Sumset size races for measurable sets
Melvyn B. Nathanson
TL;DR
This work develops continuous analogues of Nathanson's sumset size races for measurable sets in locally compact abelian groups. It combines a discrete-to-continuous transfer via Kravitz-type finite integer sets and a geometric construction in non-torsion groups to realize prescribed patterns of the Haar measure across dilations, proving that target orderings of $\mu(hA_i)$ can be achieved for $h=1$ to $H$. It also provides a real-line construction that realizes specified differences $\mu(hA_{i})-\mu(hA_{i+1})$ and shows how to scale these constructions to achieve arbitrary positive measures. The results deliver strong continuous counterparts to discrete sumset size race theorems, with clear methods for embedding combinatorial data into measurable sets and for solving associated linear systems to realize arbitrary target patterns.
Abstract
Let $G$ be a locally compact abelian group with Haar measure $μ$. 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 $\left( μ(hA_1),\ldots, μ(hA_n) \right)$ 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 $μ(hA_{i+1}) - μ(hA_i) = m_{i,h}$ for all $i$ and $h$.