Size of exceptional sets in weakly mixing systems

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 $A,B\in \mathcal{B}$, such that for every increasing function $h:\mathbb{N}\to {\mathbb{R}}_{>0}$ diverging to infinity, $\mathrm{\mid }J\cap [0,n]\mathrm{\mid }\le (\log n{)}^{h(n)}$ 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 $\mathrm{\mid }J\cap [0,n]\mathrm{\mid }\ge (\log n{)}^t$ 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 Cesàro weak-mixing averages satisfy a rate $o(b_N)$, then $J_{A,B}$ may be chosen so that $\mathrm{\mid }{J}_{A,B}\cap [0,N]\mathrm{\mid }=o(N{b}_N)$. 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.