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 $\epsilon > 0$, and with high probability it outputs a value $0 \leq v \leq 1$ that is $\pm \epsilon$-close to $\mathrm{Vol}(A' + A')$ for some perturbation $A' \subseteq A$ of $A$ satisfying $\mathrm{Vol}(A \setminus A') \leq \epsilon.$ 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 $2^{\Omega(n)}$ queries. In contrast, we give an algorithm whose query complexity depends only on $\epsilon$ and is completely independent of the ambient dimension $n$.

中文翻译：

---

近似总集大小

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发挥着重要作用，在该领域的许多核心结果中都有它们的特征。本文的主要成果是针对和集大小估计问题的次线性时间算法。更详细地说，我们的算法被授予 oracle 访问（的指标函数）任意 $A \subseteq \mathbb{F}_2^n$ 和精度参数 $\epsilon > 0$，并且很有可能它会输出一个值 $0 \leq v \leq 1$ 即 $\pm \epsilon$-接近 $\mathrm{Vol}(A' + A')$ 一些扰动 $A' \subseteq A$ of $A$ 满足 $ \mathrm{Vol}(A \setminus A') \leq \epsilon. $ 很容易看出，如果不放松处理 $A'$ 而不是 $A$，任何估计 $\mathrm{Vol}(A+A)$ 到任何非平凡精度的算法都必须使 $2^{\Omega (n)}$ 查询。相比之下，我们给出了一个算法，其查询复杂度仅取决于 $\epsilon$，并且完全独立于环境维度 $n$。