An improvement on Furstenberg's intersection problem

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