Avoiding a star of three-term arthmetic progressions

TL;DR

This paper establishes an upper bound on the size of subsets in finite vector spaces that avoid a specific configuration of multiple three-term arithmetic progressions sharing a middle term, advancing combinatorial understanding.

Contribution

It introduces a novel adaptation of Sauermann's method to bound subsets avoiding k-stars of 3-APs in F_p^n.

Findings

Derived an explicit upper bound for subset sizes avoiding k-stars of 3-APs.

Extended combinatorial techniques to handle complex progression configurations.

Provided a new methodological approach in additive combinatorics.

Abstract

We provide an upper bound of the size of a subset A of F_p^n that does not admit a k-star of 3-APs (three-term arithmetic progressions). Namely, the subset A is assumed to contain no configuration of k 3-APs, sharing the middle term, such that all 2k+1 terms are distinct. In the proof, we adapt a new method in the recent work of Sauermann.