A new matrix potential game for self-affine sets yields non-empty intersections, Hausdorff dimension lower bounds, and finite-pattern theorems for classes of carpets with zero thickness.
On finitely many base $q$ expansions
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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.
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The matrix potential game and structures of self-affine sets
A new matrix potential game for self-affine sets yields non-empty intersections, Hausdorff dimension lower bounds, and finite-pattern theorems for classes of carpets with zero thickness.