标题： 矩阵势博弈与自仿集的结构 显示英文标题

标题： The matrix potential game and structures of self-affine sets

摘要： 我们提出了一种势博弈的新变体，并证明了$\R^n$的某些紧子集，包括一大类自仿集，在我们的博弈中是获胜的。 我们证明了具有足够强获胜条件的集合是非空的，提供了它们的豪斯多夫维数的下界，展示了它们具有良好的交集性质，并给出了在给定$M \in \N$的情况下，它们包含每个至多有$M$个元素的集合的相似复制的条件。 我们的博弈在自仿集上的应用是新的，并补充了Yavicoli等人（Math. Z. 2022和Int. Math. Res. Not. IMRN 2023）对自相似集的近期工作。 显示英文摘要

摘要： We present a new variant of the potential game and show that certain compact subsets of $\R^n$, 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 $M \in \N$, 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.