Translation invariant equations and the method of Sanders
type: Item
instance of: scholarly article
title: Translation invariant equations and the method of Sanders
zbMATH Open document ID: 1277.11007
author: Thomas F. Bloom
published in: Bulletin of the London Mathematical Society
publication date: 2012-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.1110
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 6092635
zbMATH Keywords: translation invariant equations
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: additive combinatorics
MaRDI profile type: Publication
OpenAlex ID: W2139192171
arXiv ID: 1107.1110
DOI: 10.1112/BLMS/BDS045
review text: The author considers a linear equation NEWLINE\[NEWLINEc_1 x_1+\cdots+ c_s x_s= 0,\quad c_j\neq 0,\quad s\geq 3\quad\text{and}\quad c_1+\cdots+ c_s= 0.\tag{1}NEWLINE\]NEWLINE Extending the result of Sanders. The author proves:NEWLINENEWLINE Theorem. The sine of the largest subset of \(A\in\{1,\dots, N\}\) having no solutions of equation (1) differ from the form \((a_1,\dots, a)\in A^s\) is NEWLINE\[NEWLINE|A|= O\left(N\left({(\log\log N)^s\over\log N}\right)^{s-2}\right).NEWLINE\]NEWLINE A similar result is obtained in the ring of polynomials \(\mathbb F_q[t]\) over a finite field.
reviewed by: Ilya D. Shkredov
description: scientific article; zbMATH DE number 6092635