Popular progression differences in vector spaces

2017/08/28 by Fox, Jacob, Pham, Huy Tuan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1708.08482

Abstract

Green proved an arithmetic analogue of Szemerédi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of \mathbbF_pⁿ with n sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive ε. For a fixed odd prime p, we prove that the required dimension grows as an exponential tower of p's of height Θ(log(1/ε)). This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary.

Cited by

• Quantitative bounds in a popular polynomial Szemerédi theorem

Related

• Tower-type bounds for Roth's theorem with popular differences
• On distribution of three-term arithmetic progressions in sparse subsets of F_pⁿ
• On the number of three-term arithmetic progressions in a dense subset of F_qⁿ
• Effective Bounds for Restricted 3-Arithmetic Progressions in \mathbbF_pⁿ
• Narrow arithmetic progressions in the primes