On finitely many base $q$ expansions

TL;DR

This paper identifies specific intervals in (1,2) where points with exactly m base q expansions exist and have positive Hausdorff dimension, using a theorem by Falconer and Yavicoli.

Contribution

It provides the first explicit collection of intervals in (1,2) where the set of points with exactly m base q expansions is nonempty and has positive Hausdorff dimension.

Findings

Explicit intervals in (1,2) for m-base q expansions.

Existence of points with exactly m expansions in these intervals.

Positive Hausdorff dimension of these sets.

Abstract

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any $q$ in one of these intervals, the set of points with exactly $m$ base $q$ expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of $\mathbb{R}^d$ has positive Hausdorff dimension under certain conditions.