Generalizing the Distribution of Missing Sums in Sumsets

Given a finite set of integers \(A\), its sumset is \(A+A:= \{a_i+a_j \mid a_i,a_j\in A\}\). We examine \(|A+A|\) as a random variable, where \(A\subset I_n = [0,n-1]\), the set of integers from 0 to \(n-1\), so that each element of \(I_n\) is in \(A\) with a fixed probability \(p \in (0,1)\). Recen...

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Bibliographic Details
Published in:arXiv.org 2021-08
Main Authors: Chu, Hung V, King, Dylan, Luntzlara, Noah, Martinez, Thomas C, Miller, Steven J, Shao, Lily, Sun, Chenyang, Xu, Victor
Format: Article
Language:English
Subjects:
Closed form solutions
Exact solutions
Graph theory
Integers
Mathematical analysis
Numerical analysis
Random variables
Online Access:Get full text
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Summary:Given a finite set of integers \(A\), its sumset is \(A+A:= \{a_i+a_j \mid a_i,a_j\in A\}\). We examine \(|A+A|\) as a random variable, where \(A\subset I_n = [0,n-1]\), 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 O'Bryant studied the case in which \(p=1/2\) and found a closed form for \(\mathbb{E}[|A+A|]\). Lazarev, Miller, and O'Bryant 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 \(p\in (0,1)\) and establish good bounds for \(\mathbb{E}[|A+A|]\) and \(m_{n\,;\,p}(k)\). We continue to investigate \(m_{n\,;\,p}(k)\) by studying \(m_p(k) = \lim_{n\to\infty}m_{n\,;\,p}(k)\), 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_{p}(1)
ISSN:2331-8422