  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.
