Density modulo 1 and Hardy fields

In this article we extend Boshernitzan’s result on density modulo 1 of sequences arising from functions belonging to a Hardy field. We also merge these results with Furstenberg’s ×2×3 theorem. We prove, for example, that, given a vector f of subpolynomial functions in a Hardy field, such that \((\mathbf{f}(n))_{n=1}^{\infty}\) $$(\mathbf{f}(n){)}_{n=1}^{\mathrm{\infty }}$$ is dense modulo 1 in R^d, the sequence (2^m3ⁿα, f(n))_m,n≥1 is dense modulo 1 in R^d+1 for irrational α. Some negative results concerning Furstenberg’s theorem are obtained as well.