Equidistribution of subset sums

We answer a question of Katona and Makar-Limanov, by showing that in an abelian group of order $2h$ the $h$-element subset sums are asymptotically (as $h\to \mathrm{\infty }$) equidistributed. In fact we prove a more general result where the order of the group can be arbitrary, also providing a bound for the ``error term’'.