On the spacing of zeros of paraorthogonal polynomials for singular measures

Eyal Seelig; Jonathan Breuer

On the spacing of zeros of paraorthogonal polynomials for singular measures

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Pith's one-line read This paper proves that the Hausdorff dimension of a measure on the unit circle controls how fast the zeros of its paraorthogonal polynomials repel each other, and that purely singular continuous measures can still exhibit exact clock…

desk verdict Solid OPUC transfer of OPRL zero-spacing results; the claimed sign error is a misreading, but two minor typos need fixing. read the letter →

arxiv 1908.06737 v2 pith:YCMJ4I5R submitted 2019-08-19 math.SP math.CA

Jonathan Breuer , Eyal Seelig This is my paper · ORCID

classification math.SPmath.CA MSC 42C0528A7847B36

keywords paraorthogonalpolynomialsunitcircleHausdorffdimensionzerospacingclockbehaviorChristoffel-DarbouxkernelVerblunskycoefficientstransfermatrices

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a link between how continuous a measure on the unit circle is, measured by Hausdorff dimension, and how strongly the zeros of the associated paraorthogonal polynomials repel each other. The main theorem says that if the measure gives zero weight to all sets of Hausdorff dimension at most α, then for almost every point on the circle the rescaled gap between the two zeros flanking that point, multiplied by $n^{{2/α−1}}$, tends to infinity. A second theorem shows the boundary is subtle: there exist purely singular continuous measures for which every choice of paraorthogonal polynomials still has exact clock spacing, with adjacent zeros separated by 2π/n. If true, these results make the smoothness of the measure a quantitative driver of zero repulsion, while showing that singularity alone cannot force irregular spacing.

What carries the argument

The load-bearing identity is the transfer-matrix gap lower bound (Theorem 2.3): for two consecutive zeros z'=$e^{{iθ_{-1}}$^{(n)}} and z''=$e^{{iθ_0^{(n)}}$} flanking z=$e^{{iΘ}}$, |$θ_0^{{(n)}}$−θ_{-1}^{(n)}| ≥ (Σ_{k=0}^{n−1}||T_k(z)||²)^{-1}, where T_k(z) are the 2×2 transfer matrices from the Szegő recurrence. This converts the question of zero spacing into a question about growth of transfer-matrix norms, to which subordinacy theory applies. For the construction, the central object is a sequence of Verblunsky coefficients that is sparse, with non-zero entries separated by rapidly growing gaps, and decaying to zero; the proof shows the Christoffel–Darboux kernel of the resulting measure converges to the sine kernel uniformly in the angle, and the appendix's unit-circle Freud–Levin–Lubinsky theorem converts that convergence into clock spacing.

What would settle it

Construct a measure μ that is supported on a self-similar Cantor set of Hausdorff dimension α and gives zero weight to all sets of dimension at most α, compute the zeros of the paraorthogonal polynomials for a fixed β, and check whether $n^{{2/α−1}}$($θ_0^{{(n)}}$(Θ)−θ_{-1}^{(n)}(Θ)) diverges to infinity for μ-almost every Θ; if a point of positive μ-measure has a sequence of n with finite liminf of this quantity, Theorem 1.1(2) fails.

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Extended reading notes

Core claim

The central discovery is a quantitative lower bound on local zero spacing for paraorthogonal polynomials on the unit circle, controlled by the Hausdorff dimension of the underlying measure. Concretely, for γ>1 the set of points where liminf n^γ($θ_0^{{(n)}}$(Θ)−θ_{-1}^{(n)}(Θ))<∞ has μ-measure supported on a set of Hausdorff dimension at most 2/(1+γ). Therefore if μ gives zero weight to sets of dimension at most α, then for μ-almost every Θ the gaps satisfy $n^{{2/α−1}}$($θ_0^{{(n)}}$(Θ)−θ_{-1}^{(n)}(Θ))→∞. The paper also constructs purely singular continuous measures whose Verblunsky coefficients are sparse and decaying, and proves that their Christoffel–Darboux kernels have sine-kernel asymptotics uniformly in the angle, which forces n(θ_{j+1}^{(n)}(Θ)−$θ_j^{{(n)}}$(Θ))→2π for every Θ and every j.

Load-bearing premise

The dimension part of the main theorem assumes that two previously established results about how quickly certain recurrence solutions grow and what that growth implies about set size apply to every probability measure on the unit circle, without the paper restating their hypotheses.

Editorial extensions

If this is right

For any measure giving zero weight to sets of Hausdorff dimension at most α, the rescaled gap n^{2/α−1}(θ_0^{(n)}(Θ)−θ_{-1}^{(n)}(Θ)) tends to infinity for μ-almost every Θ.
For the absolutely continuous part of any measure, the weaker bound limsup n(θ_0^{(n)}(Θ)−θ_{-1}^{(n)}(Θ)) > 0 holds at almost every point.
There exist purely singular continuous measures on the circle such that, for every choice of paraorthogonal polynomials, n(θ_{j+1}^{(n)}(Θ)−θ_j^{(n)}(Θ))→2π at every point Θ.
Singularity of the underlying measure therefore does not rule out maximal repulsion; the constructed measures satisfy the strongest possible clock behavior.
The appendix supplies a unit-circle version of the Freud–Levin–Lubinsky principle: sine-kernel asymptotics of the Christoffel–Darboux kernel imply clock spacing for every corresponding paraorthogonal sequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the dimension bound 2/(1+γ) is sharp; a natural test is to construct a measure whose support has Hausdorff dimension exactly 2/(1+γ) and check whether the liminf of the rescaled gap is finite on a positive-measure set.
The sparse-Verblunsky construction suggests a quantitative trade-off: if the gaps between nonzero coefficients grow only polynomially instead of rapidly, one might expect a slower approach to sine-kernel asymptotics, and computing that rate would give a finer picture of the singularity–repulsion boundary.
The remark that the real-line example is absolutely continuous with respect to every h_α with α<1 invites the same question for the circle measures; if true, these singular continuous measures are as continuous as possible while still singular, reinforcing the message that pure singularity does not constrain spacing.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read. The paper is a clean transfer of the OPRL machinery to OPUC. The genuinely new pieces are Theorem 1.1(2) — the first Hausdorff-dimension-based lower bound on POPUC zero spacing — and the construction in Theorem 1.2/Theorem 3.1, which gives singular continuous measures on the circle with universal clock behavior. The proof of the Last–Simon type bound (Theorem 2.3) is new and is a nice simplification of the existing strategy; the sparse-Verblunsky construction also avoids the CD formula, which is a genuine simplification over [2].

I looked at the alleged sign error in the proof of Theorem 1.1(2). It does not hold up: the printed 'δ = 1/γ − ε' is ambiguous, but the displayed identity 2δ/(1+δ) = 2/(1+γ−ε) forces δ = 1/(γ−ε), which makes the exponent (1−δγ)/2 + η negative and the liminf ratio 0. So the subordinacy step goes through as written, modulo a typo in the typesetting of δ. Same for the Section 4.1 constant: the bound (3.2) gives |φ^{(ℓ)}|² ≤ ((1+|v|)/(1−|v|)) |φ^{(ℓ+1)}|², not the squared version displayed. That is a real typo, but harmless because the prefactor B_{ℓ+1} → 0 and the missing factor is bounded for large ℓ. Both should be corrected, but neither is load-bearing.

The reliance on external results is fair: [26, Thm 4.3.16] and Thms 10.8.5/10.8.7 are standard and apply to arbitrary probability measures on the circle; the growth bound also follows elementarily from the CD kernel trace, so I don't worry about hidden hypotheses. The citation pattern is honest, including the debt to [2] and the real-line results.

Minor quibbles: the 'sufficiently sparse' condition in Theorem 3.1 is deliberately vague, though the recursive construction makes it clear; the remark about the unit-circle analog of Zlatoš' result is explicitly unsupported, which is fine. The paper does what it claims.

Bottom line: this deserves a serious referee. I'd recommend acceptance after minor revisions. It will be cited by people working on zero spacing of paraorthogonal polynomials and on universality for singular measures.

Referee Report

0 major / 6 minor

Summary. The paper proves two theorems about the spacing of zeros of paraorthogonal polynomials on the unit circle. Theorem 1.1 shows that if a probability measure mu gives zero weight to sets of Hausdorff dimension at most alpha, then for mu-almost every Theta the rescaled zero gap n^gamma (theta_0^(n)(Theta)-theta_{-1}^(n)(Theta)) tends to infinity for gamma = 2/alpha - 1; more generally, the set where liminf of this gap is finite is supported on a set of Hausdorff dimension at most 2/(1+gamma). Theorem 1.2 constructs purely singular continuous measures for which clock spacing n(theta_{j+1}^(n)(Theta)-theta_j^(n)(Theta)) tends to 2 pi at every point Theta, for every sequence of paraorthogonal polynomials. The proofs combine a transfer-matrix lower bound on gaps (Theorem 2.3), OPUC subordinacy theory, and a sparse Verblunsky-coefficient construction for which the Christoffel-Darboux kernel has sine-kernel asymptotics.

Significance. If the results are correct, they give a quantitative connection between measure continuity in the Hausdorff-dimension sense and zero repulsion for POPUC, which appears to be new, and they show that pure singularity of the measure does not preclude maximal repulsion. The transfer-matrix bound in Theorem 2.3 is a clean and useful OPUC analog of the Last-Simon bound, and the appendix provides a missing OPUC version of the Freud-Levin-Lubinsky theorem. The proofs are structured, use standard tools, and the main external inputs are cited precisely. I found no load-bearing error; the concern about a sign error in the choice of delta does not survive a careful reading of the formula as delta = 1/(gamma-epsilon).

minor comments (6)

[Section 2] Please typeset the parameter in the proof of Theorem 1.1(2) as delta = 1/(gamma-epsilon). If a reader parses it as (1/gamma)-epsilon, the exponent (1-delta gamma)/2 is positive and the displayed liminf would not be zero; the dimension computation 2 delta/(1+delta) = 2/(1+gamma-epsilon) confirms the intended reading, but the ambiguity should be removed.
[Lemma 2.1] In the display after (2.9), the vectors in the sum should be w'_{m-1} and w'_m (with primes), matching the statement in (2.7); the current display appears to omit the primes.
[Section 4.1] The factor obtained from (3.2) should be (1+|v_{\ell+1}|)/(1-|v_{\ell+1}|), not (1+|v_{\ell+1}|^2)/(1-|v_{\ell+1}|^2). The error does not affect convergence because both factors tend to 1 as v_{\ell+1}->0, but the displayed inequality is incorrect as written.
[Section 4] To conclude Theorem 1.2 from [26, Theorem 12.5.2], the recursive sparseness condition should explicitly include N_{\ell+1}/N_\ell -> infinity, and the sequence v_\ell should be chosen with sum |v_\ell|^2 = infinity. These are compatible with the construction but should be stated.
[Section 2] The proof of Theorem 1.1(2) relies on [26, Theorem 4.3.16] and [26, Theorems 10.8.5 and 10.8.7]; please state the hypotheses of these theorems or indicate explicitly that they apply to arbitrary probability measures on the unit circle, so that the application to purely singular continuous measures is transparent.
[Appendix] The name 'Hurowitz' in the proof of Theorem 5.1 should be 'Hurwitz'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central claims are proven in-paper from stated lemmas and independent external results.

full rationale

The derivation chain is self-contained. Theorem 1.1 rests on Theorem 2.3, which is proved in the paper from Lemmas 2.1 and 2.2 together with Wong's factorization (2.1); the subsequent Hausdorff-dimension step invokes external subordinacy results from Simon's OPUC monograph and the a.e. growth bound, not any conclusion of the present paper. Theorem 1.2 is not imported from [2]: the authors construct sparse Verblunsky coefficients explicitly and prove the sine-kernel asymptotics in Theorem 3.1 directly, using [26, Theorem 12.5.2] for pure singular continuity and the appendix's Theorem 5.1 to pass from sine kernel asymptotics to clock spacing. The self-citation [2] is used only as a strategic model ('we imitate the strategy and technique of [2]') and is not load-bearing for either theorem. There is no fitted parameter renamed as a prediction and no quantity defined in terms of the target claim. One non-circular concern: the printed proof of Theorem 1.1(2) contains an apparent sign error in the choice of delta = 1/gamma - epsilon, since the displayed exponent (1 - delta*gamma)/2 + eta = epsilon*gamma/2 + eta is positive and the claimed convergence to zero does not follow as written; this is a correctness/typo issue, not a circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's original contribution is the transfer-matrix estimate in Theorem 2.3 and the sparse-perturbation construction in Theorem 3.1; the surrounding OPUC theory is imported from Simon's monograph and cited papers. No new physical or mathematical entities are posited, and no numbers are fitted to data.

assumptions (7)

domain assumption Szego recurrence (1.1) and the bijection between measures on the unit circle and Verblunsky coefficients {alpha_n}.
Used throughout the paper, imported from Simon's OPUC monograph [26].
domain assumption Wong's identity (2.1): H_n^(beta_{n-1})(z) = c (z - z_0) K_n(z, z_0) for each zero z_0 of the paraorthogonal polynomial.
Connects zeros of POPUC to zeros of the Christoffel-Darboux kernel; cited to [28].
domain assumption Essential support characterization for the absolutely continuous part: N_1 = {z : liminf (1/n) sum_{k=0}^{n-1} ||T_k(z)||^2 < infinity}.
Used in the proof of Theorem 1.1(1), cited to [26, Theorem 10.9.4].
domain assumption OPUC growth bound: for mu-almost every z and any eta > 0, ||phi dot(z)||_n <= C_eta n^(1/2+eta).
Load-bearing in Theorem 1.1(2); the paper does not restate hypotheses, cited to [26, Theorem 4.3.16].
domain assumption Subordinacy dimension bound: liminf ||phi||_n / ||psi||_n^delta = 0 implies mu is supported on a set of Hausdorff dimension at most 2 delta / (1 + delta).
Load-bearing for the dimension conclusion, cited to [26, Theorems 10.8.5 and 10.8.7].
domain assumption If v_l tends to 0, N_{l+1}/N_l tends to infinity, and sum |v_l|^2 diverges, then the associated OPUC measure is purely singular continuous.
Imported from [26, Theorem 12.5.2] to guarantee singularity in Theorem 1.2.
standard math Standard Hausdorff measure and dimension properties, including h_0 as counting measure and h_1 as Lebesgue measure.
Definitions in (1.3) and standard facts from Rogers' monograph [19].

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Pith. "Pith review of On the spacing of zeros of paraorthogonal polynomials for singular measures." pith.science (2026). https://pith.science/paper/YCMJ4I5R

@misc{pith2026190806737,
  author       = {Pith},
  title        = {Pith review of: On the spacing of zeros of paraorthogonal polynomials for singular measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCMJ4I5R}},
  note         = {Machine review of arXiv:1908.06737}
}

read the original abstract

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Breuer, Sine kernel asymptotics for a class of singular measures , J

J. Breuer, Sine kernel asymptotics for a class of singular measures , J. Approx. Theory 163 (2011), 1478–1491

work page 2011

[2] [1]

Avila, Y

A. Avila, Y. Last and B. Simon, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum , Analysis and PDE 3 (2010), 81–108

work page 2010

[3] [3]

Breuer and D

J. Breuer and D. Weissman, Level repulsion for Schr¨ odinger operators with singular continuous spectrum , J. Spect. Theory 9 (2019), 429–451

work page 2019

[4] [4]

M. J. Cantero, L. Moral and L. Vel´ azquez,Measures on the unit circle and unitary truncations of unitary operators , J. Approx. Theory 139 (2006), 430–468. 27

work page 2006

[5] [5]

Castillo, F

K. Castillo, F. Marcell´ an and M N. Rebocho, Zeros of paraorthogonal polynomials and linear spectral transformations on the uni t circle, Numer. Algorithms 71 (2016), 699–714

work page 2016

[6] [6]

Freud, Orthogonal Polynomials , Pergamon Press, Oxford-New York, 1971

G. Freud, Orthogonal Polynomials , Pergamon Press, Oxford-New York, 1971

work page 1971

[7] [7]

Golinskii, Quadrature formula and zeros of para-orthogonal polynomi- als on the unit circle Acta Math

L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomi- als on the unit circle Acta Math. Hungar. 96 (2002), 169–186

work page 2002

[8] [8]

Jitomirskaya and Y

S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra, I. Half-line operators , Acta Math. 183 (1999), 171–189

work page 1999