A uniform set with fewer than expected arithmetic progressions of length 4
Combinatorics
2020-07-28 v2
Abstract
An example is presented of a subset of of density such that the largest non-trivial Fourier coefficient of the characteristic function of is very small, but the probability that a random arithmetic progression (mod ) of length 4 lies in is significantly smaller than .
Cite
@article{arxiv.2004.07598,
title = {A uniform set with fewer than expected arithmetic progressions of length 4},
author = {W. T. Gowers},
journal= {arXiv preprint arXiv:2004.07598},
year = {2020}
}
Comments
This preprint was written well over a decade ago and has been known about by experts but not made publicly available. I apologize for this and am now remedying the situation. Revised version updates the notation, corrects minor errors (pointed out by referee), and adds one or two references