论文标题
DOT产品的Falconer型估计
Falconer-type estimates for dot products
论文作者
论文摘要
我们提出了一个鲜明的鲜明示例，用于猎鹰型单点产品结果。特别是，对于任何$ d \ geq 2，对于任何$ s <\ frac {d+1} {2}，$，我们构建了满足能量估计$ i_s（μ）<\ infty的borel概率度量$μ$ 1+ε\} \ leqcε\ end {equation}与独立于$ε$的常数不存在。众所周知（\ cite {eit11}），如果$ i _ {\ frac {d+1} {2}}}}}}（μ）<\ frac <\ iffty $。因此，我们的估计值证明了在此结果中尺寸阈值的清晰度，并概括了相似的结果（\ cite {mat95}，\ cite {is16}）在点点product $ x \ cdot y $被替换为euclidean距离$ | x-y | | x-y | $ | quork $ fuld culte cully，$ | |一个对称凸面$ k $，具有光滑的边界和非变形曲率。我们的构造部分基于离散发病率理论的想法。
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.
