Additive and geometric transversality of fractal sets in the integers
Daniel Glasscock, Joel Moreira, Florian K. Richter
TL;DR
This work develops a comprehensive discrete analogue of Furstenberg’s transversality by introducing multiplicatively invariant subsets of the nonnegative integers and studying their interaction with respect to multiplicatively independent bases. By transferring tools from real fractal geometry (including $L^q$-dimension projections and Frostman-type bounds) to the integer setting, the authors obtain: (i) a classification/dichotomy for sets simultaneously invariant under base-$r$ and base-$s$; (ii) discrete intersection bounds for affine-typical projections and a discrete sumset transversality, paralleling Furstenberg–Hochman–Shmerkin–Wu results; and (iii) an integer analogue of the Lindenstrauss–Meiri–Peres iterated sumset growth. The framework connects discrete fractal geometry with symbolic dynamics and real fractal theory, yielding dimension-preserving correspondences and uniform bounds, and it opens several open questions on density, intersections, and polynomial/differentially-structured images in the integers. Overall, the paper advances the understanding of independence phenomena for additive and multiplicative structures in $ 0$, providing both sharp theorems and a roadmap for future exploration in discrete fractal geometry and arithmetic combinatorics.
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.