2025/08/15 by Richard A. Howat, Howat, Richard A., Andrew Mitchell +3
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Optimization and Variational Analysis #math.DS #math.MG
paper · pdf · doi:10.48550/arxiv.2508.11577
22 pages, 3 figures, minor revisions to improve exposition
openalex publication_date 2025/08/15 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/14 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
We present a new variant of the potential game and show that certain compact subsets of ℝn, 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 ∈ 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 recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets.