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Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindelöf hypothesis"

Kasun Fernando, Tanja I. Schindler

TL;DR

The paper develops a framework to prove central limit, Edgeworth, and mixing local central limit theorems for Birkhoff sums of unbounded, oscillatory observables over full-branch piecewise C^2 expanding maps. It introduces a chain of damped Banach spaces ${f V}_{eta}$ on which twisted transfer operators act with a Doeblin–Fortet–Lasota–Yorke-type spectral gap, enabling precise probabilistic limit results via Keller–Liverani perturbation theory. The authors apply the theory to the Boolean-type transformation and to sampling the Lindelöf hypothesis, obtaining CLT/MLCLT–type results for observables including Re ζ, Im ζ, and |ζ| along the associated dynamics, with explicit conditions on observables and parameter regimes. This work links ergodic theory, spectral analysis, and analytic number theory, providing quantitative refinements (Edgeworth expansions) beyond the classical SLLN in a setting relevant to number-theoretic sampling problems. The results offer a rigorous pathway to understand how large, oscillatory fluctuations of arithmetic functions behave under deterministic chaotic dynamics and have potential implications for conjectures like Lindelöf via dynamical sampling schemes.

Abstract

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb{R}$. The class of observables in the CLT and the MLCLT on $\mathbb{R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for "Sampling the Lindelöf hypothesis".

Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindelöf hypothesis"

TL;DR

The paper develops a framework to prove central limit, Edgeworth, and mixing local central limit theorems for Birkhoff sums of unbounded, oscillatory observables over full-branch piecewise C^2 expanding maps. It introduces a chain of damped Banach spaces on which twisted transfer operators act with a Doeblin–Fortet–Lasota–Yorke-type spectral gap, enabling precise probabilistic limit results via Keller–Liverani perturbation theory. The authors apply the theory to the Boolean-type transformation and to sampling the Lindelöf hypothesis, obtaining CLT/MLCLT–type results for observables including Re ζ, Im ζ, and |ζ| along the associated dynamics, with explicit conditions on observables and parameter regimes. This work links ergodic theory, spectral analysis, and analytic number theory, providing quantitative refinements (Edgeworth expansions) beyond the classical SLLN in a setting relevant to number-theoretic sampling problems. The results offer a rigorous pathway to understand how large, oscillatory fluctuations of arithmetic functions behave under deterministic chaotic dynamics and have potential implications for conjectures like Lindelöf via dynamical sampling schemes.

Abstract

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on . The class of observables in the CLT and the MLCLT on include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for "Sampling the Lindelöf hypothesis".
Paper Structure (22 sections, 32 theorems, 268 equations)

This paper contains 22 sections, 32 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. The classes of dynamical systems
  5. The Banach spaces
  6. Results for the unit interval
  7. The application to the Boolean-type transformation
  8. Sampling the Lindelöf Hypothesis
  9. Review of Abstract Results for Limit Theorems
  10. Twisted Transfer Operators hat psi is
  11. Properties of twisted transfer operators
  12. DFLY Inequalities
  13. Proofs of the Main Theorems
  14. Proofs of limit theorems for expanding interval maps
  15. Proofs of limit theorems for the Boolean-type transformation
  16. ...and 7 more sections

Key Result

Theorem 2.5

Suppose $\chi$ is continuous and the right and left derivatives of $\chi$ exist on $\mathring{I}$, $\chi$ is not a coboundary and there exist constants $a,b>0$ such that Assume Then, the following Central Limit Theorem holds:

Theorems & Definitions (86)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Example 2.10
  • ...and 76 more