Continued fractions and heavy sequences

We initiate the study of the sets $\mathcal{H}(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally,

$$\mathcal{H}(c) \stackrel{\text{def}}{=} \big\{\alpha\in \mathbb{... ...n\mid \langle k\alpha\rangle <c\}\big)\geq cn, \text{ for all } n\geq1\big\},$$

where $\langle x\rangle =x-[x]$ stands for the fractional part of $x\in\mathbb{R}$.

We prove that, for rational $c$, the sets $\mathcal{H}(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the numbers $\alpha\in\mathcal{H}_m \stackrel{\text{def}}{=} \mathcal{H}(\tfrac1m)$ in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by $m$. The characterization implies that $x\in\mathcal{H}_m$ if and only if $\frac 1{mx} \in\mathcal{H}_m$, for $x>0$. We are unaware of a direct proof of this equivalence without making use of the mentioned characterization of the sets $\mathcal{H}_m$.

We also introduce the dual sets $\widehat{\mathcal{H}}_m$ 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 $\mathcal{H}_m$:

The motivation for the present study comes from Y. Peres's ergodic lemma.