Additive and geometric transversality of fractal sets in the integers

TL;DR

This paper explores additive and geometric independence of fractal-like sets of integers structured by multiplicatively independent bases, extending concepts from fractal geometry and dynamical systems to the discrete setting.

Contribution

It introduces a new class of fractal sets in integers, classifies invariant subsets, and proves analogues of Furstenberg's transversality conjectures for these sets.

Findings

Classification of all -invariant subsets of positive integers.

Analogues of Furstenberg's conjectures on intersections and sumsets.

Dimension formulas for iterated sumsets of -invariant sets.

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 - analogues 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: -a classification of all subsets of the positive integers that are simultaneously $\times r$- and $\times s$-invariant; -integer analogues of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection $A\cap B$ and the sumset $A+B$ of $\times r$- and $\times s$-invariant sets $A$ and $B$ when $r$ and $s$ are multiplicatively independent; and -a description of the dimension of iterated sumsets $A+A+\cdots+A$ for any $\times r$-invariant set $A$. We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the $L^q$-dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.