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Size of exceptional sets in weakly mixing systems

Jiyun Park, Kangrae Park

TL;DR

This work develops a quantitative theory of exceptional sets in weakly mixing systems by linking the size of J_{A,B} to the rate of decorrelation via Cesàro bounds, establishing general o(N b_N) control when the rate is o(b_N). It introduces a return-time framework with distributions D_l and recursions, enabling universal constructions of exceptional sets for broad classes of systems, notably restrictive tight maps and the Chacon transformation, with near-optimal bounds |J∩[0,N]|≤(log N)^{h(N)} and sharp lower bounds of (log N)^t. The results specialize to Chacon-type rank-one maps and generalize to flows and Z^d-actions, yielding concrete exceptional-set estimates for interval exchange transformations and substitution tilings. The findings substantially sharpen prior weak-mixing bounds, providing explicit construction methods, optimality results, and a unified approach across multiple weakly mixing models, while raising open questions about extending to broader tight-map families and improving bounds in specific settings.

Abstract

For any weakly mixing system $(X,\mathscr{B},μ,T)$ and any $A,B\in\mathscr{B}$, it is well known that there exists a density-zero exceptional set $J_{A,B}\subseteq\mathbb{N}$ along which $μ(A\cap T^{-n}B)\toμ(A)μ(B)$ for $n\notin J_{A,B}$. In this paper, we investigate finer quantitative properties of the exceptional set. First, assuming a given rate of weak mixing, we derive an explicit upper bound on $|J_{A,B}\cap[0,n]|$ in terms of that rate. We apply this result to interval exchange transformations and to a class of substitution dynamical systems. Next, we analyze a broad family of cutting and stacking transformations (of which the Chacon map is a special case): we construct a universal exceptional set $J$ and show that for any increasing function $h\colon\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ diverging to infinity, one can choose $J\subseteq\mathbb{N}$ satisfying $|J\cap[0,n]|\le(\log n)^{h(n)}$ for all $n$, uniformly over all Lebesgue-measurable $A,B\subseteq[0,1]$. We prove that this is optimal, in the sense that for any $t>0$, there exist measurable sets $A,B\subseteq[0,1]$ and a constant $N>0$ such that $|J_{A,B}\cap[0,n]|\ge(\log n)^{t}$ for all $n>N$.

Size of exceptional sets in weakly mixing systems

TL;DR

This work develops a quantitative theory of exceptional sets in weakly mixing systems by linking the size of J_{A,B} to the rate of decorrelation via Cesàro bounds, establishing general o(N b_N) control when the rate is o(b_N). It introduces a return-time framework with distributions D_l and recursions, enabling universal constructions of exceptional sets for broad classes of systems, notably restrictive tight maps and the Chacon transformation, with near-optimal bounds |J∩[0,N]|≤(log N)^{h(N)} and sharp lower bounds of (log N)^t. The results specialize to Chacon-type rank-one maps and generalize to flows and Z^d-actions, yielding concrete exceptional-set estimates for interval exchange transformations and substitution tilings. The findings substantially sharpen prior weak-mixing bounds, providing explicit construction methods, optimality results, and a unified approach across multiple weakly mixing models, while raising open questions about extending to broader tight-map families and improving bounds in specific settings.

Abstract

For any weakly mixing system and any , it is well known that there exists a density-zero exceptional set along which for . In this paper, we investigate finer quantitative properties of the exceptional set. First, assuming a given rate of weak mixing, we derive an explicit upper bound on in terms of that rate. We apply this result to interval exchange transformations and to a class of substitution dynamical systems. Next, we analyze a broad family of cutting and stacking transformations (of which the Chacon map is a special case): we construct a universal exceptional set and show that for any increasing function diverging to infinity, one can choose satisfying for all , uniformly over all Lebesgue-measurable . We prove that this is optimal, in the sense that for any , there exist measurable sets and a constant such that for all .
Paper Structure (23 sections, 54 theorems, 194 equations, 3 figures)

This paper contains 23 sections, 54 theorems, 194 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Main Ideas and Outline of Paper
  5. Definitions and Notation
  6. Weak Mixing and Exceptional Sets
  7. Proof of Proposition \ref{['prop:weak-conv-rate']}
  8. Constructing Exceptional Sets
  9. The Chacon Transformation
  10. The Chacon Transformation
  11. The Recursive Formula
  12. Properties of $D_l$
  13. Tight Maps
  14. Rank-one transformations
  15. Properties of $D_l$
  16. ...and 8 more sections

Key Result

Proposition 1.1

Fix $p\in\mathbb{N}$ and let $b_T\to 0$ be a positive sequence (or function). Let $(X,\mathscr{B},\mu,T)$ be a measure‑preserving transformation and let $A,B\in\mathscr{B}$ satisfy Then there exists an exceptional set $J_{A,B}\subseteq\mathbb{N}$ such that as $n\to\infty$ with $n\notin J_{A,B}$.

Figures (3)

  • Figure 1: Construction of Chacon Transformation in step $0$ and $1$ ($\tau_1$)
  • Figure 2: Construction of Chacon Transformation in step $n\geq 2$
  • Figure 3: $D_l$ for $l=0,1,2,3$

Theorems & Definitions (114)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Proposition \ref{['prop:weak-conv-rate']}
  • Proposition 2.2
  • proof
  • ...and 104 more