paper

An improvement on Furstenberg's intersection problem

arXiv:1811.11073

Abstract

In this paper, we study a problem posed by Furstenberg on intersections between $\times 2, \times 3$ invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if $A_{2}, A_{3} \subset [0, 1]$ are closed and $\times 2, \times 3$ invariant respectively, assuming that $dim A_{2} + dim A_{3} < 1$ then $A_{2} \cap (u A_{3} + v)$ is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters $u, v$ such that $u$ and $u^{- 1}$ are both bounded away from $0$ .

Accepted in Transactions of the american mathematical society