Michal Rams (IMPAN, Warsaw)

Title: Badly approximable numbers for irrational rotation

Abstract: The paper by Lim, de Saxce, and Shapira was the first to investigate the set $B(\alpha, c)=\{x; L(x,\alpha)>c\}$ for varying $c\leq 0$, they proved that for almost every $\alpha$ $\dim_H B(\alpha,c) < 1$ for all positive $c$ and they also gave a sufficient condition for $\alpha$ under which $\dim_H B(\alpha, c) =1$ for sufficiently small $c$. We prove that $\dim_H B(\alpha,c) < 1$ for all $c>0$ if and only if $\liminf_{k\to\infty} q_k^{1/k}<\infty$ (where $q_k$ are the denominators in the continuous fraction form of $\alpha$). Moreover, we prove that if $\dim_H B(\alpha, c) =1$ for some $c>0$ then also  $\dim_H B(\alpha, 1/432) =1$ (that is, we have a dichotomy: either $\dim_H B(\alpha,c)<1$ for all $c>0$ or $\dim_H B(\alpha, c) =1$ for all $c\leq 1/432$). The number 1/432 is probably not optimal, for example if $\lim_k a_k=\infty$ (where $a_k$ are the continuous fraction coefficients of $\alpha$) then $\dim_H B(\alpha, c) =1$ for all $c\leq 1/16$.