Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients

TL;DR

This paper investigates subsets of F_p^n lacking three-term arithmetic progressions, establishing lower bounds on their large Fourier coefficients, and extends previous work by focusing on significant Fourier coefficients with more complex proofs.

Contribution

It introduces new lower bounds for large Fourier coefficients of progression-free subsets in F_p^n, advancing understanding of their Fourier spectrum.

Findings

Lower bounds for large Fourier coefficients established

Progression-free subsets can have significant Fourier coefficients

More sophisticated proof techniques developed

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.