Generalizing the Distribution of Missing Sums in Sumsets

Given a finite set of integers $A$, its sumset is $A+A:= {a_i+a_j mid a_i,a_jin A}$. We examine $|A+A|$ as a random variable, where $Asubset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p in (0,1)$. Recently, Martin and OBryant studied the case in which $p=1/2$ and found a closed form for $mathbb{E}[|A+A|]$. Lazarev, Miller, and OBryant extended the result to find a numerical estimate for $text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n,;,p}(k) := mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $mathbb{E}[|A+A|]$ and $text{Var}(|A+A|)$ for all $pin (0,1)$ and establish good bounds for $mathbb{E}[|A+A|]$ and $m_{n,;,p}(k)$. We continue to investigate $m_{n,;,p}(k)$ by studying $m_p(k) = lim_{ntoinfty}m_{n,;,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and OBryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)<m_{1/2}(8)$. 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 $pgeq 0.68$ 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 OBryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $mathbb{P}(iin B mid iin A) = p_1$ and $mathbb{P}(iin B mid i otin A) = p_2$. We provide some preliminary results using the extension of this framework.