Generalizations of Furstenberg's Diophantine result

Abstract

We prove two generalizations of Furstenberg's Diophantine result regarding density of an orbit of an irrational point in the one-torus under the action of multiplication by a non-lacunary multiplicative semi-group of $\mathbb{N}$. We show that for any sequences $\{a_n\},\{b_n\}\subset \mathbb{Z}$ for which the quotients of successive elements tend to $1$ as $n$ goes to infinity, and any infinite sequence $\{c_n\}$, the set $\{a_n{b}_m{c}_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo $1$ for every irrational $x$. Moreover, by ergodic-theoretical methods, we prove that if $\{a_n\},\{b_n\}$ are sequence having smooth $p$-adic interpolation for some prime number $p$, then for every irrational $x$, the sequence $\{p^n{a}_m{b}_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo 1.

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