On the Diophantine properties of lambda-expansions

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

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