论文标题
概括丢失总和中的分布
Generalizing the Distribution of Missing Sums in Sumsets
论文作者
论文摘要
给定一组有限的整数$ a $，其集合为$ a+a：= \ {a_i+a_j \ mid a_i，a_j \ in a \} $。我们检查$ | a+a | $作为一个随机变量，其中$ a \ subset i_n = [0，n-1] $，从0到$ n-1 $的整数，因此$ i_n $的每个元素均在$ a $ a $ a $ a中，固定概率$ p \ in（0,1）$。最近，Martin和O'Bryant研究了$ P = 1/2 $的情况，并找到了$ \ Mathbb {e} [| a+a |] $的封闭式。 Lazarev，Miller和O'Bryant扩展了结果，以找到$ \ text {var}（| a+a |）$的数值估计，并在$ a+a+a $ a $ a+a $，$ m_ {n \，; \，; \，; \，p}（k）（k）中的丢失总和的数量进行界定。他们的主要工具是一个图理论框架，我们现在将其概括为$ \ mathbb {e} [| a+a+a |] $和$ \ text {var}（| a+a |）$ in（0,1）$，并为$ p \ in（0,1）$建立$ \ mathbb {e | a | a | a | a | a | a |] $ m_ {n \，; \，p}（k）$。 我们继续研究$ m_ {n \，; \，p}（k）$，通过研究$ m_p（k）= \ lim_ {n \ to \ infty} m_ {n \ n \，; \，; \，p}（k）$，已被Zhao证明存在。 Lazarev，Miller和O'Bryant证明，对于$ p = 1/2 $，$ m_ {1/2}（6）> m_ {1/2}（7）<m_ {1/2}（8）$。这种分布不是单峰的，据说在7时具有“ divot”。我们报告结果调查了该divot为$ p $各种，并且通过理论和数值分析都证明，对于$ p \ geq 0.68 $，divot售价为$ 1 $;也就是说，$ m_ {p}（0）> m_ {p}（1）<m_ {p}（2）$。 最后，我们将最初由Lazarev，Miller和O'Bryant引入的图理论框架扩展到相关的总和$ a+b $，其中$ b $与$ \ Mathbb {p}（p}（p}（i \ in Mid I \ in A）= p_1 $和$ him b}的概率$ \ mathbb {p}（i \ in Mid i \ in Med MathB}） A）= P_2 $。我们使用该框架的扩展提供了一些初步结果。
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 $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)<m_{1/2}(8)$. 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 $p\geq 0.68$ 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 $\mathbb{P}(i\in B \mid i\not\in A) = p_2$. We provide some preliminary results using the extension of this framework.
