近似和集大小
Approximating Sumset Size
摘要：
给定$n$维布尔超立方体$\mathbb{F}_2^n$的子集$A$，和集$A+A$是集合$\{a+a': a, a' \in A\}$，其中加法在$\mathbb{F}_2^n$中。和集在加法组合学中起着重要的作用，它们在该领域的许多中心结果中起着重要作用。本文的主要结果是和集大小估计问题的一个次线性时间算法。更详细地说，我们的算法被赋予oracle访问任意$A \subseteq \mathbb{F}_2^n$（的指示函数）和精度参数$ε> 0$，并且对于满足$\mathrm{Vol}(A \setminus A') \leq ε.$的$A$的某些扰动$A' \subseteq A$，它以高概率输出值$0 \leq v \leq 1$，该值$0 \leq v \leq 1$是$\pm ε$ -接近于$\mathrm{Vol}(A' + A')$。容易看出，在不放松处理$A'$而不是$A$的情况下，用于将$\mathrm{Vol}(A+A)$估计到任何非平凡精度的任何算法必须进行$2^{Ω(n)}$查询。相比之下，我们给出了一个算法，其查询复杂度只取决于$ε$和周围的尺寸$n$是完全独立的。
英文摘要：
Given a subset $A$ of the $n$-dimensional Boolean hypercube $\mathbb{F}_2^n$, the sumset $A+A$ is the set $\{a+a': a, a' \in A\}$ where addition is in $\mathbb{F}_2^n$. Sumsets play an important role in additive combinatorics, where they feature in many central results of the field. The main result of this paper is a sublinear-time algorithm for the problem of sumset size estimation. In more detail, our algorithm is given oracle access to (the indicator function of) an arbitrary $A \subseteq \mathbb{F}_2^n$ and an accuracy parameter $ε> 0$, and with high probability it outputs a value $0 \leq v \leq 1$ that is $\pm ε$-close to $\mathrm{Vol}(A' + A')$ for some perturbation $A' \subseteq A$ of $A$ satisfying $\mathrm{Vol}(A \setminus A') \leq ε.$ It is easy to see that without the relaxation of dealing with $A'$ rather than $A$, any algorithm for estimating $\mathrm{Vol}(A+A)$ to any nontrivial accuracy must make $2^{Ω(n)}$ queries. In contrast, we give an algorithm whose query complexity depends only on $ε$ and is completely independent of the ambient dimension $n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
