On the fractional parts of lacunary sequences - Dubickas, Arturas (MATHEMATICA SCANDINAVICA, 2006)
In this paper, we prove that if $t_0, t_1, t_2, …$ is a lacunary sequence, namely, $t_{n+1}/t_n≥ 1+r^{-1}$ for each $n≥ 0$, where $r$ is a fixed positive number, then there are two positive constants $c(r)=\max(1-r, 2(3r+6)^{-2})$ and $ξ=ξ(t_0, t_1,…)$ such that the fractional parts $\{ξ t_n\}$, $n=0,1,2,…$, all belong to a subinterval of $[0,1)$ of length $1-c(r)$. Some applications of this theorem to the chromatic numbers of certain graphs and to some fast growing sequences are discussed. We prove, for instance, that the number $\sqrt{10}$ can be written as a quotient of two positive numbers whose decimal expansions contain the digits $0$, $1$, $2$, $3$ and $4$ only.