Sumset size races for measurable sets

Abstract

Let $G$ be a locally compact abelian group with Haar measure $\hat{I}¼$. For integers $n\ge 2$ and $H\ge 2$ and for any $n$-tuples ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_H\in {\mathbf{N}}^n$, there exist measurable subsets $A_1,\dots ,A_n$ of $G$ such that the $n$-tuple $\left(\hat{I}¼(h{A}_1),\dots ,\hat{I}¼(h{A}_n)\right)$ has the same relative order as the $n$-tuple ${\mathbf{u}}_h$ for all $h=1,\dots ,H$. For integers $m_{i,h}$ for $i=1,\dots ,n-1$ and $h=1,\dots ,H$, there are Lebesgue measurable sets $A_1,\dots ,A_n$ in $\mathbf{R}$ such that $\hat{I}¼(h{A}_{i+1})-\hat{I}¼(h{A}_i)=m_{i,h}$ for all $i$ and $h$.

12 pages