Bernoulli decomposition and arithmetical independence between sequences

Abstract

In this paper we study the following set\[A=\{p(n)+2^nd \mod 1: n\geq 1\}\subset [0.1],\] where $p$ is a polynomial with at least one irrational coefficient on non constant terms, $d$ is any real number and for $a\in [0,\infty )$, $a\mathrm{mod}1$ is the fractional part of $a$. By a Bernoulli decomposition method, we show that the closure of $A$ must have full Hausdorff dimension.

11 pages; Version accepted for publication in ETDS