On the Diophantine properties of lambda-expansions

TL;DR

This paper investigates the Diophantine approximation properties of lambda-expansions for numbers, establishing bounds on Hausdorff dimensions of certain approximation sets and showing their optimality for almost all lambda in a specific interval.

Contribution

It provides new results on the Hausdorff dimension of sets related to lambda-expansions, demonstrating optimal bounds for almost all lambda in (1/2, 2/3).

Findings

Upper bound of Hausdorff dimension is optimal for almost all lambda in (1/2, 2/3).

Lower bound is optimal for a countable set of lambda values.

Sets are in Falconer's intersection classes for Hausdorff dimension.

Abstract

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's intersection classes for Hausdorff dimension $s$ for some $s$ such that $- \frac{1}{\alpha} \frac{\log \lambda}{\log 2} \leq s \leq \frac{1}{\alpha}$. We show that for almost all $\lambda \in (1/2, 2/3)$, the upper bound of $s$ is optimal, but for a countable infinity of values of $\lambda$ the lower bound is the best possible result.