Abstract.
In the late 1960's, Hillel Furstenberg proposed a series of conjectures in seeking to make precise the heuristic that there is no common structure between digit expansions in distinct prime bases. While his famous x2, x3 conjecture in ergodic theory remains unsolved, recent solutions to the equally momentous transversality conjectures of his, which concern the dimension of sumsets and intersections of p- and q- invariant sets, now shed new light on old problems. In this talk, I will explain how to use tools from fractal geometry and uniform distribution to establish natural analogues of Furstenberg's transversality conjectures in the integers.
This talk is based on joint work with Daniel Glasscock and Joel Moreira.