The matrix potential game and structures of self-affine sets

Source abstract

Abstract We present a new variant of the potential game and show that certain compact subsets of $\mathbb{R}^{n}$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \mathbb{N}$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the work of Yavicoli Int. Math. Res. Not. (2023) and Yavicoli et al. Math. Z. 2022 for self-similar sets.