The smash sum is the unique sum of open sets satisfying a natural list of axioms

TL;DR

This paper proves the uniqueness of a specific sum operation on open sets satisfying certain axioms, establishing it as the natural scaling limit of the Diaconis-Fulton smash sum.

Contribution

It demonstrates that, up to measure-zero differences, there is a unique sum operation on open sets fulfilling a natural set of axioms, linking it to the Diaconis-Fulton smash sum.

Findings

Uniqueness of the sum operation up to measure-zero sets

Identification of the sum as the scaling limit of the Diaconis-Fulton smash sum

Establishment of a natural axiomatic framework for open set sums

Abstract

A sum of open sets is a map taking two bounded open sets $A,B$ and producing a new open set $A \oplus B$. We prove that, up to sets of measure zero, there is only one such sum satisfying a natural list of axioms. It is the scaling limit of the Diaconis-Fulton smash sum.