Generalizing the Distribution of Missing Sums in Sumsets

Abstract

We examine |A + A| as a random variable, where A ⊂ In = [0, n − 1], the set of integers from 0 to n − 1, so that each element of In is in A with a fixed probability p ∈ (0, 1). Recently, Martin and O'Bryant studied the case in which p = 1/2 and found a closed form for E[|A + A|]. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for Var(|A + A|) and bounds on mn ; p(k) := P(2n − 1 − |A + A| = k). Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for E [|A+A|] and Var(|A + A|) for all p ∈ (0, 1) and establish good bounds for E[|A + A|] and mn ; p(k).We continue to investigate mn ; p(k) by studying mp(k) = limn→∞ mn ; p(k), proven to exist by Zhao. Lazarev, Miller, and O'Bryant 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 p ≥ 0.68 there is a divot at 1; that is,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 probabilitiesWe provide some preliminary results using the extension of this framework. CONTENTS 1. Introduction 2. Generalizations of [MO] 3. Graph-Theoretic Framework 4. Expected Value 5. Variance 6. Divot Computations 6.1. An Upper Bound on m p (k) 6.2. A Lower Bound on m p (k) 7.