Open Access
Translator Disclaimer
Spring 2002 On the pseudo-random properties of $n^c$
Christian Mauduit, Joël Rivat, András Sárközy
Illinois J. Math. 46(1): 185-197 (Spring 2002). DOI: 10.1215/ijm/1258136149

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.

Citation

Download Citation

Christian Mauduit. Joël Rivat. András Sárközy. "On the pseudo-random properties of $n^c$." Illinois J. Math. 46 (1) 185 - 197, Spring 2002. https://doi.org/10.1215/ijm/1258136149

Information

Published: Spring 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1070.11031
MathSciNet: MR1936084
Digital Object Identifier: 10.1215/ijm/1258136149

Subjects:
Primary: 11K45
Secondary: 11K36

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign

JOURNAL ARTICLE
13 PAGES


SHARE
Vol.46 • No. 1 • Spring 2002
Back to Top