Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients
Combinatorics
2007-07-11 v1 Number Theory
Abstract
Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j, which is the jth largest Fourier coefficient of f. This result is similar in spirit to that appearing in an earlier paper [1] by the author; however, in that paper the focus was on the ``small'' Fourier coefficients, whereas here the focus is on the ``large'' Fourier coefficients. Furthermore, the proof in the present paper requires much more sophisticated arguments than those of that other paper.
Keywords
Cite
@article{arxiv.0707.1496,
title = {Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients},
author = {Ernie Croot},
journal= {arXiv preprint arXiv:0707.1496},
year = {2007}
}
Comments
This is a preliminary draft. Later drafts will have more references and cleaner proofs