Counting sets with small sumset and applications

TL;DR

This paper analyzes the number of small sumset sets within certain bounds, improves existing results, and applies findings to random Cayley graphs and sumset properties in probabilistic settings.

Contribution

It provides a near-complete enumeration of sets with small sumsets and applies these results to graph theory and probabilistic sumset analysis.

Findings

Number of sets with small sumset determined up to a factor of $2^{o(k)} N^{o(1)}$

Random Cayley graph on $ extbf{Z}/N extbf{Z}$ has no large clique or independent set beyond $(2+o(1)) ext{log}_2 N$

Probability that sumset misses exactly $k$ elements is asymptotically $(2+o(1))^{-k/2}$

Abstract

We study the number of $k$-element sets $A \subset \{1,\ldots,N\}$ with $|A + A| \leq K|A|$ for some (fixed) $K > 0$. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of $2^{o(k)} N^{o(1)}$ for most $N$ and $k$. As a consequence of this and a further new result concerning the number of sets $A \subset \mathbf{Z}/N\mathbf{Z}$ with $|A +A| \leq c |A|^2$, we deduce that the random Cayley graph on $\mathbf{Z}/N\mathbf{Z}$ with edge density~$\frac{1}{2}$ has no clique or independent set of size greater than $\big( 2 + o(1) \big) \log_2 N$, asymptotically the same as for the Erd\H{o}s-R\'enyi random graph. This improves a result of the first author from 2003 in which a bound of $160 \log_2 N$ was obtained. As a second application, we show that if the elements of $A \subset \mathbf{N}$ are chosen at random, each with probability $1/2$, then the probability that $A+A$ misses exactly $k$ elements of $\mathbf{N}$ is equal to $\big( 2 + o(1) \big)^{-k/2}$ as $k \to \infty$.