Title: On the independence number of sparser random Cayley graphs Show Chinese title

Title: 关于更稀疏随机Cayley图的独立数

Abstract: The Cayley sum graph $\Gamma_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $\mathbb{Z}_n$ with $p = 1/2$, then the independence number of $\Gamma_A$ is asymptotically equal to $\alpha(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$\alpha(\Gamma_A) = (1 + o(1)) \alpha(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$. One of the tools in our proof is a geometric-flavoured theorem that generalises Fre\u{i}man's lemma, the classical lower bound on the size of high dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant. Show Chinese abstract

Abstract: Cayley求和图$\Gamma_A$的集合$A \subseteq \mathbb{Z}_n$定义为顶点集$\mathbb{Z}_n$，并且两个不同的顶点$x, y \in \mathbb{Z}_n$之间有一条边，如果$x + y \in A$。 格林和莫里斯证明了，如果集合$A$是$\mathbb{Z}_n$的一个$p$-随机子集且满足$p = 1/2$，那么$\Gamma_A$的独立数几乎必然等于$\alpha(G(n, 1/2))$。 我们的主要定理是他们结果在$p = o(1)$上的首次扩展：我们证明，当$p \ge (\log n)^{-1/80}$时，$$\alpha(\Gamma_A) = (1 + o(1)) \alpha(G(n, p))$$以高概率成立。 我们证明中的一个工具是一个几何风格的定理，该定理推广了 Freĭman 的引理，这是高维和集大小的经典下界。 我们还给出了这个结果的一个常数因子内的简短证明；这个版本在牺牲常数较差的情况下，给出了我们主要定理的一个更简单的证明。