Effective Bounds for Restricted 3-Arithmetic Progressions in \mathbbF_pⁿ

2023/08/12 by Bhangale, Amey, Khot, Subhash, Minzer, Dor · 2 citations
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.06600

Abstract

For a prime p, a restricted arithmetic progression in \mathbbF_pⁿ 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⊆ \mathbbF_pⁿ that is free of restricted arithmetic progressions? We show that the density of any such a 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.

Cited by

• Words with Repeated Letters in a Grid
• The Lens of Abelian Embeddings

Related

• New lower bounds for three-term progression free sets in \mathbbF_pⁿ
• Asymptotic upper bounds on progression-free sets in ℤ_pⁿ
• Popular progression differences in vector spaces
• Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields
• On the number of three-term arithmetic progressions in a dense subset of F_qⁿ