Subsets of F_pⁿ without three term arithmetic progressions have several large Fourier coefficients

2007/07/10 by Ernie Croot, Croot, Ernie
Mathematics · #05D99 #Advanced Topology and Set Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.CO #math.NT #msc:05D99

paper · pdf · doi:10.48550/arxiv.0707.1496

This is a preliminary draft. Later drafts will have more references and cleaner proofs

arxiv created 2007/07/10 · openalex publication_date 2007/07/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that f : F_pⁿ -> [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.

Citations

• On the Decay of the Fourier Transform and Three Term Arithmetic Progressions

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