Non-trivial solutions to a linear equation in integers
Number Theory
2015-06-26 v2 Combinatorics
Abstract
For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A has O(N^{1/2-epsilon}) elements. This answers a question of I. Ruzsa.
Cite
@article{arxiv.math/0703767,
title = {Non-trivial solutions to a linear equation in integers},
author = {Boris Bukh},
journal= {arXiv preprint arXiv:math/0703767},
year = {2015}
}
Comments
5 pages, typos corrected, stronger result given