高维中的超平面相交与距离集

中文摘要 AI 辅助

我们将Ren和Wang在$\u211d^2$中关于点与线相交的界（见\uc1ecite{RenWan23}）推广到更高维度。我们展示了如何利用这一相交界来改进$\u211d^3$和$\u211d^4$中Falconer距离集问题的最佳已知结果。我们证明，若$d=3$或$d=4$，且$E\subset \R^d$是维数$\dim_H(E) > d/2$的Borel集，则\n\begin{equation*} \sup_{x\in E} \dim_H(\Delta_x(E)) \geq 2/3, \end{equation*}\n其中$\Delta_x(E)$是$E$关于$x$的钉扎距离集。我们还展示了如何利用该相交界将平面Furstenberg集界推广到$\R^d$中的集合，这些集合包含一个$t$维超平面族，每个超平面包含一个$s$维点集，适用于任意$d\ge 2$、$s \in (d-2, d-1]$和$t \in (0, d]$。

英文摘要

We generalize Ren and Wang's incidence bound between points and lines in $\R^2$ \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in $\R^3$ and in $\R^4$. We show that if $d=3$ or $d=4$, and $E\subset \R^d$ is a Borel set of dimension $\dim_H(E) > d/2$, then \begin{equation*} \sup_{x\in E} \dim_H(Δ_x(E)) \geq 2/3, \end{equation*} where $Δ_x(E)$ is the pinned distance set of $E$ with respect to $x$. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in $\R^d$ that contain a $t$-dimensional set of hyperplanes, each of which contains an $s$-dimensional set of points, for any $d\ge 2$, $s \in (d-2, d-1]$ and $t \in (0, d]$.

发表机构

• University of Chicago（芝加哥大学）

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