Generalizations of Furstenberg's Diophantine result

TL;DR

This paper extends Furstenberg's Diophantine results by demonstrating density of certain multiplicative orbits on the one-torus, using sequences with specific properties and ergodic theory techniques.

Contribution

It introduces new generalizations of Furstenberg's theorem, including conditions on sequences and the use of ergodic methods for density results.

Findings

Density of orbits for sequences with quotients tending to 1

Density results for sequences with $p$-adic interpolation

Application of ergodic theory to multiplicative orbit density

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.