Additive transversality of fractal sets in the reals and the integers

By juxtaposing ideas from fractal geometry and dynamical systems, Hillel 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 analogues of some of the notions and results surrounding Furstenberg's work in the discrete setting of the integers. 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 we investigate the additive independence between these fractal sets when they are structured with respect to different bases. We obtain - an integer analogue of a result of Furstenberg regarding the classification of all sets that are simultaneously $\times 2$ and $\times 3$ invariant; - an integer analogue of a result of Lindenstrauss-Meiri-Peres on iterated sumsets of $\times r$-invariant sets; - an integer analogue of Hochman and Shmerkin's solution to Furstenberg's sumset conjecture regarding the dimension of the sumset $X+Y$ of a $\times r$-invariant set $X$ and a $\times s$-invariant set $Y$. To obtain the latter, we provide a quantitative strengthening of a theorem of Hochman and Shmerkin which provides a lower bound on the dimension of $\lambda X + \eta Y$ uniformly in the scaling-parameters $\lambda$ and $\eta$ at every finite scale. Our methods yield a new combinatorial proof of the theorem of Hochman and Shmerkin that avoids the machinery of local entropy averages and CP-processes.