Density modulo 1 and Hardy fields

Abstract 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 $$(\mathbf{f}(n))_{n=1}^{\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:msubsup> <mml:mo>)</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> is dense modulo 1 in R d , the sequence (2 m 3 n α , 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.