Approximation properties of $\beta$-expansions II

Abstract

Given $\beta \in (1,2)$ and $x\in [0,\frac{1}{\beta -1}]$, a sequence $({\epsilon }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ is called a $\beta$-expansion for $x$ if $$x=\sum _{i=1}^{\infty }\frac{{\epsilon }_i}{{\beta }^i}.$$ In a recent article the author studied the quality of approximation provided by the finite sums ${\sum }_{i=1}^n{\epsilon }_i{\beta }^{-i}$ \cite{Bak}. In particular, given $\beta \in (1,2)$ and $\Psi :\mathbb{N}\to {\mathbb{R}}_{\ge 0},$ we associate the set $$W_{\beta }(\Psi ):=\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }\bigcup _{({\epsilon }_i{)}_{i=1}^n\in \{0,1{\}}^n}[\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i},\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i}+\Psi (n)].$$ Alternatively, $W_{\beta }(\Psi )$ is the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $({\epsilon }_i{)}_{i=1}^n$ satisfying the inequalities $$0\le x-\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i}\le \Psi (n).$$ If ${\sum }_{n=1}^{\infty }{2}^n\Psi (n)<\infty$ then $W_{\beta }(\Psi )$ has zero Lebesgue measure. We call a $\beta \in (1,2)$ approximation regular, if ${\sum }_{n=1}^{\infty }{2}^n\Psi (n)=\infty$ implies $W_{\beta }(\Psi )$ is of full Lebesgue measure within $[0,\frac{1}{\beta -1}]$. The author conjectured in \cite{Bak} that almost every $\beta \in (1,2)$ is approximation regular. In this paper we make a significant step towards proving this conjecture. The main result of this paper is the following statement: given a sequence of positive real numbers $({\omega }_n{)}_{n=1}^{\infty },$ which satisfy ${\lim }_{n\to \infty }{\omega }_n=\infty$, then for Lebesgue almost every $\beta \in (1.497\dots ,2)$ the set $W_{\beta }({\omega }_n\cdot 2^{-n})$ is of full Lebesgue measure within $[0,\frac{1}{\beta -1}]$. Here the sequence $({\omega }_n{)}_{n=1}^{\infty }$ should be interpreted as a sequence tending to infinity at a very slow rate.