  We study groups which satisfy Gardner's equidecomposition conjecture for
uniformly distributed sets. We prove that an amenable group has this property
if and only if it does not admit $(\mathbb{Z}/2\mathbb{Z})
*(\mathbb{Z}/2\mathbb{Z})$ as a quotient by a finite subgroup. Our technical
contribution is an algorithm for rounding Borel flows for actions of amenable
groups.
