On complex positive definite functions on Z_n vanishing on squares
Number Theory
2011-07-19 v3
Abstract
We generalize the Sarkozy-Furstenberg theorem on squares in difference sets of integers, and show that, given any positive definite function f:Z_N->C with density at least r(N), where r(N)=O((\log N)^{-c}), there is a perfect square s<=N/2 such that f(s) is non-zero. We do not rely on the usual analysis of the dichotomy of randomness and periodicity of a set and iterative application of the Hardy-Littlewood method. Instead, we find a bound for the van der Corput property of the set of squares.
Keywords
Cite
@article{arxiv.0811.1360,
title = {On complex positive definite functions on Z_n vanishing on squares},
author = {Sinisa Slijepcevic},
journal= {arXiv preprint arXiv:0811.1360},
year = {2011}
}
Comments
The paper has been completely revised and posted in a new form and with a new title, as accepted for publishing