On the Erdős-Turán inequality on uniform distribution. I
type: Item
instance of: scholarly article
title: On the Erdős-Turán inequality on uniform distribution. I
zbMATH Open document ID: 0639.10033
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: The fundamental inequality of Erdős and Turán is a quantitative version of the famous Weyl criterion for uniform distribution mod 1. In the present note the author introduces a new general version of the Erdős-Turán inequality. The main result of the paper is the following Theorem. Let a function f satisfy the one-sided Lipschitz condition on [0,1] with constant L, and let \(f(0)=f(1)\). Then for any positive integer m, we have \[ [f] < \frac{4L}{m+1}+ \frac{4}{\pi} \sum^{m}_{h=1} (\frac{1}{h}- \frac{1}{m+1})| \hat f(h)|, \] where \(\hat f(h)\) is the Fourier-Stieltjes transform of f, and where [f] represents the oscillation of f.
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 4042022
zbMATH Keywords: discrepancy
zbMATH Keywords: uniform distribution
zbMATH Keywords: Erdős-Turán inequality
zbMATH Keywords: Fourier- Stieltjes transform
zbMATH Keywords: oscillation
author: Petko D. Proinov
reviewed by: L. Kuipers
MaRDI profile type: Publication
full work available at URL: pjaa.64.27
OpenAlex ID: W2035548967
cites work: On some new questions on the distribution of prime numbers
cites work: Q4192121
cites work: Q4767350
cites work: Berry-Esseen bounds and a theorem of Erdős and Turan on uniform distribution mod 1
DOI: 10.3792/PJAA.64.27
description: scientific article; zbMATH DE number 4042022