The asymptotic uniform distribution of subset sums

Let G be a finite abelian group of order n, and for each a∈G and integer 1≤h≤n let F_a(h) denote the family of all h-element subsets of G whose sum is a. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families F_a(h) (as a ranges over G) become asymptotically equal as n→∞ when [Formula presented]. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every [Formula presented].