Discrepancy estimates for generalized polynomials
type: Item
instance of: scholarly article
title: Discrepancy estimates for generalized polynomials
zbMATH Open document ID: 1423.11131
author: Anirban Mukhopadhyay
author: Olivier Ramaré
author: G. K. Viswanadham
published in: Monatshefte für Mathematik
publication date: 2019-03-01T00:00:00Z
review text: Let $[y]$ denote the integral part of $y$ and let $\{y\}$ be the fractional part of $y$. Let $\left(x_n\right)_{n\geq 0}$ be a sequence of real numbers and $N$ be any positive integer. The discrepancy of $\left(x_n\right)_{n\geq 0}$, denoted by $D_{N}(x_n)$, is defined by \[ D_{N}(x_n)=\sup_{0\leq a<b\leq 1}\left|\frac{\#\left\{n\leq N: \ \{x_n\}\in[a,b)\right\}}{N}-(b-a)\right|. \] Let $p(x)\in\mathbb{R}[x]$ be a monic polynomial of degree $d\geq 2$, and let $\alpha,\beta$ be non-zero real numbers satisfying certain conditions. The authors obtain an upper bound for the discrepancy of the sequence $\displaystyle \left(\left[p(n)\alpha\right]\beta\right)_{n\geq 0}$.
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K31
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 7031540
zbMATH Keywords: discrepancy
zbMATH Keywords: generalized polynomial
zbMATH Keywords: irrationality measure
MaRDI profile type: Publication
full work available at URL: s00605-017-1119-x
OpenAlex ID: W2765559756
cites work: Q4692762
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cites work: Discrepancy estimates for some linear generalized monomials
cites work: Q4767350
cites work: Well Distributed Sequences of Integers
DOI: 10.1007/S00605-017-1119-X
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description: scientific article; zbMATH DE number 7031540