On growth rates of infinite and finite sumsets

Abstract

We study growth rates of infinite and finite sumset patterns in sets of positive density. In the infinite setting, we show that no such rate exists, answering a question of Kra, Moreira, Ritcher, and Robertson. Namely, for any proposed growth rate $\mathcal{H}:\mathbb{N}\to \mathbb{N}$ tending to infinity, we construct a set $A$ of lower density $1$ such that whenever $B,C\subseteq \mathbb{N}$ are infinite and $B+C\subseteq A$ we have the minimum of $|B\cap [N]|$ and $|C\cap [N]|$ is less than $\mathcal{H}(N)$ for infinitely many $N$. In the finitary setting, we prove that for all $\delta \in (0,1)$, for all sufficiently large $N$, for all subsets $A$ of $\{1,\dots ,N\}$ of proportion $\delta$, one can always find sumset patterns $B+C\subseteq A$ with $|B|$ and $|C|$ of order $\log N$, partially resolving a conjecture of Kra, Moreira, Richter, and Robertson. Moreover, we generalize our second result to the case of the $k$-fold sum $B_1+B_2+\dots +B_k\subseteq A$.

18 pages. Comments are welcome!