An Approximation by Lacunary Sequence of Vectors

Abstract

Let $(t_k)_{k=0}^{\infty}$ be a sequence of real numbers satisfying $t_0 \ne 0$ and $|t_{k+1}| \geq (1+1/M) |t_{k}|$ for each k ≥ 0, where M ≥ 1 is a fixed number. We prove that, for any sequence of real numbers $(\xi_k)_{k=0}^{\infty}$, there is a real number ξ such that $\|t_k \xi-\xi_k\|>1/(80M \log(28M))$ for each k ≥ 0. Here, $\|x\|$ denotes the distance from $x \in \R$ to the nearest integer. This is a corollary derived from our main theorem, which is a more general matrix version of this statement with explicit constants.