let ⊂ R be a Parreau-Widom set of | | > 0. Assume that σ ess (J) = and denote by {x k } the eigenvalues of J outside , if any. On condition that k g(x k ) < ∞, we have if and only if (5.2) lim sup n→∞ a 1 · · · a n Cap( ) n > 0.
According to Prop. 3, the eigenvalue sum k g(x n,k ) is bounded above, uniformly in n, and the relative entropy S(dµ n ) is ≤ 0 for all n. Hence the right-hand side of (5.3) is bounded below and thus (5.4) lim inf n→∞ a 1 · · · a n Cap( ) n > 0.
So much the more, (5.2) is true.
To prove that lim sup > 0 implies the Szegő condition, let J be the Jacobi matrix of dµ and setJ = J in (4.5). ThenJ (n) reduces to J if we coefficient strip n times. By Prop. 3 and because log f is integrable with respect to dµ , cf. (4.2), the iterated step-by-step sum rule (3.33) applies toJ (n) . Since dµ has no eigenvalues and S(dµ ) = 0, we arrive at (5.5) log a 1 · · · a n Cap( ) n = k g x (n) k where dμ n is the spectral measure ofJ (n) . Clearly,J (n) converges strongly to J as n → ∞ and thus dμ n w − → dµ. As relative entropy is weakly upper semi-continuous, we therefore have lim sup n→∞ S(dμ n ) ≤ S(dµ).