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Pointwise Weyl law for graphs from quantized interval maps

We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pako\'nski et al. [J. Phys. A: Math. Gen. 34 9303-9317 (2001)] as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law, as a consequence we obtain for these models a strengthening of the quantum ergodic theorem from Berkolaiko et al. [Commun. Math. Phys. 273 137-159 (2007)], and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.


Introduction
Quantum graphs have been used as models of idealized one-dimensional structures in physics for many decades, and more recently, as simplified models for studying complex phenomena such as Anderson localization and quantum chaos [6]. The first evidence for quantum chaotic behavior in quantum graphs was given by Kottos and Smilansky [27,28], who showed numerically that the spectral statistics of certain families of quantum graphs behave like those of a random matrix ensemble, and further investigated this relationship using an exact trace formula. In view of the Bohigas-Giannoni-Schmidt conjecture [10], this led to the investigation of quantum graphs as a model for quantum chaos. Additional results regarding convergence of spectral statistics to those of random matrix theory include [7,8,19,20,45], among others.
In this article, we will consider a quantization method for certain ergodic piecewise-linear 1D interval maps S : [0, 1] → [0, 1], introduced by Pakoński,Życzkowski, and Kuś in [37] as a model for quantum chaos on graphs. Precise conditions for these interval maps will be described in Section 2, but one can consider just the doubling map, S(x) = 2x (mod 1), as a model example. The quantization method associates to S a family of unitary matrices U n , where U n is an n × n matrix, and n ∈ N is taken in a subset of allowable dimensions. These unitary matrices can be viewed as describing quantum evolution on directed graphs. The quantization is done in two steps: one first discretizes the map S, producing a family of Markov transition matrices P n , where P n is an n × n matrix corresponding to a classical Markov chain on a graph with n states. If P n is unistochastic, meaning there is a unitary matrix U n with the entrywise relation |(U n ) xy | 2 = (P n ) xy , then the U n are called a quantization of the classical map S. They are considered "quantizations" in the sense they satisfy a classical-quantum correspondence principle (Egorov theorem [5]), which relates unitary conjugation by the U n to the classical dynamics of S in the limit as an effective semiclassical parameter, in this case the reciprocal of the dimension, 1/n, tends to zero.
The matrices U n will be sparse and can in principle be very simple and non-random, such as the particular quantization for the doubling map given in (1.2). Surprisingly then, if the classical map S is chaotic, then for large n the resulting U n 's tend to have level spacings that numerically look Wigner-Dyson [37,44,45], as well as eigenvector coordinates that numerically look Gaussian ( Figure 1). This is despite their simple, sparse structure uncharacteristic of a typical CUE Haar unitary matrix. This behavior is however consistent with major conjectures in quantum chaos, that quantum systems corresponding to classically chaotic ones should exhibit random matrix ensemble spectral statistics (BGS conjecture [10]) and have eigenvectors that behave like Gaussian random waves [9] in the semiclassical limit.
More specifically, as investigated in [37,44,45], if the graphs correspond to classically chaotic systems, then when averaged over a choice of phase in the quantization, the spectral properties of these U n , such as the level spacings, spectral rigidity, and spectral form factor, appear to behave like those of CUE random matrices. As for eigenvector statistics, quantum ergodicity for these graphs with classically ergodic S was proved by Berkolaiko, Keating, and Smilansky in [5]. They showed that in the large dimension limit, nearly all eigenvectors of U n equidistribute over their coordinates: for sequences of allowable dimensions n, there is a sequence of sets Λ n ⊆ [n] := {1, . . . , n} with lim n→∞ #Λn n = 1 so that for all sequences (j n ) n with j n ∈ Λ n , and appropriate quantum observables O n (φ), where ψ (n,j) is the jth eigenvector of U n . This is the analogue for these graphs of the Shnirelman-Zelditch-de-Verdière quantum ergodic theorem [42,48,16], which was originally stated for ergodic flows on compact Riemannian manifolds. Quantum ergodicity has also been extended to other settings such as torus maps [12,29,30,34,49] and other graphs [4,3,1]; see also [2] for an overview and additional references.
In addition to the equidistribution from the quantum ergodic theorem, eigenfunctions from a classically ergodic system are expected to follow Berry's random wave conjecture [9], which asserts that the eigenfunctions should behave like Gaussian random waves in the large eigenvalue limit. For graphs, instead of the large eigenvalue limit, one considers as usual the large dimension limit. In this limit, [21,22] used supersymmetry methods to study the eigenfunction statistics for quantum graphs, specifically Gaussian moments, in view of the random wave conjecture.
In the specific discrete models from interval maps that we consider, one expects that the empirical distribution of the coordinates {ψ (n,j) x } n x=1 of an eigenvector of U n should behave like a random complex Gaussian N C (0, 1 n ) for most eigenvectors. This is consistent with both the random matrix ensemble behavior and the random wave conjecture. As an example, consider the doubling map S(x) = 2x (mod 1). For n ∈ 2N, the Markov matrices P n along with a particularly simple unitary quantization U n , are where the non-specified entries are all zeros. Numerically, for large n, the vast majority of eigenvectors of U n have coordinate value distributions that look complex Gaussian N C (0, 1 n ). Typical histograms for the coordinates of an eigenvector of U n from (1.2) are shown in Figure 1.
Motivated by the above, we will study the eigenvectors of such unitary quantizations U n constructed from allowable interval maps S by proving a pointwise Weyl law, which consists of estimates on the diagonal elements of spectral projection matrices. Because we allow for shrinking spectral windows in the pointwise Weyl law, this will have two additional implications concerning eigenvectors. First, we will be able to strengthen the quantum ergodic theorem from [5], and second, we will be able to construct random quantizations V n of S whose eigenvectors have the approximately Gaussian value statistics-while we do not prove Gaussian behavior for the eigenvectors of the original U n (except in a very special case where the eigenspaces end up highly degenerate, see Section 8), we will show there are many random quantizations V n near U n that do have the desired Gaussian eigenvector coordinates. These quantizations V n are not quantizations in the strict sense of |(V n ) xy | 2 = (P n ) xy from [37,5], but they will satisfy |(V n ) xy | 2 = (P n ) xy + o(1) as well as an Egorov theorem, so they are still quantizations of S in the sense that they satisfy a classical-quantum correspondence principle, recovering the classical dynamics in the semiclassical limit n → ∞. plotted against the pdf of N (0, 1 20 000 ). Right: 2D histogram in C of the coordinates (ψ x ) 10 000 x=1 . Since this is fairly spherically symmetric, the overall choice of phase for the eigenvector does not significantly impact the shapes of the first two plots.
Traditionally, a pointwise Weyl law gives the leading order asymptotics of the spectral projection ker- , for x in M a compact Riemannian manifold. For the unitary matrices U n , we look at the spectral projection onto arcs on the unit circle, P I = j:θ (n,j) ∈I |ψ (n,j) ψ (n,j) | where I ⊆ R/(2πZ). Then a pointwise Weyl law analogue would be a statement of the form j:θ (n,j) ∈I(n) |ψ (1)) for n → ∞ and appropriate intervals I(n). We will show this holds for sequences of intervals I(n) shrinking at certain rates, and for at least n(1−o(|I(n)|)) coordinates x. The coordinates for which this statement may not hold correspond to short periodic orbits in the graphs corresponding to S.
We then present the two applications of this pointwise Weyl law. The first is the strengthening of the quantum ergodic theorem to apply to sets of eigenvectors in bins {ψ (n,j) : θ (n,j) ∈ I(n)} with shrinking I(n). The second concerns random perturbations of the matrix U n to produce a family of random matrices V n (β [n] ) whose eigenvectors have the approximately Gaussian N C (0, 1 n ) eigenvector statistics. These eigenvectors will also tend to satisfy a version of quantum unique ergodicity (QUE), a notion introduced by Rudnick and Sarnak in [39], and where all eigenvectors are considered in the limit (1.1), rather than just those in a sequence of limiting density one sets.
Acknowledgements. The author would like to thank Ramon van Handel for suggesting this problem and providing many helpful discussions and feedback, and for pointing out the simpler proof of Proposition 8.1. The author would also like to thank Peter Sarnak for helpful discussion and suggesting the use of the Beurling-Selberg function as the particular smooth approximation. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2039656. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Set-up and main results
2.1. Set-up. Here we state the assumptions on the map S and matrices P n . Let S : [0, 1] → [0, 1] be a piecewise-linear map that satisfies the following conditions: (i) S is (Lebesgue) measure-preserving, µ(A) = µ(S −1 (A)) for any measurable set A.
(ii) There exists a partition M 0 of [0, 1] into M 0 equal intervals (called atoms) A 1 , . . . , A M0 , with S linear on each atom A j . (iii) The endpoints E 0 = E(M 0 ) of the atoms have left and right limits satisfying lim x→e ± 0 S(x) ∈ E 0 for e 0 ∈ E 0 . This means the linear segments in S begin and end in the grid E 0 × E 0 . With (i) and (ii), this ensures the slope of S on each atom must be an integer. For convenience, also assume S(e 0 ) takes one of the values of these one-sided limits. (iv) The absolute value of the slope of S on each atom is at least two, i.e. the slope is never ±1. Conditions (i), (ii), and (iii) are essentially the same as in [37,5]. Condition (iv) is there instead of the ergodicity assumption. It allows for some non-ergodic S such as those corresponding to block matrices of various ergodic maps. Two examples of allowable ergodic S are the doubling map and the "four legs map" shown in Figure 2. In general, conditions for ergodicity of S would follow from results on piecewise expanding Markov maps, see for example Chapter III in the textbook [32]. For n ∈ M 0 Z, partition [0, 1] into n equal atoms, E x = ( x−1 n , x n ) for x = 1, . . . , n, and define the corresponding n × n Markov transition matrix P n by The matrix P n looks at where S sends an atom E x , and assigns a uniform probability 1 |S (z)| to each atom E y that S can reach from E x . To generate the family of corresponding unitary matrices U n as done in [37,5], it is required that the P n be unistochastic, so that there are unitary matrices U n with the entrywise relation |(U n ) xy | 2 = (P n ) xy . In general, characterizing which bistochastic matrices are unistochastic is difficult; however see [37,52,5] for some conditions and examples. Note that the relation |(U n ) xy | 2 = (P n ) xy does not uniquely define U n if it exists, as one can always add additional phases without changing unitarity or the entrywise relation. For example, given any Φ ∈ [0, 2π) n and defining the diagonal matrix e iΦ := diag(e iΦ1 , . . . , e iΦn ), then e iΦ U n is also unitary and satisfies the same entrywise norm-squared relation.
Finally, let L 0 be the least common multiple of the slopes in S, and let K(n) be the largest power of L 0 that divides n/M 0 , so n = M 0 L K(n) 0 r and r does not contain any factors of L 0 . The purpose of K(n) will be to keep track of how many powers of S we can take, while still ensuring S behaves nicely with the partition into n atoms.

2.2.
Pointwise Weyl law and overview. With the above definitions, we state the main result: Consider a sequence (n k ) k so that K(n k ) → ∞, and suppose each n k × n k Markov matrix P n k is unistochastic with corresponding unitary matrix U n k . Let (I(n k )) be a sequence of intervals in R/(2πZ) satisfying Then denoting the eigenvalues and eigenvectors of U n k by (e iθ (n k ,j) ) j and (ψ (n k ,j) ) j respectively, there is a sequence of subsets G n k ⊆ [n k ] with sizes #G n k = n k (1 − o(|I(n k )|)) so that for all x ∈ G n k , where the error term o(1) can be taken to depend only on n k , |I(n k )|, and K(n k ), and is independent of x ∈ G n k . Additionally, G n k can actually be chosen independent of I(n k ) and |I(n k )|.
3) cannot in general be improved to hold for all coordinates x ∈ [n k ], as Section 9 will show. The coordinates x that we exclude from G n k correspond to those with short periodic orbits in the graphs associated to P n k and U n k . This is reminiscent of the relationship between geodesic loops and the size of the remainder in the Weyl law [18,26] or pointwise Weyl law [40,43,13], in the usual setting on manifolds.
(ii) The condition (2.2) that K(n k )|I(n k )| → ∞ is essentially optimal under the stated hypotheses, as we will see by considering the doubling map when n = 2 K (Section 8). In this case, K(n) = log 2 n, and if |I(n)| < π 2 log 2 n , then one can take an interval that avoids the spectrum entirely and thus produces a spectral projection matrix filled with zeros. (iii) This condition that |I(n k )| does not shrink too fast appears from error terms from only considering powers of U n k up to an Ehrenfest time K(n k ) ∼ log n k . This time is a common obstruction in semiclassical problems, and even in these discrete models, our analysis does not go beyond this time. If the lengths |I(n k )| are larger than the bare minimum required to satisfy (2.2), then more precise remainder terms than just o(1) are obtained from the proof, cf. equation (4.1).
The proof details of Theorem 2.1 will be specific to our discrete case, where we have sparse matrices U n k and can analyze matrix powers and paths in finite graphs. We will start by just taking a smooth approximation of the indicator function of the interval I(n k ), and estimating the left side of (2.3) by a Fourier series in terms of powers of U n k . However, the properties of S ensure that we understand powers of U n k well up to time K(n k ). This allows us to identify and exclude the few coordinates x that have short loops before a set cut-off time. Using properties of the powers of U n k again, the remaining coordinates will then produce small enough Fourier coefficients that (2.3) holds.
Summing (2.3) over all x (separating x ∈ G n k from x ∈ G n k ) produces a Weyl law analogue that counts the number of eigenvalues in a bin.
Corollary 2.2 (Weyl law analogue). Let S, (n k ) k , U n k , and I(n k ) be as in Theorem 2.1, including (2.2). Then as k → ∞, (2.4) #{j : θ (n k ,j) ∈ I(n k )} = n k |I(n k )| 2π where the remainder term depends on |I(n k )| but is independent of the particular location of I(n k ).
In the following subsections, we discuss implications of Theorem 2.1 on eigenvectors of U n . We present the two applications, the first a strengthening of the quantum ergodic theorem for this model, and the second a construction of random perturbations of U n with approximately Gaussian eigenvectors. For the first application, using Theorem 2.1 with shrinking intervals |I(n k )|, rather than the usual local Weyl law, in the standard proof of quantum ergodicity naturally produces a stronger quantum ergodicity statement. For the second, we take random unitary rotations of bins of eigenvectors, and apply results on the distribution of random projections from [17,15,35] to show the resulting eigenvectors have approximately Gaussian value statistics.
2.3. Application to quantum ergodicity in bins. To state a quantum ergodic theorem, we first define quantum observables as in [5], as discretized versions of a classical observable h ∈ L 2 ([0, 1]). Given n ∈ N and h ∈ L 2 ([0, 1]), define its quantization O n (h) to be the n × n diagonal matrix with entries Note that 1 n tr O n (h) = 1 0 h, the analogue of the local Weyl law. Quantum ergodicity for this model, as proved in [5], states there is a sequence of sets Λ n k ⊆ [n k ] with lim n k →∞ #Λn k n k = 1 such that for all sequences (j n k ) k , j n k ∈ Λ n k and h ∈ C([0, 1]), This is equivalent to the decay of the quantum variance, as k → ∞. Using Theorem 2.1 and an Egorov property from [5], we will prove the following concerning quantum ergodicity in bins.
Theorem 2.3 (Quantum ergodicity in bins). Let S satisfy (i)-(iv) and also be ergodic. Let (n k ) k , U n k , and I(n k ) be as in Theorem 2.1, including (2.2). Then for any Lipschitz h : [0, 1] → C, This decay of the quantum variance in a bin implies there is a sequence of sets Λ n k ⊆ {j : θ (n k ,j) ∈ I(n k )} with #Λn k #{j:θ (n k ,j) ∈I(n k )} → 1 such that (2.6) holds for all sequences (j n k ) k , j n k ∈ Λ n k and continuous h : [0, 1] → C. Since we allow |I(n)| → 0, the bin sizes are o(n) (density 0) by the Weyl law analogue, and so Theorem 2.3 guarantees that the density 0 subsequence excluded from the original quantum ergodic theorem cannot accumulate too strongly in a region I(n) of the unit circle.
2.4. Application to random quantizations with Gaussian eigenvectors. The second application of Theorem 2.1 will be to construct random perturbations of U n with eigenvectors that look approximately Gaussian. To construct the random perturbations, we use the estimates on the spectral projection matrix combined with results on low-dimensional projections from [17,35,15] to first prove: Theorem 2.4 (Gaussian approximate eigenvectors). Let S, (n k ) k , U n k , and I(n k ) be as in Theorem 2.1, in particular assume (2.2) holds. Denote the eigenvalues and eigenvectors of U n k by (e iθ (n k ,j) ) j and (ψ (n k ,j) ) j respectively. Then letting φ (n k ) be a unit vector chosen randomly according to Lebesgue measure from span(ψ (n k ,j) : θ (n k ,j) ∈ I(n k )), the empirical distribution µ (n k ) of the scaled coordinates √ n k φ , . . . , √ n k φ (n k ) n k , converges weakly in probability to the standard complex Gaussian N C (0, 1) as k → ∞. In fact, there are absolute numerical constants C, c > 0 so that for any f : We then construct random perturbations V n k (β [n k ] ) of U n k by grouping the eigenvalues of U n k into bins and randomly rotating the eigenvectors within each bin. This idea of rotating small sets of eigenvectors was used in different models in [50,47,33,14] to construct random orthonormal bases with quantum ergodic or quantum unique ergodic properties. In our setting, Theorem 2.4 will also show that the coordinates of these randomly rotated eigenvectors look approximately Gaussian. The matrices V n k (β [n k ] ) also satisfy the entrywise relations |(V n k (β [n k ] )) xy | 2 = (P n k ) xy + o(1) as well as a weaker Egorov property relating them to the classical dynamics, so that they can be viewed as a quantization of the classical map S. Thus while we do not prove approximate Gaussian behavior for the quantizations U n k with |(U n k ) xy | 2 = (P n k ) xy , we prove it for the family of random matrices V n k (β [n k ] ), which are alternative quantizations of the original classical dynamics of S.
Note also that S is not required to be ergodic here. In particular, we can take the direct sum of two ergodic maps S 1 and S 2 , whose resulting block matrix U n k has eigenvectors localized on just half of the coordinates. Then U n k will not have equidistributed or Gaussian eigenvectors, though the randomly perturbed matrices V n k (β [n k ] ) still will.
Theorem 2.5 (Random quantizations with Gaussian eigenvectors). Let S satisfy (i)-(iv), and let (n k ) k be a sequence with K(n k ) → ∞ and with each Markov matrix P n k unistochastic. Then there exists a family of random unitary matrices V n k (β [n k ] ) in some probability spaces (Ω n k , P n k ) with the following properties: 2.5. The doubling map. Finally, in Sections 7 and 8 we study the case when S is the doubling map 2x (mod 1) and the specific quantization U n is the orthogonal one in (1.2). We study this case using similar arguments as in the general case, but with stronger estimates from analyzing binary trees and bit shifts specific to the doubling map. Theorem 2.1 will hold with any sequence of even n ∈ 2N, not just those with K(n) → ∞. Additionally, when n = 2 K , the spectrum of this specific quantization U n is degenerate with multiplicities asymptotically 2 K 4K , and most every eigenbasis looks Gaussian (Theorem 8.4).
2.6. Outline. Section 3 contains some lemmas concerning properties of the map S and the corresponding Markov matrices P n . Section 4 is the proof of the pointwise Weyl law analogue, Theorem 2.1. The first application, on quantum ergodicity in bins, is proved in Section 5. Section 6 covers the second application on random perturbations of U n with approximately Gaussian eigenvectors. Sections 7 and 8 deal with the specific map the doubling map, especially with the degenerate case of dimension a power of two. An example where the pointwise Weyl relation (2.3) fails for a particular coordinate choice x is given in Section 9.

Properties of the map S and matrices P n
In this section we gather some results about the relationship between the map S and the Markov matrices P n . The following lemma contains properties from [5] and [37], stated here for a specific condition involving K(n). Lemma 3.1 (powers of S, [5,37]). Assume (i)-(iii) and let the partition size be n ∈ M 0 Z with atoms E 1 , . . . , E n . Then for 1 ≤ ≤ K(n) + 1, (a) S is linear with integer slope on each atom E x , and for endpoints e ∈ E, the right and left limits satisfy lim y→e ± S (y) ∈ E.
is a union of several adjacent atoms and some endpoints.
The condition here with K(n) can be more restrictive than needed in [5], but is a concrete example of allowable powers and dimensions n. For completeness with these concrete conditions, we include most of the proofs below. Proof.
(a) Both parts are done recursively. For example, if S −1 is linear on atoms E x , then for the first part of (a), it suffices to show for each x, S −1 (E x ) ⊆ A j for one of the atoms A j of the "base" partition M 0 (depending on x), since then composition with S shows The first part follows from the above inclusions as well; note that if (P n ) xτ1 (P n ) τ1τ2 · · · (P n ) τ −1 y = 0, then S (E x ) ⊃ S −1 (E τ1 ) ⊃ · · · ⊃ E y . The unique path part is Lemma 2 from [5]: the proof is to suppose there are z 1 , z 2 ∈ E x with S (z 1 ), S (z 2 ) ∈ E y but with S r (z 1 ) ∈ E 1 and S r (z 2 ) ∈ E 2 for some 1 < r < and E 1 = E 2 . Then pick w ∈ E y , and by part (b), then there is The next lemma shows how K(n) is used to ensure that small powers of P n interact nicely with the partition of [0, 1] into n atoms.
That is, for 1 ≤ ≤ K(n) + 1, we can compute P n by drawing S and applying the same procedure we used to define P n from S.
equal atoms ensures S is linear on each atom. Since 1 ≤ ≤ K(n) + 1, then M 0 · L −1 0 divides n so for these , the map S is linear on each atom of the size n partition and the value 1 |(S ) (z)| is the same for any z ∈ E x . The matrix elements of P n are (P n ) xy = τ :x→y (P n ) xτ1 (P n ) τ1τ2 · · · (P n ) τ −1 y . By Lemma 3.1(d), for 1 ≤ ≤ K(n) + 1 and fixed x, y, this sum over τ collapses to either zero or just a single term (P n ) xτ1 · · · (P n ) τ −1 y . If this is nonzero, then by the definition of P n , The following lemma demonstrates the sparseness of the matrices P n for times before K(n). Essentially, this is because for these times, the nonzero entries of the matrix P n are placed by drawing S and an n × n grid in [0, 1] 2 , and then placing a nonzero entry in each position in the grid that S passes through. As n increases, the grid becomes finer and the (one-dimensional) graph of S in [0, 1] 2 cannot pass through a very large fraction of the boxes. Proof. Pick an atom E x . Since the maximum slope magnitude of S is s max , the interval S (E x ) has length at most s max · |E x | and intersects at most s max atoms E y . Thus by Lemma 3.2 the xth row of P n has at most s max nonzero entries, so in total P n has at most n · s max nonzero entries.
Also by Lemma 3.2, nonzero diagonal elements of P n occur exactly when , so that the nonzero diagonal elements (P n ) xx occur exactly when S intersects the xth square ( x n , x+1 n )×( x n , x+1 n ) (Figure 3). Choose an interval atoms. This is the coarsest partition for S for which we can guarantee by Lemma 3.1(a) that S is linear on each atom. Since ≤ K(n) + 1, the partition into n atoms E x is a refinement of this one. If the slope of S is negative on I i , then S can intersect at most one box in the diagonal Q. If the slope of S is positive and at least two on I i , then it can intersect at most two boxes in Q (see Figure 3): Consider the slope one lines t ± 1 n in [0, 1] 2 , which bound a parallelogram R ⊃ Q. Project the line segment S (I i ) ∩ R onto the x-axis. If S on I i has slope m > 1, then one can compute this projection is an interval of length ≤ 1 n 2 m−1 . For m ≥ 3, this bound is ≤ 1 n so S (I i ) ∩ R can intersect at most two 1 n × 1 n boxes in Q. For m = 2, the length can be 2 n , but S (I i ) ∩ R can still only intersect at most two boxes in Q, by using that Remark 3.1. Although the above argument works for slope −1, we do not allow slope −1 in S since powers of S could then have segments with slope +1.

Proof of Theorem 2.1 pointwise Weyl law
In this section we prove Theorem 2.1 using a Fourier series approximation of the projection matrix, and properties of the Markov matrix P n and quantization U n to identify potentially bad coordinates x. For notational convenience, we will use n instead of n k . The idea is to approximate the spectral projection matrix P I(n) := j:θ (n,j) ∈I(n) |ψ (n,j) ψ (n,j) | using a Fourier series in powers of U n , The constant term in the Fourier series is already the leading term in the pointwise Weyl law. So it remains to show all the other terms are small, which can be done by splitting up the |j| ≥ 1 terms into three regions pictured as follows: Fourier series convergence Remove coords. with short loops For |j| > K(n), we have little knowledge of (U j n ) xx , so these terms are controlled simply by convergence of the Fourier series, or decay of the Fourier coefficients. For |j| before what is essentially the Ehrenfest time K(n), we have good knowledge of U j n . We will split this region 1 ≤ |j| ≤ K(n) into two regions at a cut-off r( K(n)). Before the cut-off time r( K(n)), we remove all the coordinates x that have short loops x → x of length j, since these coordinates can produce large Fourier series terms through the large entries (U j n ) xx . There are not many of these coordinates by Lemma 3.3. For r( K(n)) < |j| ≤ K(n), we use the relationship between U j n and P j n , specifically Lemma 3.1(d), to control the size of (U j n ) xx . Before starting the proof, we make some additional remarks about the proof and statement.
, and for x ∈ G n that (4.1) (ii) To ensure the right side of (4.1) is o(|I(n)|) and #G n = n(1 − o(|I(n)|)), choose r so that r( K(n)) → ∞, K(n) − r( K(n)) − log L0 1 |I(n)| → ∞, as K(n) → ∞. Since |I(n)| · K(n) → ∞ by (2.2), then eventually 1 |I(n)| < K(n), so condition (4.2) is always met if For example, r( K(n)) = K(n)/2 or r( K(n)) = log K(n) always satisfy the conditions on r. (iii) We are interested in sequences of intervals I(n) where |I(n)| → 0. For the proof, we will assume that |I(n)| is bounded away from 2π. If |I(n)| is near 2π, apply the Theorem to the complement I(n) c or to a larger interval around I(n) c that satisfies (2.2), to conclude j:θ (n,j) ∈I(n) as |I(n) c | → 0.  Thus also using property (iii), I,δ ( )e i x + g Recall we also defined p I(n) (e it ) = χ I(n) (t) and the projection P I(n) = p I(n) (U n ), so that by the spectral theorem, By (4.5) and the spectral theorem again, The identity term |I(n)| 2π (1±2π|I(n)| −1 K(n) −1 ) Id has the values we want already since |I(n)| −1 K(n) −1 → 0 by (2.2), so to show (4.1) we want to show the rest of the terms are small. Since then for any x, y ∈ [n], the (x, y) element of the non-identity terms can be bounded as (4.9)

4.3.
Removing potentially bad points. Here we use properties of U n and P n from Section 3 to remove coordinates x where (4.9) may be large. For 1 ≤ ≤ K(n) + 1, by Lemma 3.1(d) there is at most one path of length from a given x to itself (or to another y), so Since all slopes of S are at least 2 in absolute value, then all the slopes of S are at least 2 in absolute value, so by Lemma 3.2, |(U n ) xx | ≤ 2 − /2 for all x ∈ [n]. In order to make the sum 2 K(n) =1 |(U n ) xx | in (4.9) small then, we only need to be concerned with smaller , since |(U n ) xx | decays exponentially in . As we will see, by Lemma 3.3, for small , (U n ) xx = 0 for most coordinates x, so we can pick a cut-off for small and just throw out any coordinates x where (U n ) xx = 0 below this cut-off.
Let r : N → N satisfy r(k) < k and (4.2); this will determine the cut-off for which are "small". Define the set of potentially bad coordinates as using assumption (4.2) for the last equality. For x ∈ G n := [n] \ B n , then Then for x ∈ G n , since |I(n)| K(n) → ∞ by (2.2). By (4.11), #G n ≥ n(1 − o(|I(n)|)).

Remark 4.2.
The same method to estimate the diagonal entries of the projection matrix P I(n) can also be used to estimate the off-diagonal entries of P I(n) . In this case, the constant term in the Fourier series expansion is zero, and then one can use similar arguments to show that the higher order terms are small. Alternatively, one can also obtain some bounds using that by the Weyl law, x,y∈[n] |(P I(n) ) xy | 2 = tr((P I(n) ) 2 ) = tr P I(n) = n|I(n)| 2π (1 + o(1)).

Quantum ergodicity in bins
In this section we prove Theorem 2.3 concerning quantum ergodicity in bins {j : θ (n k ,j) ∈ I(n k )}, following the standard proof of quantum ergodicity that uses the Egorov property.
Theorem 5.1 (Egorov property, [5]). Suppose S satisfies conditions (i)-(iv) and has a corresponding n × n unitary matrix U n with eigenvectors (ψ (n,j) ) n j=1 . Let O n (h) be the quantum observable corresponding to h : [0, 1] → C. If h is Lipschitz continuous on each image S(E x ), and n ∈ M 0 L 0 Z, then where the norm on the left side is the operator norm.
If t ≤ K(n) + 1, then by the same recursive argument as in Lemma 3.1(a), S t−1 is linear on each S(E x ), so h • S t−1 is Lipschitz on S(E x ) with Lipschitz constant ≤ h Lip L t−1 0 . Then iterating (5.1) t times yields, If say t ≤ K(n) 2 , then L t 0 n, so the error bound is small, and the Egorov property (5.2) relates the quantum dynamics U t n O n (h)U −t n to the classical dynamics h • S t for t well before the Ehrenfest time T E := K(n) log n. V n k := 1 #{j : θ (n k ,j) ∈ I(n k )} j:θ (n k ,j) ∈I(n k ) ψ (n k ,j) , O n k (h)ψ (n k ,j) 2 , which we will show tends to zero as k → ∞. For a function g : Using that ψ (n k ,j) are eigenvectors of U n k , followed by the Egorov property and averaging over t Then by Cauchy-Schwarz, For this quantization method, just a sup norm bound shows so that , then for t ≤ T , S t is linear on every E x so that 1 Applying the above and Theorem 2.1 yields 1 #{j : θ (n k ,j) ∈ I(n k )} j:θ (n k ,j) ∈I(n k ) using the L 2 ergodic theorem as T = K(n k ) 2 → ∞.
The passage from decay of the quantum variance (2.7) to the density one statement is by the usual method (for details see for example Theorem 15.5 in the textbook [51]). To start, by Chebyshev-Markov with ε = V 1/4 n k , Theorem 2.3 implies for a single Lipschitz function h, there is the sequence of sets Λ n k (h) ⊆ {j : θ (n k ,j) ∈ I(n k )} with such that for all sequences (j n k ) k with j n k ∈ Λ n k (h), For a countable set of Lipschitz functions (h ) , since finite intersections of sets Λ n k satisfying (5.7) also satisfy (5.7), we can assume Λ n k (h +1 ) ⊆ Λ n k (h ) for all n k . Then for each h , let N ( ) > 0 be large enough so that for n k ≥ N ( ), Take N ( ) increasing in and let Λ ∞ n k := Λ n k (h ) for N ( ) ≤ n k < N ( +1), so (5.8) holds for sequences in Λ ∞ n k and h in the countable set. Then take (h ) to be a countable set of Lipschitz functions that are dense in (C([0, 1]), · ∞ ), so that for any h ∈ C([0, 1]), The terms on the right side are bounded by h − h ∞ or are o(1) as k → ∞.

Random Gaussian eigenvectors
In this section we prove Theorems 2.4 and 2.5 on random unitary rotations of bins of eigenvectors. We are interested in the value statistics of a random unit vector v chosen from V = span{ψ (n,j) : θ (n,j) ∈ I(n)}. These coordinate values are just the one-dimensional projections of v onto the standard basis vectors {e 1 , . . . , e n }. The behavior of such low-dimensional projections of high-dimensional vectors has been well-studied since the 1970s for its applications in analyzing large data sets; see for example the survey [25] for an overview of the early history and motivation of "projection pursuit" methods. The marginals of high-dimensional random vectors are often known to look approximately Gaussian, with precise conditions first proved by Diaconis and Freedman in [17]. We will apply a quantitative version due to Meckes [35] and Chatterjee and Meckes [15] to show the desired Gaussian coordinate value behavior for an orthonormal basis of such randomly chosen vectors. 6.1. Random projections and bases. First we start with Theorem 2.4, which concerns the coordinate values of a single random unit vector in V = span{ψ (n,j) : θ (n,j) ∈ I(n)}. Let M V be the n × (dim V ) matrix whose columns are the basis vectors (ψ (n,j) ) of V . Then P I(n) := M V M * V is the projection onto this space, and a random unit vector φ in the span is chosen according to ω ∼ g/ g 2 ∼ M V u, where g ∼ N C (0, P I(n) ) and u ∼ Unif(S dim V −1 C ). The coordinates φ 1 = φ 1 , e x , φ 2 , . . . , φ n are u, M * V e 1 , u, M * V e 2 , . . . , u, M * V e n which is a 1-dimensional projection in the direction u ∈ C dim V of the data set {M * V e 1 , . . . , M * V e n } ⊂ C dim V . Since n x=1 |φ x | 2 = 1, we will use the scaled data set √ n{M * V e 1 , . . . , M * V e n }. The following theorem from [35] also using [15] is a quantitative version of the theorem from [17]. Theorem 6.1 (Complex version of Theorem 2 in [35]). Let {x j } n j=1 be deterministic vectors in C d , normalized so that σ 2 = 1 nd n i=1 x i 2 = 1 and suppose n n j=1 δ θ,xj on C. There are absolute numerical constants C, c > 0 so that for θ ∼ Unif(S C d−1 ), any bounded L-Lipschitz f : C → C, and ε > 2L(A+3) d−1 , there is the quantitative bound where Z ∼ N C (0, 1). In particular, if A = o(d) and B = o(d), then µ (n) converges weakly in probability to N C (0, 1) as n → ∞.
Proof. The proof is the same as the real version in [35], except that the (multi-dimensional) Theorem 6.2 written below from [15] replaces the single-variable version. The proof idea from [35] is to let Then one uses Theorem 6.2, a generalization of Stein's method of exchangeable pairs for abstract normal approximation, to bound |EF (θ) − Ef (Z)| with W = θ, x I where I ∼ Unif[n], and then one can apply Gaussian concentration (Lemma 6.3, cf. [11, §5.4]) to F which is (L √ B)-Lipschitz.
Theorem 6.2 (Theorem 2.5 for C in [15]). Let W be a C-valued random variable and for each ε > 0 let W ε be a random vector such that L(W ) = L(W ε ), with the property that lim ε→0 W ε = W almost surely. Suppose there is a function λ(ε) and measurable Γ, Λ such that as ε → 0, Applying Theorem 6.1 to our case immediately yields the following. There are absolute numerical constants C, c > 0 so that for f : C → C bounded L-Lipschitz and ε > 2L(A+3) d−1 , where Z ∼ N C (0, 1). Remark 6.1. As will be shown by (6.13) and (6.11), Theorem 2.

Proof.
Let v 1 , . . . , v d be an orthonormal basis for V (ν) , and let M V be the n × d matrix with those vectors as columns. Apply Theorem 6.1 to the data set We can also take B = 1 since for any θ ∈ S d−1 Theorem 6.4 provides a bound for the probability that a single randomly chosen vector does not look Gaussian. Because the quantitative bound (6.7) decays quickly, a simple union bound gives a bound on finding an entire orthonormal basis that looks Gaussian (Corollary 6.6 below). This family of random orthonormal bases will then be used to construct the unitary matrices V n k (β [n k ] ) in Theorem 2.5. Proof. Let µ be normalized Haar measure on U (d). Then for any x ∈ S d−1 C , σ(A) = µ(g ∈ U (d) : g(x) ∈ A). By union bound, letting {e j } be the standard basis, where Z ∼ N C (0, 1).
Proof. The numbers d need not be the dimensions of Then Theorem 6.4 implies for ε > 2L(2A+3) 6.2. Proof of Theorem 2.5. Choose κ(n k ) ∈ N so that if we divide [0, 2π] up into κ(n k ) equal sized intervals I 1 (n k ), . . . , I κ(n k ) (n k ), then (2.2) holds for |I(n k )| = 2π κ(n k ) . Let ψ (n k ,j) be the jth eigenvector of U n k . Like the method used in [14], construct V n k (β [n k ] ) by taking a random unitary rotation (according to Haar measure) of the eigenvectors {ψ (n k ,j) : θ (n k ,j) ∈ I (n k )} within each interval. Then perturb any degenerate eigenvalues to be simple, while still keeping them in the same bin. Denote the resulting eigenvectors of V n k (β [n k ] ) by φ [n k ,j] (β) . (a) Let U n k be the perturbation of U n k obtained by reassigning all eigenvalues in the same bin I (n k ) to a single value e iΘ in the bin. Then since the reassigned eigenvalues are still in the same bin. Also, U n k − V n k (β [n k ] ) ≤ C 2π κ(n k ) by the same computation, since U n k has degenerate eigenspaces that can be rotated to match the eigenvectors of V n k (β [n k ] ). Thus for any random . The Egorov property for U n k , Theorem 5.1, then implies the weaker Egorov property for V n k (β (b) To show Gaussian behavior, we first show there is ε(n k ) → 0 so that for any bounded Lipschitz f : C → C, as k → ∞, where Z ∼ N C (0, 1). A density argument followed by tightness will then complete the proof of (b).
To show (6.12), note that for any W [ ] = span{ψ (n k ,j) : θ (j) ∈ I (n k )} and P [ ] the orthogonal projection onto W [ ] , the pointwise Weyl law Theorem 2.1 implies (6.13) so the quantity A in Corollary 6.6 can be taken to be o(n k |I (n k )|). Let µ Now let (f ) be a countable set of Lipschitz functions with compact support that are dense in C c (C), and set k |I(n k )| 1/2 ) → 0, since by (2.2), n 1/2 k |I(n k )| 1/2 2 log n k . For a sequence of matrices ( V n k ) k with V n k ∈ Π n k , let µ [n k ,j] be the scaled coordinate distribution of the jth eigenvector φ [n k ,j] of V n k . By definition of Π n k , we know for any f that f dµ [n k ,jn k ] → Ef (Z) as k → ∞, for any sequence (j n k ) k with j n k ∈ [1 : n k ]. Denseness of (f ) shows that this holds for all f ∈ C c (C) as well. Then ( µ [n k ,jn k ] ) k is tight, and with the vague convergence we get weak convergence of µ [n k ,jn k ] to N C (0, 1). (c) To show QUE, like in (b), we first show there is ε(n k ) → 0 so that for any bounded Lipschitz h : [0, 1] → C, as k → ∞, This is done by the same argument presented in [14] using the Hanson-Wright inequality [38]. After proving (6.16), part (c) follows from density like in (b   [14], Theorem 4.1 for details), which by the pointwise Wey law Theorem 2.1 is 1 0 h + R(n k , h) with |R(n k , h)| ≤ R(n k ) h ∞ , some R(n k ) → 0. In particular, for any ε > 2|R(n k , h)|, Then taking ε(n k ) = max 2R(n k ), 1 (n k |I(n k )|) 1/4 and applying a union bound like with Lemma 6.5 yields (6.16), using that eventually min(ε(n k ) 2 , ε(n k )) = ε(n k ) 2 .
Next, taking (h ) to be a countable dense set of Lipschitz functions in C([0, 1]), let Then P[Γ c n k ] ≤ C 1 n 2 k exp(−C 2 n 1/2 k |I(n k )| 1/2 ) → 0, and denseness shows that for sequences ( V n k ) k with V n k ∈ Γ n k and with eigenvectors denoted by κ(n k ) , since for any matrices U and V with entries | · | ≤ 1,

The doubling map with any even n
Recall if S : [0, 1] → [0, 1] is the doubling map, then for any n ∈ 2N, the n × n Markov matrix P n along with a specific quantization U n can be taken as in (1.2). For general maps S, in Theorem 2.1, we restricted to dimensions n ∈ M 0 Z with K(n) → ∞. This ensured that enough powers of P n behaved nicely with the partitions (Lemma 3.2). For the doubling map, M 0 = 2 and we can take all n ∈ 2Z, not just those with the largest power of two dividing n tending to infinity. The statement is as follows (note that the quantization U n does not have to be the orthogonal one in (1.2)).
Theorem 7.1 (pointwise Weyl law analogue for the doubling map). For n ∈ 2N, let P n be the matrix in (1.2), and let U n satisfy |(U n ) xy | 2 = (P n ) xy . Denote the eigenvalues and eigenvectors of U n by (e iθ (n) j ) j and (ψ (n,j) ) j respectively for j ∈ [n]. Let (I(n)) be a sequence of intervals in R/(2πZ) satisfying Then there is a sequence of subsets G n ⊆ [n] with sizes #G n = n(1 − o(|I(n)|)) so that for all x ∈ G n , where the error term o(1) depends only on n, |I(n)|, and #G n , and is independent of x ∈ G n . Additionally, G n k can be chosen independent of I(n k ) or |I(n k )|.
The proof is the same as Theorem 2.1, except that Lemma 3.3, which bounds the number of nonzero entries on the diagonal of P n , is proved differently. To analyze the matrix powers P n , instead of viewing them in terms of S , we count paths of length in the directed graph associated with the Markov matrix P n . The proof of Theorem 7.1 then follows from the following lemma and by replacing all instances of K(n) + 1 by K := log 2 n in the proof of Theorem 2.1.
Lemma 7.2 (number of nonzero entries for the doubling map). For n ∈ 2N, let P n be as in (1.2) and let 1 ≤ ≤ K. Consider the directed graph with n nodes 1, 2, . . . , n, whose adjacency matrix is 2P n . Then: (i) For any coordinates x, y, there is at most one path of length from x to y in the graph.
(ii) The diagonal of P n contains at most 2 · 2 nonzero entries.
(iii) In total, P n has exactly n · 2 nonzero entries.
Proof. All possible paths starting at a node x and of length can be represented as paths in a binary tree of height with root node x. (Figure 4.) The nodes 1, 2, . . . , n of the graph may be listed multiple times in the binary tree. If we always put the descendant 2x − 1 on the left and put 2x on the right, then the list of nodes in each row of the tree will be consecutive increasing in Z/nZ. Thus if ≤ K := log 2 n , the th row of the tree will contain 2 ≤ n nodes, so that for any two nodes x and y, there is at most one path of length from x to y, proving part (i).  Applying part (i), the total number of nonzero entries on the diagonal of P n is the total number of paths of length ≤ K with the same start and end point x. Similarly, the total number of nonzero entries in P n is the total number of length paths from any x to any y. The collection of all paths of length can be represented by the paths in a forest of n binary trees each of depth , one tree for each possible starting node x ∈ [n]. (Figure 5.) The th row contains 2 · n numbers, showing part (iii). These 2 · n numbers at the bottom of the forest are 2 copies of the sequence (1, 2, . . . , n). To show (ii), we will show that for each copy C j of (1, 2, . . . , n), there can be at most two paths with the same start and end point that end in this copy.
Let F (j) be the set of starting nodes that have descendants in the jth copy C j of (1, 2, . . . , n). (The last node in F (j) may overlap with the first node in F (j + 1).) Consider just the paths that start in F (j) and end in C j , and suppose there is a length path x → x. We claim that only either x − 1 or (c) If R (x) = x, then only also x + 1 has a path x + 1 → x + 1. (a) x Thus in total there are at most 2 · 2 paths of length that start and end at the same point, proving part (ii).
8. The doubling map with n = 2 K When n = 2 K , the corresponding graphs from the doubling map are the de Bruijn graphs on two symbols. Orbits in these graphs have been studied in the context of quantum chaos in for example [44,31,23,24]. In these dimensions, the particular matrices U n from (1.2), despite coming from the doubling map, exhibit some behavior like that of integrable systems. Any choice of eigenbasis still satisfies the quantum ergodic theorem since the doubling map is ergodic, but the eigenvalues of U n in these dimensions are degenerate and evenly spaced in the unit circle. As a result of the degeneracy, we will be able to show that random eigenbases look approximately Gaussian. This will follow from properties of the spectral projection matrix of an eigenspace combined with the results on random projections used in Section 6.
We start by showing the eigenvalues of U n are 4Kth roots of 1 if K is even, and 4Kth roots of −1 if K is odd. Proof. Part (b) follows from (a) and unitarity (orthogonality) of U n . For part (a), view the doubling map on [0, 1] as the left bit shift on a sequence {0, 1} N corresponding to the binary expansion of x ∈ [0, 1]. If we partition [0, 1] into 2 K atoms E i = ( i 2 K , i+1 2 K ), i = 0, . . . , 2 K − 1, then we can identify atom E i with the length K bit string corresponding to the binary expansion of i 2 K . Then z ∈ E i iff the first K digits of its binary expansion match the length K bit string for E i . The Markov matrix P n then takes an atom indexed by i = (i 1 , . . . , i K ) and sends it to the atoms indexed by (i 2 , . . . , i K−1 , 0) and (i 2 , . . . , i K−1 , 1), the result of the left bit shift. Thus for 1 ≤ ≤ K, there is at most one length path from i = (i 1 , . . . , i K ) to j = (j 1 , . . . , j K ), which is described by the sequence (i 1 , . . . , i , j 1 , . . . , j K ) and requires i +1 , i +2 , · · · , i +(K− ) = j 1 , j 2 , . . . , j K− . Note this recovers Lemma 3.1(d).
Now considering the signs in U 2 K and viewing the indices i, j as length K bit strings, if there is an edge i → j, then Thus if there is a length K path τ : i → j, then since (τ m ) 1 = i m and (τ m ) K = j m . This is the structure of a tensor product, Remark 8.1. Proposition 8.1 can also be proved (although with significantly more effort) by analyzing paths in the corresponding de Bruijn graph. As in Section 7, possible paths can be described using trees, but the edges in the trees carry a sign to keep track of the negative signs in the matrix U 2 K .
Since the eigenvalues of U 2 K are 4K-th roots of 1 or −1, instead of eigenvectors from eigenvalues in an interval I(n) ⊆ R/(2πZ) like in Theorem 2.1, we are just interested in all the eigenvectors from a single eigenspace. A stronger version of Theorem 2.1 for this specific case controls the spectral projection onto a single eigenspace.
Theorem 8.2 (Eigenspace projection when n = 2 K ). For n = 2 K , let U n be as in (1.2), and let P (n,j) be the projection onto its jth eigenspace. Let r(K) : N → N be any function satisfying r(K) < K, r(K) → ∞, and K − r(K) − log 2 K → ∞ as K → ∞. Then there are sets G K ⊆ [1 : , (8.5) such that the following hold as K → ∞.
(a) For x ∈ G K and any j, For pairs (x, y) ∈ GP K and any j, | e y , P (n,j) e x | ≤ 10 · 2 −r(K)/2 4K . (8.7) Using (8.6) and summing over all x, we also obtain: (1)). Returning to eigenvectors, Theorem 8.2(a) applied to Corollary 6.6 shows that taking a random basis within each eigenspace produces approximately Gaussian eigenvectors. Theorem 8.4 (Gaussian eigenvectors when n = 2 K ). For K ∈ N, let (ψ (2 K ,j) ) 2 K j=1 be a random ONB of eigenvectors chosen according to Haar measure in each eigenspace of U n . Let µ (2 K ,j) := be the empirical distribution of the scaled coordinates of ψ (2 K ,j) . Then there is ε(K) → 0 so that for any bounded Lipschitz f : C → C with L ≤ 1, as K → ∞, where Z ∼ N C (0, 1). In particular, each µ (2 K ,j) converges weakly in probability to N C (0, 1) as K → ∞.
The rest of this section is the proof of Theorem 8.2.
8.1. Polynomial for eigenspace projection. Instead of using trigonometric polynomials to approximate the spectral projection matrix like in the proof of Theorem 2.1, we use a polynomial with zeros at 4K-th roots of 1 or −1 to get exact formulas. Let U n be as in (1.2) with n = 2 K . First consider K even, so U 4K n = I and the eigenvalues of U n are 4K-th roots of unity. Since x 4K −1 x−1 = 1 + x + x 2 + · · · + x 4K−1 is zero at all 4K-th roots of unity except for x = 1, the polynomial is zero at all 4K-th roots of unity except for e 2πij/(4K) , where it takes the value 4K. Writing the spectral projection onto the eigenspace of e 2πij/(4K) is If K is odd, then U 4K . We will only need to know where the entries of U m n are nonzero, which follows from matrix multiplication: Lemma 8.5 (Powers up to K). Let n = 2 K . For m ≤ K, let A m be the set of real n × n matrices A such that A m consists of matrices whose nonzero entries are ±1 arranged in 2 m descending "staircases" with steps of length 2 m . Then for m ≤ K − 1 and A ∈ A m , A · √ 2U n ∈ A m+1 .
In particular, since 2 1/2 U n ∈ A 1 , then for m ≤ K, 2 m/2 U m n ∈ A m . Between 2K and 3K, U m n has a flipped staircase structure: Corollary 8.6 (Powers from 2K to 3K). Let n = 2 K . For m ≤ K, let B m be the set of n × n matrices B such that the matrix A defined by A ij := B (n−i)j is in A m . Then for m ≤ K − 1 and B ∈ B m , B · √ 2U n ∈ B m+1 .
Thus for I(n) = [−π/2, π/2], the sequence of coordinates (x n ) n with just x n = 1 does not satisfy the pointwise Weyl law. Note the coordinate x = 1 was one of the "bad" points that was removed during the proof of the pointwise Weyl law, since it always has the very short periodic loop consisting of just itself.
To approximate (P I(n) ) 11 from below, we use the piecewise linear approximation h ∆ on R/(2πZ) in Figure 8, defined by Since this I(n) = [−π/2, π/2] is not shrinking, we do not need further smoothness of the approximation, and the piecewise linear h ∆ has Fourier coefficients that are easy to work with. We only need continuity and absolutely summable Fourier coefficients, so that the Fourier series converges uniformly to h ∆ . For convenience we also use the same notation h ∆ or χ [−π/2,π/2] to denote the corresponding function on the unit circle in C (via R/2πZ t ↔ e it ∈ S 1 ). Since pointwise h ∆ (t) ≤ χ [−π/2,π/2] (t) for any ∆ ≥ 0, then by the spectral theorem (h ∆ (U n )) xx ≤ χ [−π/2,π/2] (U n ) xx = (P I(n) ) xx for any coordinate x ∈ [n].