Some remarks on barycentric-sum problems over cyclic groups

TL;DR

This paper investigates the barycentric Olson constants in cyclic groups, establishing new bounds and properties for subsets with specific additive relations, advancing understanding in additive combinatorics.

Contribution

It introduces new results on the k-th barycentric Olson constants for cyclic groups, expanding the theoretical framework of additive combinatorics in finite abelian groups.

Findings

Derived new bounds for barycentric Olson constants in cyclic groups

Identified conditions for subsets to contain barycentric configurations

Extended previous results on Olson constants in abelian groups

Abstract

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.