Continued fractions and heavy sequences

Abstract:

We initiate the study of the sets $\mathcal {H}(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq 1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally, \[ \mathcal {H}(c) \eqdef \big \{\alpha \in \mathbb {R} \mid \mathbf {card}\big (\{1\leq k\leq n\mid \langle k\alpha \rangle <c\}\big )\geq cn, \text { for all } n\geq 1\big \}, \] 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 $m\geq 1$, we obtain a surprising characterization of the numbers $\alpha \in \mathcal {H}_m \eqdef \mathcal {H}(\tfrac 1m)$ 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 $\big ([ky]\big )_{k\geq 1}$ consistently hits the set $m\mathbb {Z}$ with the at least expected frequency $\frac 1m$ and establish the connection with the sets $\mathcal {H}_m$: If $xy=m$ for $x,y>0$, then $x\in \mathcal {H}_m\iff y\in \widehat {\mathcal {H}}_m$. The motivation for the present study comes from Y. Peres’s ergodic lemma.