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Texas A&M University
Mathematics

Number Theory Seminar

Date: November 10, 2022

Time: 2:30PM - 3:30PM

Location: BLOC 302

Speaker: Hung Viet Chu, University of Illinois at Urbana-Champaign

  

Title: Divots in the distribution of missing sums in sumsets & more-sum-than-difference sets

Abstract: We begin by talking about missing sums in sumsets. For a finite set $A$ of integers, define the sumset $A + A := \{a+b: a, b\in A\}$ and the difference set $A - A := \{a-b: a, b\in A\}$. Consider the probability model where $A$ is formed by choosing each integer in $\{0, 1, \ldots, n-1\}$ with probability $p$. Note that if $A = \{0,1,\ldots, n-1\}$, then $|A + A| = 2n-1$. Hence, the random variable $X := 2n-1 - |A+A|$ counts the number of missing sums in the sumset of $A$. Let $\mathbb{P}_{p, n}(|X| = k)$ denote the probability that we observe $k$ missing sums. This probability depends on both $p$ and $n$. In 2011, Zhao proved that $\mathbb{P}_{p}(|X| = k) := \lim_{n\rightarrow\infty}\mathbb{P}_{p, n}(|X|= k)$ exists. In 2013, Lazarev, Miller, and O’Bryant showed an interesting behavior in the distribution of missing sums in the uniform model: $$\mathbb{P}_{1/2}(|X| = 6) \ >\ \mathbb{P}_{1/2}(|X| = 7) \ <\ \mathbb{P}_{1/2}(|X| = 8).$$ This result says that in the uniform model, it is less likely to miss $7$ sums than both to miss $6$ sums and to miss $8$ sums. We call $7$ a divot when $p = 1/2$ and investigate whether a divot can happen earlier when $p$ varies. We proved $$\mathbb{P}_{p}(|X| = 0) \ >\ \mathbb{P}_{p}(|X| = 1) \ <\ \mathbb{P}_{p}(|X| = 2),\forall p \geqslant 0.68.$$ Next, we move on to talk about more-sum-than-difference (MSTD) sets, which are sets $A$ satisfying $|A + A| > |A - A|$. In 2007, Martin and O’Bryant proved a surprising result that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{0, 1, \ldots, n-1\}$ as $n\rightarrow\infty$. We proved that for any $k\geqslant 2$, it is possible to partition $\{0, 1, \ldots, n-1\}$ into $k$ MSTD subsets and provide bounds on the smallest $n$ for such partitions to take place. This answered a question by Asada, Manski, Miller, and Suh in 2017. The key idea in both of the above results is fringe analysis, which centers around the observation that it is much more likely to miss a sum in the two end