The uniform Gardner conjecture and rounding Borel flows

TL;DR

This paper investigates the Gardner conjecture in the context of amenable groups, providing a characterization involving group quotients and introducing an algorithm for rounding Borel flows.

Contribution

It offers a complete characterization of amenable groups satisfying Gardner's conjecture and presents a novel algorithm for rounding Borel flows in group actions.

Findings

Amenable groups satisfy Gardner's conjecture iff they lack certain group quotients.

Established an algorithm for rounding Borel flows in amenable group actions.

Provided a new perspective on equidecomposition problems in group theory.

Abstract

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.