Limiting Behavior in Missing Sums of Sumsets

Abstract

We study $|A + A|$ as a random variable, where $A \subseteq \{0, \dots, N\}$ is a random subset such that each $0 \le n \le N$ is included with probability $0 < p < 1$, and where $A + A$ is the set of sums $a + b$ for $a,b$ in $A$. Lazarev, Miller, and O'Bryant studied the distribution of $2N + 1 - |A + A|$, the number of summands not represented in $A + A$ when $p = 1/2$. A recent paper by Chu, King, Luntzlara, Martinez, Miller, Shao, Sun, and Xu generalizes this to all $p\in (0,1)$, calculating the first and second moments of the number of missing summands and establishing exponential upper and lower bounds on the probability of missing exactly $n$ summands, mostly working in the limit of large $N$. We provide exponential bounds on the probability of missing at least $n$ summands, find another expression for the second moment of the number of missing summands, extract its leading-order behavior in the limit of small $p$, and show that the variance grows asymptotically slower than the mean, proving that for small $p$, the number of missing summands is very likely to be near its expected value.

Figures (6)

• Figure 1: Probabilities of non-inclusion of $n$ into $A + A$ for $0 \le n \le 2N$ and $p = 1/2$, $N = 40$.
• Figure 2: Probabilities of missing more than $n$ summands on the left fringe for $p = 1/2$ and $N = 200$. Monte Carlo simulation (MC) with $1 \times 10^6$ trial runs, theoretical upper bound (UB) from Corollary \ref{['corr:better bound on prob of missing many']}, and theoretical lower bound (LB) from \ref{['eq:missing more than n lower bound']} are shown. Here, $\beta$ is the slope of $\log(\mathbb{P}\left(Y \ge n\right))$ as $n$ increases, equal to $\log{\sqrt{1 - p}}$ for LB and $\log{\lambda_1}$ for UB (see eq. \ref{['eq:defn lambda12']}). For MC, $\beta$ is estimated numerically by fitting a least-squares line, which entails not only random error but also systematic error because the original curve seems to be concave.
• Figure 3: Second moment $\mathbb{E}\left[Y^2\right]$ of the number of missing summands on the left fringe: theoretical prediction \ref{['eq:second moment']}, Monte Carlo values, and the $4 p^{-4}$ approximation \ref{['eq:2p-4 approx']} (left); discrepancies between Monte Carlo values $\mathbb{E}\left[Y^2\right]_{\rm mc}$ and the infinite sum $\mathbb{E}\left[Y^2\right]_{\rm is}$ from \ref{['eq:second moment']}, together with what we expect the discrepancies to be based on simulation size and $N$ (right).
• Figure 4: The cumulative distribution function of $Y$, normalized by $\mathbb{E}\left[Y\right]$, for $N = 800$ and $p = 0.05, 0.08, 0.16, 0.24, 0.32$. (Monte Carlo simulation.)
• Figure 5: Snail diagram for $m = 17$, $n = 13$.

Theorems & Definitions (59)

• Theorem 1.0
• Corollary 1.0
• Corollary 1.0
• Proposition 1.0
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof