In this question it is discussed that for a set $$A=\{2^n3^m| n,m=0,1,2,\ldots\}$$ there exists an irrational number $\alpha$ such that the set $\alpha\cdot A$ (being naturally enumerated) is not uniformly distributed modulo 1. For which general conditions on $A$ does such $\alpha$ exist? For example, is it sufficient that $A$ has zero upper density, or maybe even that $A$ has upper density less than 1?