点积的Falconer型估计
Falconer-type estimates for dot products
摘要：
我们提出了一个家庭的清晰度的例子猎鹰型单点产品的结果。特别地，对于任何$s<\frac{d+1}{2},$的$d\geq 2,$，我们构造满足能量估计$I_s(μ)<\infty,$的Borel概率测度$μ$，但估计\begin{equation} (μ\times μ)\{(x,y):1\leq x\cdot y \leq 1+ε\} \leq Cε\end{equation}不与$ε$无关的常数保持。已知（\cite{EIT11}），如果$I_{\frac{d+1}{2}}(μ)<\infty$，则这样的估计总是与$C$独立于$ε$。因此，我们的估计证明了在这个结果中的尺寸阈值的锐度，并推广了类似的结果（\cite{Mat95}，\cite{IS16}）建立的情况下，当点积$x \cdot y$被替换为欧几里德距离函数$|x-y|$，或更一般地说，${||x-y||}_K$，距离来自规范引起的对称凸体$K$具有光滑的边界和非零曲率。我们的结构部分基于来自离散关联理论的想法。
英文摘要：
We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $μ$ satisfying the energy estimate $I_s(μ)<\infty,$ yet the estimate \begin{equation} (μ\times μ)\{(x,y):1\leq x\cdot y \leq 1+ε\} \leq Cε\end{equation} does not hold with constants independent of $ε$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $ε$ if $I_{\frac{d+1}{2}}(μ)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and generalizes similar results (\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x \cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more generally, ${||x-y||}_K$, the distance that comes from the norm induced by a symmetric convex body $K$ with a smooth boundary and non-vanishing curvature. Our constructions are partially based on ideas that come from discrete incidence theory.
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。
