Elementary proof of Furstenberg’s Diophantine result

Abstract

We present an elementary proof of a diophantine result (due to H. Furstenberg) which asserts (in a special case) that for every irrational α \alpha the set { 2 m 3 n α | m , n ≥ 0 } \{ {2^m}{3^n}\alpha |m,n \geq 0\} is dense modulo 1. Furstenberg’s original proof employs the theory of disjointness of topological dynamical systems.