Large Deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-R\"ussmann frequency

Kai Tao; Wenmeng Geng

arxiv: 1906.11136 · v1 · pith:TYT7ESHZnew · submitted 2019-06-26 · 🧮 math.DS · math-ph· math.MP· math.SP

Large Deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-R\"ussmann frequency

Wenmeng Geng , Kai Tao This is my paper

Pith reviewed 2026-05-25 15:08 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MPmath.SP

keywords large deviation theoremsDirichlet determinantsquasi-periodic Jacobi operatorsBrjuno-Rüssmann frequencyBirkhoff ergodic theoremsubharmonic functionseigenvalue distribution

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The pith

The Brjuno-Rüssmann function, through its smallest deviation, controls large deviation estimates for Dirichlet determinants of analytic quasi-periodic Jacobi operators.

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

The paper first proves a strong Birkhoff ergodic theorem for subharmonic functions under shifts generated by Brjuno-Rüssmann frequencies on the torus. It then transfers this theorem to obtain large deviation bounds on the finite-scale Dirichlet determinants of analytic quasi-periodic Jacobi operators whose frequency lies in the same class. The bounds are expressed in terms of the smallest deviation, a quantity that directly encodes the degree of irrationality of the frequency. As a direct consequence the paper derives a limiting distribution for the eigenvalues of the associated finite Jacobi matrices with Dirichlet boundary conditions. A reader would care because these quantitative controls on deviations are prerequisites for statements about localization and spectral measures in quasi-periodic Schrödinger-type operators.

Core claim

The Brjuno-Rüssmann function that reflects the irrationality of the frequency plays the key role in the large deviation theorems for the finite-scale Dirichlet determinants of analytic quasi-periodic Jacobi operators via the smallest deviation; the same dependence appears in the resulting distribution of eigenvalues for the operators with Dirichlet boundary conditions.

What carries the argument

The Brjuno-Rüssmann function together with its associated smallest deviation, which quantifies irrationality and makes the strong Birkhoff ergodic theorem for subharmonic functions available.

If this is right

Large deviation theorems hold for the finite-scale Dirichlet determinants with explicit dependence on the smallest deviation.
The eigenvalue distribution for finite Jacobi matrices with Dirichlet boundaries likewise depends on the smallest deviation.
The results apply uniformly to all frequencies in the Brjuno-Rüssmann class.
The ergodic theorem for subharmonic functions is the sole new analytic ingredient required for the deviation estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same smallest-deviation control could be used to obtain quantitative estimates on the density of states or on the Lyapunov exponent once the determinants are related to transfer matrices.
Numerical checks of the deviation bounds become feasible for concrete Brjuno frequencies such as the golden mean, providing a direct test of the predicted scaling.
If analogous ergodic theorems exist for other Diophantine classes, the large-deviation statements would extend immediately to those frequencies.

Load-bearing premise

The frequency must lie in the Brjuno-Rüssmann class, which is needed for the strong Birkhoff ergodic theorem on subharmonic functions to hold and transfer to the determinants.

What would settle it

An explicit Brjuno-Rüssmann frequency and an explicit analytic Jacobi operator for which the observed deviation of a Dirichlet determinant from its mean exceeds the bound stated in terms of the smallest deviation.

read the original abstract

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-R\"ussmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-R\"ussmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends large-deviation estimates to analytic quasi-periodic Jacobi operators under the Brjuno-Rüssmann frequency condition by proving a strong Birkhoff theorem for subharmonic functions first.

read the letter

The new result is the large deviation theorem for the finite-scale Dirichlet determinants when the frequency is Brjuno-Rüssmann, together with the resulting eigenvalue distribution for the Dirichlet Jacobi matrices. The authors first establish the strong ergodic theorem for subharmonic functions under that shift on the torus, then transfer it to the log of the determinants, with the smallest deviation (controlled by the Brjuno-Rüssmann function) as the key quantity. This is a direct extension of existing techniques rather than a routine reapplication, and the abstract keeps the logical steps clear without circularity or hidden fitting parameters. The role of the frequency's arithmetic properties is stated explicitly and used consistently. The work is technically grounded in the sense that the hypothesis is named upfront and the chain from ergodic theorem to large deviations to eigenvalue counts follows the standard pattern in this subfield. The main limitation is that the actual proofs are not visible here, so any subtle issues in applying the subharmonic ergodic theorem to the Jacobi cocycle cannot be checked directly; nothing in the description indicates a load-bearing flaw, but the transfer step is where the technical risk sits. This is a paper for people already working on large deviations and spectral theory for quasi-periodic operators. A reader who follows the Brjuno-Rüssmann or related arithmetic conditions will get a concrete refinement. It deserves a serious referee because the claims are stated cleanly, the assumptions are explicit, and the result adds a new case to the literature even if the scope stays within specialists.

Referee Report

0 major / 1 minor

Summary. The paper first establishes a strong Birkhoff ergodic theorem for subharmonic functions under the Brjuno-Rüssmann shift on the torus. It then applies this theorem to derive large deviation estimates for the logarithms of finite-scale Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency. The estimates are controlled by the Brjuno-Rüssmann function via the smallest deviation. As an application, the paper obtains results on the distribution of eigenvalues of the associated Jacobi operators with Dirichlet boundary conditions.

Significance. If the central claims hold, the work extends large-deviation control for quasi-periodic Jacobi operators beyond the usual Diophantine regime to the broader Brjuno-Rüssmann class, with explicit dependence on the arithmetic properties of the frequency. This could be useful for quantitative spectral theory of analytic cocycles.

minor comments (1)

The abstract refers to 'the smallest deviation' without a precise definition or reference to its relation to the Brjuno-Rüssmann function; a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, which accurately reflects the main contributions: a strong Birkhoff ergodic theorem for subharmonic functions under the Brjuno-Rüssmann shift, followed by large-deviation estimates for finite-scale Dirichlet determinants of analytic quasi-periodic Jacobi operators, with explicit control via the smallest deviation, and the resulting eigenvalue distribution results. The recommendation is listed as uncertain, but the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper first proves a strong Birkhoff ergodic theorem for subharmonic functions under the Brjuno-Rüssmann shift, then applies the result to obtain large deviation estimates for the Dirichlet determinants. This is a standard sequential derivation with no reduction of the target theorems to fitted parameters, self-definitions, or load-bearing self-citations; the Brjuno-Rüssmann condition is stated explicitly as the hypothesis enabling the ergodic theorem, and the smallest deviation appears as a derived controlling quantity rather than an input renamed as output.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results in ergodic theory and analytic function theory together with the new ergodic theorem proved in the paper; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Strong Birkhoff ergodic theorem holds for subharmonic functions under Brjuno-Rüssmann shifts on the torus
This is the first result established and then applied to the Jacobi operators.

pith-pipeline@v0.9.0 · 5652 in / 1161 out tokens · 25625 ms · 2026-05-25T15:08:46.930851+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmann shift on the Torus... the Brjuno-Rüssmann function... plays the key role... via the smallest deviation δ_n^0
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4... δ_n^0 = ... log Δ(n) / (Δ^{-1}(C_ω n))^{1-} ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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Avila and S

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A.Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies. In preparation

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Avila, J

A. Avila, J. You and Z. Zhou, Sharp Phase transitions for the almost Mathieu operator. Duke\ Math.\ J. 166 (2017), 2697-2718

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J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J.\ Statist.\ Phys., 108 (2002), 1203--1218

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Ann.\ of Math

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R.Han, Dry Ten Martini problem for the non-self-dual extended Harper's model. Trans.\ Amer.\ Math.\ Soc.\ 370 (2018), 197-217

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R.Han and S.Zhang, Optimal Large Deviation Estimates and H\"older Regularity of the Lyapunov Exponents for Quasi-periodic Schr\"odinger Cocycles. arXiv:1803.02035v1

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Jitomirskaya, D.A

S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles. Ergodic\ Theory\ Dynam.\ Systems, 29 (2009),1881-1905

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Jitomirskaya, D.A

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older continuity of the Lyapunov exponent for analytic quasiperiodic Schr\

J.You and S.Zhang, H\"older continuity of the Lyapunov exponent for analytic quasiperiodic Schr\"odinger cocycles with week Liouville frequency. Ergod.\ Th.\ Dynam.\ Sys.\ , 34 , (2014), 1395 - 1408

work page 2014

[1] [1]

Avila and S

A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann.\ Math.\ 170 (2009), 303-342

work page 2009

[2] [2]

Inv.\ Math.\ 210 (2017), 1-57

A.Avila, S.Jitomirskaya and C.A.Marx, Spectral theory of extended Harpers model and a question by Erd\"os and Szekeres. Inv.\ Math.\ 210 (2017), 1-57

work page 2017

[3] [3]

A.Avila, Y. Last, M. Shamis and Q. Zhou, On the abominable properties of the Almost Mathieu operator with well approximated frequencies. In preparation

work page

[4] [4]

Avila, J

A. Avila, J. You and Z. Zhou, Sharp Phase transitions for the almost Mathieu operator. Duke\ Math.\ J. 166 (2017), 2697-2718

work page 2017

[5] [5]

Bourgain and S

J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J.\ Statist.\ Phys., 108 (2002), 1203--1218

work page 2002

[6] [6]

Ann.\ of Math

J.Bourgain and M.Goldstein, On nonperturbative localization with quasi-periodic potential. Ann.\ of Math. 152 (2000), no. 3, 835-879

work page 2000

[7] [7]

Schlag, Anderson localization for Schr\"odinger operators on Z ^2 with potentials given by the skew-shift

J.Bourgain, M Goldstein and W. Schlag, Anderson localization for Schr\"odinger operators on Z ^2 with potentials given by the skew-shift. Comm.\ Math.\ Phys.\ 220 (2001), no. 3, 583--621

work page 2001

[8] [8]

I.Binder and M.Voda, An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size n contained in an interval of size n^ -C . J. Spec. Theory, 3 (2013),1-45,2013

work page 2013

[9] [9]

I.Binder and M.Voda, On optimal separation of eigenvalues for a quasiperiodic Jacobi matrix. Comm. Math. Phys., 325 ,(2014),1063-1106,2014

work page 2014

[10] [10]

Goldstein, D

M. Goldstein, D. Damanik, W. Schlag, and M. Voda, Homogeneity of the spectrum for quasi-perioidic Schr\"odinger operators. J.\ Eur.\ Math.\ Soc.\ , 20 (2018), 3073-3111

work page 2018

[11] [11]

older continuity of the integrated density of states for quasiperiodic Schr\

M.Goldstein and W.Schlag, H\"older continuity of the integrated density of states for quasiperiodic Schr\"odinger equations and averages of shifts of subharmonic functions. Ann.\ of Math. 2 154 (2001), no. 1, 155--203

work page 2001

[12] [12]

Goldstein and W.Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues

W. Goldstein and W.Schlag, Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues. Geom.\ Funct.\ Analysis.\ 18 , (2008), 755-869

work page 2008

[13] [13]

Michael Goldstein and Wilhelm Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic S chr\"odinger equations. Ann. of Math. (2), 173(1):337--475, 2011

work page 2011

[14] [14]

Trans.\ Amer.\ Math.\ Soc.\ 370 (2018), 197-217

R.Han, Dry Ten Martini problem for the non-self-dual extended Harper's model. Trans.\ Amer.\ Math.\ Soc.\ 370 (2018), 197-217

work page 2018

[15] [15]

older Regularity of the Lyapunov Exponents for Quasi-periodic Schr\

R.Han and S.Zhang, Optimal Large Deviation Estimates and H\"older Regularity of the Lyapunov Exponents for Quasi-periodic Schr\"odinger Cocycles. arXiv:1803.02035v1

work page arXiv

[16] [16]

Jitomirskaya, D.A

S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles. Ergodic\ Theory\ Dynam.\ Systems, 29 (2009),1881-1905

work page 2009

[17] [17]

Jitomirskaya, D.A

S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Localization for a family of one-dimensional quasiperiodic operators of magnetic origin. Ann.\ Henri.\ Poincar.\ 6 (2005),103-125

work page 2005

[18] [18]

Journal of Fixed Point Theory and Applications.\ 10 , (2011),129-146

S.Jitomirskaya and C.A.Marx, Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities. Journal of Fixed Point Theory and Applications.\ 10 , (2011),129-146

work page 2011

[19] [19]

Levin, Lectures on entire functions

Ya.B. Levin, Lectures on entire functions. Transl.\ of Math.\ Monographs, vol. 150. AMS, Providence, RI, 1996

work page 1996

[20] [20]

Tao, H\"older continuity of Lyapunov exponent for quasi-periodic Jacobi operators

K. Tao, H\"older continuity of Lyapunov exponent for quasi-periodic Jacobi operators. Bulletin de la SMF , 142 , (2014), 635-671

work page 2014

[21] [21]

Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

K. Tao, Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles. http:arxiv.org/abs/1805.00431

work page internal anchor Pith review Pith/arXiv arXiv

[22] [22]

older continuity of the Lyapunov exponent for analytic quasiperiodic Schr\

J.You and S.Zhang, H\"older continuity of the Lyapunov exponent for analytic quasiperiodic Schr\"odinger cocycles with week Liouville frequency. Ergod.\ Th.\ Dynam.\ Sys.\ , 34 , (2014), 1395 - 1408

work page 2014