Diophantine approximation and run-length function on \beta-expansions

TL;DR

This paper studies the behavior of run-length functions in eta-expansions, linking them to Diophantine approximation, and determines the Hausdorff dimension of certain level sets, revealing complex size and residual properties.

Contribution

It provides the first Hausdorff dimension results for level sets of run-length ratios in eta-expansions, connecting Diophantine approximation with fractal geometry.

Findings

Hausdorff dimension of level sets $E_{a,b}$ is explicitly determined.

The set $E_{0,1}$ is residual despite having zero Hausdorff dimension.

Analysis extends to parameter space, revealing intricate structure.

Abstract

For any $\beta > 1$, denoted by $r_n(x,\beta)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $\beta$-expansion of $x\in[0,1]$. The limit superior (respectively limit inferior) of $\frac{r_n(x,\beta)}{n}$ is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level set $$E_{a,b}=\left\{x \in [0,1]: \liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=a,\ \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=b\right\}\ (0\leq a\leq b\leq1).$$ Furthermore, we show that the extremely divergent set $E_{0,1}$ which is of zero Hausdorff dimension is, however, residual. The same problems in the parameter space are also examined.