论文标题
远端组的近似亚组中的无产品集
Product-free sets in approximate subgroups of distal groups
论文作者
论文摘要
回想一下，如果$ x^2 \ cap x = \ varnothing $，则$ g $的子集$ x $是'无产品'，即如果$ xy \ notin x $ for All $ x，y \ in x $。令$ g $为远端结构可定义的组。我们证明有常数$ c> 0 $和$δ\ in（0,1）$，这样每个有限的子集$ x \ subseteq g $与$ \ {1 \} $不同的，包含的无产品子集至少$δ| x | x | x | x |^{c+1}/| x^2 |^2 |^c $。特别是，$ g $的每个有限$ k $ - 与$ \ {1 \} $不同的子组包含一个无产品密度子集，至少包含$Δ/k^c $。证明很短，从Ruzsa微积分以及Chernikov和Starchenko的远端规则性引理的迭代应用。
Recall that a subset $X$ of a group $G$ is 'product-free' if $X^2\cap X=\varnothing$, ie if $xy\notin X$ for all $x,y\in X$. Let $G$ be a group definable in a distal structure. We prove there are constants $c>0$ and $δ\in(0,1)$ such that every finite subset $X\subseteq G$ distinct from $\{1\}$ contains a product-free subset of size at least $δ|X|^{c+1}/|X^2|^c$. In particular, every finite $k$-approximate subgroup of $G$ distinct from $\{1\}$ contains a product-free subset of density at least $δ/k^c$. The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma.
