Sumset size races for measurable sets

TL;DR

This paper demonstrates the flexibility in the sizes of sumsets of measurable sets within locally compact abelian groups, showing that any prescribed order or differences can be realized through appropriate set constructions.

Contribution

It establishes that for any given orderings and differences, there exist measurable sets in groups and the real line that realize these configurations in their sumsets.

Findings

Constructs measurable sets with prescribed sumset size orderings.

Shows existence of sets with specified sumset size differences.

Extends sumset size control to general locally compact abelian groups.

Abstract

Let $G$ be a locally compact abelian group with Haar measure $\mu$. For integers $n \geq 2$ and $H \geq 2$ and for any $n$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_H \in \mathbf{N}^n$, there exist measurable subsets $A_1,\ldots, A_n$ of $G$ such that the $n$-tuple $\left( \mu(hA_1),\ldots, \mu(hA_n) \right)$ has the same relative order as the $n$-tuple $\mathbf{u}_h$ for all $h = 1,\ldots, H$. For integers $m_{i,h}$ for $i =1,\ldots, n-1$ and $h = 1,\ldots, H$, there are Lebesgue measurable sets $A_1,\ldots, A_n$ in $\mathbf{R}$ such that $\mu(hA_{i+1}) - \mu(hA_i) = m_{i,h}$ for all $i$ and $h$.