Additive and geometric transversality of fractal sets in the integers

Abstract

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $\times r$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: