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arXiv · 0707.1496

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

Ernie Croot
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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.

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Ernie Croot. 2007-07-10. Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients. https://arxiv.org/abs/0707.1496

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