Diophantine approximation and run-length function on β-expansions

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 $$E_{a,b}=\left\{x\in [0,1]:\underset{n\to \infty }{\mathrm{liminf}}\frac{r_n(x,\beta )}{n}=a,\underset{n\to \infty }{\mathrm{limsup}}\frac{r_n(x,\beta )}{n}=b\right\}(0\le a\le b\le 1).$$ 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.

24 pages, 4 theorems, 2 corollary