On the Structure of Sets with Few Three-Term Arithmetic Progressions
Abstract
Fix a density d in (0,1], and let Fpn be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of Fpn having the minimal number of three-term progressions, subject to the constraint |S| is at least dpn. We show that S must have some structure, and that up to o(pn) elements, it is a union of a small number of cosets of a subspace of dimension n-o(n).
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