Generalizations of Furstenberg's Diophantine result (1607.00670v1)

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.