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 $A,B\in\mathscr{B}$, 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.