Continued fractions and heavy sequences

Abstract

We initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the sequence $(kx{)}_{k\ge 1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\ge c$). More formally, \[ H(c)=\{α\in \mathbf R\mid {\rm card}(\{1\leq k\leq n\mid < kα><c\})\geq cn, {for all}n\geq1\}. \] where $<x>=x-[x]$ 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 $m\ge 1$, we obtain a surprising characterization of the numbers $\alpha \in H_m=H(\frac{1}{m})$ 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 $([ky]{)}_{k\ge 1}$ consistently hits the set $m\mathbb{Z}$ with the at least expected frequency $\frac{1}{m}$ 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.

9 pages