通过马尔可夫过程生成和推广MSTD集合
Generating and generalizing MSTD sets through Markov processes
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中文总结 AI 辅助
本文通过马尔可夫链框架推广经典MSTD问题，证明和主导、差主导及平衡集合的概率趋于正极限，并给出期望差的闭式表达式。
中文摘要 AI 辅助
经典的“和多于差”（MSTD）问题研究有限集合$A\subset\{0,1,\ldots,n\}$，其中$|A+A|>|A-A|$，这里$A+A=\{a_1+a_2:a_1,a_2\in A\}$，$A-A=\{a_1-a_2:a_1,a_2\in A\}$。由于加法满足交换律而减法不满足，曾有猜想认为当$n\to\infty$时，从$\{0,1,\ldots,n\}$的幂集中均匀随机选取的子集$A$几乎都是差主导的，因此当Martin和O'Bryant证明有一定百分比的集合是和主导时，这令人惊讶。我们通过引入马尔可夫链框架极大地推广了这一模型，其中经典的MSTD模型现在只是一个特例。设$(X_i)_{i=0}^n$为定义在$\{0,1\}$上的平稳两状态马尔可夫链，转移概率为$P(0,0)=p$和$P(1,1)=q$，其中$p,q\in(0,1)$。当且仅当$X_i=1$时，我们将$i$包含在$A$中，并定义$A=\{i\in\{0,\ldots,n\}:X_i=1\}$。当连续的包含决策独立时，即当$p=1-q$时，就恢复了通常的独立伯努利模型。特别地，均匀随机子集模型对应于$p=q=1/2$。利用MSTD文献中的边缘-中间方法，我们证明了中间和与差以高概率被填满，因此$|A+A|$与$|A-A|$的比较再次由端点边缘决定。通过边缘操作，我们证明了和主导、差主导和平衡集合的概率在$n\to\infty$时趋于严格正的极限。我们还给出了在$p$和$q$的一系列取值下，有限$n$时这三个概率的数值估计。通过组合方法，我们找到了当$n\to\infty$时$\mathbb{E}[|A-A|-|A+A|]$的闭式表达式。
英文摘要
The classical More Sums Than Differences (MSTD) problem studies finite sets $A\subset\{0,1,\ldots,n\}$ for which $|A+A|>|A-A|$, where $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A-A=\{a_1-a_2:a_1,a_2\in A\}$. As addition is commutative and subtraction is not, it was conjectured that as $n\to\infty$ almost all subsets $A$ chosen uniformly from the power set of $\{0,1,\ldots,n\}$ are difference-dominated, and it was thus a surprise when Martin and O'Bryant proved a positive percentage of sets are sum-dominant. We greatly generalize this model by introducing a Markov-chain framework, where the classical MSTD model is now just a special case. Let $(X_i)_{i=0}^n$ be a stationary two-state Markov chain on $\{0,1\}$ with transition probabilities $P(0,0)=p$ and $P(1,1)=q$, where $p,q\in(0,1)$. We include $i$ in $A$ exactly when $X_i=1$, and define $A=\{i\in\{0,\ldots,n\}:X_i=1\}$. The usual independent Bernoulli model is recovered when consecutive inclusion decisions are independent, equivalently when $p=1-q$. In particular, the uniformly random subset model corresponds to $p=q=1/2$. Using the fringe-middle method from the MSTD literature, we show that the middle sums and differences are filled with high probability, so the comparison between $|A+A|$ and $|A-A|$ is again governed by endpoint fringes. By fringe manipulation, we prove that the probabilities of sum-dominant, difference-dominant, and balanced sets tend to strictly positive limits as $n\to\infty$. We also give numerical estimates of these three probabilities for finite $n$ over a range of values of $p$ and $q$. Through combinatorial methods, we find a closed-form expression for $\mathbb{E}[|A-A|-|A+A|]$ as $n\to\infty$.
