Continued fractions and heavy sequences

Abstract

We initiate the study of the sets $H(c)$, $0 =x-[x]$ stands for the fractional part of $x\in \mathbb R$. We prove that, for rational $c$, the sets $H(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the numbers $α\in H_m= H(\frac1m)$ in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by $m$. The characterization implies that $x\in H_m$ if and only if $\frac 1{mx} \in H_m$, for $x>0$. We are unaware of a direct proof of this equivalence, without making a use of the mentioned characterization of the sets $H_m$. We also introduce the dual sets $\hat H_m$ of reals $y$ for which the sequence of integers $\big([ky]\big)_{k\geq1}$ consistently hits the set $m\mathbb Z$ with the at least expected frequency $\frac1m$ and establish the connection with the sets $H_m$: {2mm} If $xy=m$ for $x,y>0$, then $x\in H_m$ if and only if $y\in \hat H_m$. The motivation for the present study comes from Y. Peres's ergodic lemma.