关于欧几里得空间正密度子集角点的边长
On side lengths of corners in positive density subsets of the Euclidean space
摘要：
我们将Cook，Magyar和Pramanik [3]关于$\mathbb{R}^d$子集中的三项算术级数的结果推广到$\mathbb{R}^d\times\mathbb{R}^d$子集中的角点。更准确地说，如果$1<p<\infty$，$p\neq 2$，$d$是足够大的，我们表明，一个任意的可测集$A\subseteq\mathbb{R}^d\times\mathbb{R}^d$的正上Banach密度包含角落$(x,y)$，$(x+s,y)$，$(x,y+s)$，使$\ell^p$ -范数的侧$s$达到所有足够大的实际值。尽管我们严格遵循[3]的基本步骤，但证明在依赖调和分析的部分存在分歧。我们需要应用[5]中多线性估计的高维变体，我们使用[5]和[6]中的技术建立。
英文摘要：
We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is large enough, we show that an arbitrary measurable set $A\subseteq\mathbb{R}^d\times\mathbb{R}^d$ of positive upper Banach density contains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\ell^p$-norm of the side $s$ attains all sufficiently large real values. Even though we closely follow the basic steps from [3], the proof diverges at the part relying on harmonic analysis. We need to apply a higher-dimensional variant of a multilinear estimate from [5], which we establish using the techniques from [5] and [6].
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。
