Incomplete character sums and a special class of permutations
type: Item
instance of: scholarly article
title: Incomplete character sums and a special class of permutations
zbMATH Open document ID: 1065.11097
author: Stephen D. Cohen
author: Igor E. Shparlinski
author: Harald Niederreiter
author: Michael E. Zieve
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-08-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93811
full work available at URL: jtnb13-1 english.html
review text: Let \(G\) be a finite abelian group and let \(\psi\) be a permutation of \(G\). The authors study properties of the sequence defined by \(u_{n+1}=\psi(u_n)\) for some initial value \(u_0\). As \(G\) is finite, the sequence is periodic with some least period \(t\). For segments of length \(N<t\) and \(\chi\) a character of \(G\) they investigate the incomplete character sum \[ \sum_{n=0}^{N-1}\chi(u_n). \] The study involves the complete character sum \(\sum_{g\in G}\chi\left(\psi^r(g)-g\right)\) for some integer \(r\). This is particularly easy if \(\psi^r- \text{id}\) is a permutation of \(G\) as then the value is zero. This leads to the definition of \(\mathcal{R}\)-orthomorphisms, namely \(\psi\) is an \(\mathcal{R}\)-orthomorphism if \(\psi^r- \text{id}\) is a permutation for all \(r\in \mathcal{R}\). For such permutations \(\psi\) better bounds on the incomplete character sum are proven.
reviewed by: Tanja Lange
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11K45
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 2081347
zbMATH Keywords: incomplete character sum
zbMATH Keywords: permutation
zbMATH Keywords: pseudorandom sequence
zbMATH Keywords: orthomorphisms
MaRDI profile type: Publication
OpenAlex ID: W2080563171
cites work: On a trigonometric inequality of Vinogradov
cites work: Some new Latin power sets not based on groups
cites work: On the multidimensional distribution of inversive congruential pseudorandom numbers in parts of the period
cites work: ON THE DISTRIBUTION OF DIGITS IN PERIODIC FRACTIONS
cites work: Q5690468
cites work: Quasi-Monte Carlo methods and pseudo-random numbers
cites work: On the distribution of inversive congruential pseudorandom numbers in parts of the period
cites work: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers
cites work: Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus
cites work: On the distribution of pseudorandom numbers and vectors generated by inversive methods
cites work: Q4365727
cites work: Q4699468
cites work: Q4002255
DOI: 10.5802/JTNB.303
Recommended article: A CLASS OF INCOMPLETE CHARACTER SUMS
Recommended article: Characters of permutation summands
Recommended article: Q4897213
Recommended article: Congruences on “character” values of permutation summands
Recommended article: Character sums and arithmetic combinatorics
Recommended article: Q2712132
Recommended article: On certain character sums
Recommended article: Q3077324
Recommended article: Character sums and generating sets
description: scientific article; zbMATH DE number 2081347