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arxiv: 2301.09786 · v4 · submitted 2023-01-24 · 🧮 math.DS

Size of exceptional sets in weakly mixing systems

Jiyun Park , Kangrae Park This is my paper

Pith reviewed 2026-05-24 10:04 UTC · model grok-4.3

classification 🧮 math.DS
keywords exceptional setsweakly mixingChacon transformationcutting and stackingrestrictive tight mapsreturn time distributionsergodic averages
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The pith

Restrictive tight maps admit a universal exceptional set J with |J up to n| at most (log n) to any slowly diverging power, for every pair of measurable sets at once.

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

The paper constructs an explicit universal exceptional set J that works simultaneously for all measurable pairs A and B in a class of cutting-and-stacking transformations called restrictive tight maps, which includes the Chacon transformation. This J satisfies a very slow growth bound: its counting function stays below (log n) raised to any function h that tends to infinity, no matter how slowly. The authors also give a general principle that converts a rate of weak mixing into a corresponding bound on the size of the exceptional set, and they apply it to several other families of systems. They prove a matching lower bound showing that the logarithmic scale cannot be improved in general for certain maps in the class.

Core claim

In the class of restrictive tight maps we explicitly construct a universal exceptional set J subset of the naturals such that for every increasing h diverging to infinity, the cardinality of J up to n is at most (log n) to the power h(n) for all large n, and this single J controls the exceptional times uniformly for every pair of measurable sets A and B. For tight maps with no spacers above the last subcolumn the scale is sharp: for any t greater than zero there exist A and B whose every exceptional set must contain at least (log n) to the power t points up to n. A complementary principle states that if the p-th Cesaro weak-mixing averages are o(b_N) then an exceptional set of size o(N b_N)

What carries the argument

Recursive formulas for return-time distributions coming from the cutting-and-stacking construction, used to control the exceptional times uniformly across all pairs A and B.

If this is right

  • The Chacon transformation has a logarithmic-scale universal exceptional set.
  • If the p-th Cesaro weak-mixing averages decay as o(b_N), then an exceptional set of size o(N b_N) exists.
  • The same principle yields explicit exceptional-set bounds for interval exchange transformations, translation flows, and substitution systems under their known quantitative mixing rates.
  • There exists a weakly mixing one-spacer rank-one system whose exceptional sets must grow at least polynomially for some pairs.

Where Pith is reading between the lines

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

  • The construction suggests that logarithmic exceptional sets may be available in many other rank-one weakly mixing systems whenever similar recursive return-time controls can be written down.
  • The sharpness result indicates that the logarithmic scale is intrinsic to the pairwise obstructions arising from the cutting-and-stacking geometry rather than an artifact of the proof.
  • The rate-to-exceptional-set principle could be tested on additional families such as rigid rotations or Gaussian systems if quantitative weak-mixing rates become available.

Load-bearing premise

The return-time distributions of the cutting-and-stacking transformations admit recursive formulas that permit uniform control over the exceptional sets for all pairs A and B.

What would settle it

A concrete pair of measurable sets A and B inside the Chacon transformation for which every exceptional set J satisfies |J up to n| greater than (log n) to the power h(n) for some h going to infinity.

read the original abstract

We study exceptional sets for the Chacon transformation and, more generally, for a class of cutting-and-stacking transformations called restrictive tight maps. For these systems we explicitly construct a universal exceptional set \(J\subseteq\mathbb{N}\), valid uniformly for all measurable pairs \(A,B\in\mathscr{B}\), such that for every increasing function \(h:\mathbb{N}\to\mathbb{R}_{>0}\) diverging to infinity, \(|J\cap[0,n]|\le(\log n)^{h(n)}\) for all sufficiently large \(n\). The Chacon transformation considered in this paper belongs to this class, giving a logarithmic-scale universal exceptional set for Chacon. We also prove that this logarithmic scale is essentially sharp at the level of pairwise obstructions: for every tight map with no spacers above the last subcolumn, i.e. \(s_{m-1}=0\), and every \(t>0\), there exist measurable sets \(A,B\) such that every exceptional set \(J\) for \((A,B)\) satisfies \(|J\cap[0,n]|\ge(\log n)^t\) for all sufficiently large \(n\). The construction is based on recursive formulas for return-time distributions arising from the cutting-and-stacking structure. As a complementary quantitative principle, we show that if the corresponding \(p\)-th Ces\`aro weak-mixing averages satisfy a rate \(o(b_N)\), then \(J_{A,B}\) may be chosen so that \(|J_{A,B}\cap[0,N]|=o(Nb_N)\). We apply this rate-to-exceptional-set principle to several weakly mixing models, including interval exchange transformations, translation flows, and substitution dynamical systems, under the regularity assumptions of the available quantitative estimates. We also construct a separate weakly mixing one-spacer rank-one example in which exceptional-set obstructions have polynomial lower growth.

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.

Referee Report

1 major / 0 minor

Summary. The paper studies exceptional sets for the Chacon transformation and restrictive tight cutting-and-stacking maps. It claims to construct a single universal exceptional set J ⊆ ℕ that works uniformly for all measurable pairs A, B and satisfies |J ∩ [0,n]| ≤ (log n)^{h(n)} eventually for every increasing h : ℕ → ℝ>0 with h(n) → ∞. It further claims this logarithmic scale is sharp: for every t > 0 and every tight map with s_{m-1}=0 there exist A, B such that every exceptional set for (A,B) must satisfy |J ∩ [0,n]| ≥ (log n)^t eventually. A complementary rate-to-exceptional-set principle is stated for systems with quantitative weak-mixing rates, with applications to IETs, flows, and substitutions; a separate one-spacer example with polynomial lower growth is also constructed.

Significance. If the stated bounds were compatible, the explicit construction of a universal J with the claimed slow growth, together with the matching lower bounds and the Cesàro-rate principle, would give a precise quantitative picture of exceptional-set sizes across a range of weakly mixing rank-one and substitution systems.

major comments (1)
  1. [Abstract] Abstract (and the central claims): the universal J is asserted to satisfy |J ∩ [0,n]| ≤ (log n)^{h(n)} eventually for every increasing h(n) → ∞. Letting g(n) = log_{log n} |J ∩ [0,n]|, this forces limsup g(n) < ∞. The sharpness statement, however, asserts that for every t > 0 there exist A, B (on any tight map with s_{m-1}=0) such that every exceptional set for (A,B), including the universal J, must satisfy |J ∩ [0,n]| ≥ (log n)^t eventually, forcing liminf g(n) = ∞. These two requirements on the same J are incompatible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a logical inconsistency between the claimed upper bound on the universal exceptional set J and the sharpness lower bound. We agree that the statements as formulated in the abstract (and corresponding claims in the body) cannot both be true.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the central claims): the universal J is asserted to satisfy |J ∩ [0,n]| ≤ (log n)^{h(n)} eventually for every increasing h(n) → ∞. Letting g(n) = log_{log n} |J ∩ [0,n]|, this forces limsup g(n) < ∞. The sharpness statement, however, asserts that for every t > 0 there exist A, B (on any tight map with s_{m-1}=0) such that every exceptional set for (A,B), including the universal J, must satisfy |J ∩ [0,n]| ≥ (log n)^t eventually, forcing liminf g(n) = ∞. These two requirements on the same J are incompatible.

    Authors: We fully agree with the referee that the two requirements are incompatible. The universal quantifier over all diverging h in the upper bound forces limsup g(n) to be finite, while the lower bound (which applies to the same fixed universal J) forces liminf g(n) = ∞. This is an error in the formulation of the upper bound. The construction produces a single J that works uniformly for all A, B, but the growth rate achieved does not satisfy the stated 'for every h' condition simultaneously with the sharpness. We will revise the abstract, introduction, and the statements of the main theorems to remove the incompatible upper bound claim and accurately describe the growth rate of the constructed J, while retaining the sharpness result (which correctly shows that the logarithmic scale cannot be improved to any fixed power for the universal J). revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from cutting-and-stacking recursions

full rationale

The paper claims an explicit construction of universal J from recursive return-time formulas of restrictive tight maps, with the upper bound |J ∩ [0,n]| ≤ (log n)^{h(n)} derived directly from those recursions and the lower bound shown separately via existence of bad pairs A,B for maps with s_{m-1}=0. No equation reduces a result to a fitted parameter renamed as prediction, no self-citation is load-bearing for the central claim, and no ansatz or uniqueness is smuggled via prior work by the same authors. The constructions are presented as arising from the dynamical structure without reducing to their own inputs by definition. The claimed bounds are independent (though potentially inconsistent, which is a correctness issue outside circularity analysis).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of recursive formulas for return-time distributions in the cutting-and-stacking construction and on the definition of restrictive tight maps; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard definitions and properties of measure-preserving transformations, weak mixing, and exceptional sets in ergodic theory.
    Used to define J_{A,B} and the Cesaro averages throughout the abstract.

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Reference graph

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