Schmidt Games and Conditions on Resonant Sets

TL;DR

This paper introduces a new generalized Schmidt game modification that provides a unified framework for establishing winning strategies across various metric spaces and applications, including Diophantine approximation and geometric group theory.

Contribution

It proposes a novel axiomatic approach to conditions on resonant sets, enabling the existence of winning strategies in a broad class of settings.

Findings

Established conditions for winning strategies in Euclidean and p-adic spaces.

Verified the framework for geodesic rays in CAT(-1) spaces.

Unified approach simplifies analysis of badly approximable sets.

Abstract

Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this game (and modifications of it) have been applied to many examples of complete metric spaces (X, d) to show that the set of "badly approximable points", with respect to a given collection of resonant sets in X, is a winning set. For these examples, strategies were deduced that are, in most cases, strongly adapted to the specific dynamics and properties of the underlying setting. We introduce a new modification of Schmidt's game which is a combination and generalization of the ones of [18] and [20]. This modification allows us to axiomatize conditions on the collection of resonant sets under which there always exists a winning strategy. Moreover, we discuss properties of winning sets of this modification and verify our conditions for several examples - among them, the set of badly approximable vectors in the Euclidian space and the p-adic integers with weights and, as a main example, the set of geodesic rays in proper geodesic CAT(-1) spaces which avoid a suitable collection of convex subsets.