Cross sections of Borel flows with restrictions on the distance set

TL;DR

This paper characterizes when a free Borel flow can have a cross section with all inter-point distances from a specified set, providing a complete criterion for such configurations.

Contribution

It offers a necessary and sufficient condition for the existence of cross sections with prescribed distances in free Borel flows, advancing the understanding of their structure.

Findings

Established a full criterion for cross sections with given distance sets.

Solved an open problem regarding the structure of Borel flows.

Enhanced the classification of Borel flows based on their cross sections.

Abstract

Given a set of positive reals, we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.