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

Abstract

Let ${\hat{I}}^2\in (1,2)$ and $x\in [0,\frac{1}{{\hat{I}}^2-1}]$. We call a sequence $(\hat{I}{\mu }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ a ${\hat{I}}^2$-expansion for $x$ if $x=\sum_{i=1}^{\infty}ε_{i}β^{-i}$. We call a finite sequence $(\hat{I}{\mu }_i{)}_{i=1}^n\in \{0,1{\}}^n$ an $n$-prefix for $x$ if it can be extended to form a ${\hat{I}}^2$-expansion of $x$. In this paper we study how good an approximation is provided by the set of $n$-prefixes. Given $\hat{I}¨:\mathbb{N}\to {\mathbb{R}}_{\ge 0}$, we introduce the following subset of $\mathbb{R}$, $$W_β(Ψ):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^\infty\bigcup_{(ε_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{ε_i}{β^{i}}, \sum_{i=1}^n\frac{ε_i} {β^i}+Ψ(n)\Big]$$ In other words, $W_{\hat{I}}^2(\hat{I}¨)$ is the set of $x\in \mathbb{R}$ for which there exists infinitely many solutions to the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}\leq Ψ(n).$$ When ${\sum }_{n=1}^{\infty }{2}^n\hat{I}¨(n)<\infty$ the Borel-Cantelli lemma tells us that the Lebesgue measure of $W_{\hat{I}}^2(\hat{I}¨)$ is zero. When ${\sum }_{n=1}^{\infty }{2}^n\hat{I}¨(n)=\infty ,$ determining the Lebesgue measure of $W_{\hat{I}}^2(\hat{I}¨)$ is less straightforward. Our main result is that whenever ${\hat{I}}^2$ is a Garsia number and ${\sum }_{n=1}^{\infty }{2}^n\hat{I}¨(n)=\infty$ then $W_{\hat{I}}^2(\hat{I}¨)$ is a set of full measure within $[0,\frac{1}{{\hat{I}}^2-1}]$. Our approach makes no assumptions on the monotonicity of $\hat{I}¨,$ unlike in classical Diophantine approximation where it is often necessary to assume $\hat{I}¨$ is decreasing.