On the Diophantine properties of lambda-expansions

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 $-\frac{1}{\alpha }\frac{\log \lambda }{\log 2}\le s\le \frac{1}{\alpha }$. 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.

21 pages

Cited by in corpus (1)

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