Generalizing the Distribution of Missing Sums in Sumsets

Given a finite set of integers $A$, its sumset is $A+A:= {a_i+a_j \mid a_i,a_j\in A}$\mathrm{.}Weexamine$|A+A|$asarandomvariable,where$A\subset I_n = [0,n-1]$,thesetofintegersfrom0to$n-1$,sothateachelementof$I_n$ is in $A$ with a fixed probability $p\in (0,1)$. Recently, Martin and O’Bryant studied the case in which $p=1\mathrm{/}2$ and found a closed form for $\mathbb{E}[\mathrm{\mid }A+A\mathrm{\mid }]$. Lazarev, Miller, and O’Bryant extended the result to find a numerical estimate for $\text{Var}(\mathrm{\mid }A+A\mathrm{\mid })$ and bounds on the number of missing sums in $A+A$, $m_{n;p}(k):=\mathbb{P}(2n-1-\mathrm{\mid }A+A\mathrm{\mid }=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $\mathbb{E}[\mathrm{\mid }A+A\mathrm{\mid }]$ and $\text{Var}(\mathrm{\mid }A+A\mathrm{\mid })$ for all $p\in (0,1)$andestablishgoodboundsfor$$\mathbb{E}$[|A+A|]$and$m_{n,;,p}(k)$. We continue to investigate $m_{n;p}(k)$ by studying $m_p(k) = \lim_{n\to\infty}m_{n,;,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and O’Bryant proved that, for $p=1\mathrm{/}2$, $m_{1\mathrm{/}2}(6)>m_{1\mathrm{/}2}(7)<m_{1\mathrm{/}2}(8)$. This distribution is not unimodal, and is said to have a “divot” at 7. We report results investigating this divot as $p$ varies, and through both theoretical and numerical analysis, prove that for $p\ge 0.68$ there is a divot at $1$; that is, $m_p(0)>m_p(1)<m_p(2)$. Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O’Bryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $\mathbb{P}(i\in B\mid i\in A)=p_1$ and $\mathbb{P}(i\in B\mid i\in ̸A)=p_2$. We provide some preliminary results using the extension of this framework.