Effective Bounds for Restricted 3-Arithmetic Progressions in 𝔽_p^n

08/12/2023

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by   Amey Bhangale, et al.

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For a prime p, a restricted arithmetic progression in 𝔽_p^n is a triplet of vectors x, x+a, x+2a in which the common difference a is a non-zero element from {0,1,2}^n. What is the size of the largest A⊆𝔽_p^n that is free of restricted arithmetic progressions? We show that the density of any such set is at most C/(logloglog n)^c, where c,C>0 depend only on p, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was O(1/log^* n), which follows from the density Hales-Jewett theorem.