Size of exceptional sets in weakly mixing systems

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.