How do fractal sets behave in the realm of integers? This work explores the additive and geometric independence between fractal sets structured with respect to multiplicatively independent bases. It introduces a new class of fractal sets of integers, paralleling the notion of invariant sets, and investigates their properties within the discrete context of integers. By combining ideas from fractal geometry and dynamical systems, the authors build a bridge between continuous and discrete regimes. Quantitative bounds on the dimensions of projections of restricted digit Cantor measures, recently obtained by Shmerkin, are heavily relied upon for the transversality results. This research contributes to the understanding of fractal geometry and ergodic theory, offering insights into the relationship between digit expansions. It outlines a number of open questions and directions regarding fractal subsets of the integers, paving the way for future investigations in this area.
Appearing in the Journal of the London Mathematical Society, this article aligns with the journal's focus on pure mathematics and its applications. By exploring the properties of fractal sets within the integers, the study contributes to the journal's scope of publishing high-quality research in various areas of mathematical science.