A generalization of Furstenberg’s Diophantine Theorem

Abstract

We obtain a generalization of Furstenberg’s Diophantine Theorem on non-lacunary multiplicative semigroups. For example we show that the sets of sums { ( p 1 n q 1 m + p 2 n q 2 m ) α : n , m ∈ N } \{(p_1^nq_1^m + p_2^nq_2^m)\alpha :n,m \in \mathbb {N}\} and { ( p 1 n q 1 m + 2 n ) α : n , m ∈ N } \{(p_1^nq_1^m + 2^n)\alpha : n,m \in \mathbb {N}\} are dense in the circle T = R / Z \mathbb {T} = \mathbb {R}/ \mathbb {Z} for all irrational α \alpha , where ( p i , q i ) (p_i, q_i) are distinct pairs of multiplicatively independent integers for i = 1 , 2 i=1, 2 .