On the pseudo-random properties of $n^c$

Abstract

We estimate the well-distribution measure and correlation of order 2 of the binary sequence $E_N=\{e_1,\ldots,e_N\}$ defined by $e_n=+1$ if $0\leqslant\{n^c\} \lt {1}/{2}$ and $e_n=-1$ if ${1}/{2}\leqslant\{n^c\} \lt 1$, where $c$ is a real, non-integral number greater than $1$ and $\{x\}$ denotes the fractional part of $x$. We also prove an upper bound for the well-distribution measure of an arbitrary binary sequence in terms of its generating function and show that there exists no upper bound of this type for the correlation. The proof is based on the Erdős-Turán inequality, which we establish with an improved constant.