Approximation properties of ${\hat{I}}^2$-expansions II

Abstract

Given ${\hat{I}}^2\in (1,2)$ and $x\in [0,\frac{1}{{\hat{I}}^2-1}]$, a sequence $(\hat{I}{\mu }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ is called a ${\hat{I}}^2$-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 ${\hat{I}}^2\in (1,2)$ and $\hat{I}¨:\mathbb{N}\to {\mathbb{R}}_{\ge 0},$ 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, $W_{\hat{I}}^2(\hat{I}¨)$ is the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\hat{I}{\mu }_i{)}_{i=1}^n$ satisfying the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}\leq Ψ(n).$$ If ${\sum }_{n=1}^{\infty }{2}^n\hat{I}¨(n)<\infty$ then $W_{\hat{I}}^2(\hat{I}¨)$ has zero Lebesgue measure. We call a ${\hat{I}}^2\in (1,2)$ approximation regular, if ${\sum }_{n=1}^{\infty }{2}^n\hat{I}¨(n)=\infty$ implies $W_{\hat{I}}^2(\hat{I}¨)$ is of full Lebesgue measure within $[0,\frac{1}{{\hat{I}}^2-1}]$. The author conjectured in \cite{Bak} that almost every ${\hat{I}}^2\in (1,2)$ 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 $(\ddot{I}{‰}_n{)}_{n=1}^{\infty },$ which satisfy ${\lim }_{n\to \infty }\ddot{I}{‰}_n=\infty$, then for Lebesgue almost every ${\hat{I}}^2\in (1.497\dots ,2)$ the set $W_{\hat{I}}^2(\ddot{I}{‰}_n\cdot 2^{-n})$ is of full Lebesgue measure within $[0,\frac{1}{{\hat{I}}^2-1}]$. Here the sequence $(\ddot{I}{‰}_n{)}_{n=1}^{\infty }$ should be interpreted as a sequence tending to infinity at a very slow rate.