On finitely many base <i>q</i> expansions

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 non-empty and, moreover, has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli [Intersections of thick compact sets in ${\mathbb{R}}^d$ . Math. Z. 301 (3) (2022), 2291–2315, Theorem 6], which guarantees that the intersection of a family of compact subsets of $\mathbb {R}^d$ has positive Hausdorff dimension under certain conditions.