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Approximation properties of $β$-expansions II

Given $β\in(1,2)$ and $x\in[0,\frac{1}{β-1}]$, a sequence $(ε_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is called a $β$-expansion for $x$ if $$x=\sum_{i=1}^{\infty}\frac{ε_{i}}{β^{i}}.$$ In a recent article the author studied the quality of approximation provided by the finite sums $\sum_{i=1}^{n}ε_{i}β^{-i}$ \cite{Bak}. In particular, given $β\in(1,2)$ and $Ψ:\mathbb{N}\to\mathbb{R}_{\geq 0},$ we associate the set $$W_β(Ψ):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}\bigcup_{(ε_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}},\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}+Ψ(n)\Big].$$ Alternatively, $W_β(Ψ)$ is the set of $x\in \mathbb{R}$ such that for infinitely many $n\in\mathbb{N},$ there exists a sequence $(ε_{i})_{i=1}^{n}$ satisfying the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}\leq Ψ(n).$$ If $\sum_{n=1}^{\infty}2^{n}Ψ(n)<\infty$ then $W_β(Ψ)$ has zero Lebesgue measure. We call a $β\in(1,2)$ approximation regular, if $\sum_{n=1}^{\infty}2^{n}Ψ(n)=\infty$ implies $W_β(Ψ)$ is of full Lebesgue measure within $[0,\frac{1}{β-1}]$. The author conjectured in \cite{Bak} that almost every $β\in(1,2)$ 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 $(ω_{n})_{n=1}^{\infty},$ which satisfy $\lim_{n\to\infty} ω_{n}=\infty$, then for Lebesgue almost every $β\in(1.497\ldots,2)$ the set $W_β(ω_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,\frac{1}{β-1}]$. Here the sequence $(ω_{n})_{n=1}^{\infty}$ should be interpreted as a sequence tending to infinity at a very slow rate.