Some remarks on barycentric-sum problems over cyclic groups

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 .