从加权抛物面限制到k星与距离图
From weighted paraboloid restriction to $k$-stars and distance graphs
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中文总结 AI 辅助
研究与\(\mathbb{R}^n\)紧子集相关的固定k星距离集，通过重新表述联系、简化估计得到维数阈值，利用图构建机制改进相关图配置的正测度阈值，还证明了k星的非空内部结果，特殊情况\(k = 1\)时改进了固定非空内部阈值。
中文摘要 AI 辅助
本文研究与\(\mathbb{R}^n\)（\(n\geq2\)）的紧子集相关的固定k星距离集。对于\(x_1,\dots,x_k\in E\)这些固定点，固定k星距离集为\(\Delta_{x_1,\dots,x_k}^{k\text{-star}}(E) = \{(|x_1 - x|,\dots,|x_k - x|):x\in E\}\subset\mathbb{R}^k\)。我们得到了关于\(E\)的改进的豪斯多夫维数阈值，确保固定k星距离集具有正的k维勒贝格测度。主要分析输入是对\(\mathbb{R}^n\)中的k星与\(\mathbb{R}^{n + 1}\)中抛物面上的固定点积之间联系的重新表述。在我们的框架中，固定k星距离测度密度的\(L^2(\mathbb{R}^k)\)估计被简化为抛物面的加权傅里叶扩展估计，其权重根据\(E\)上的弗罗斯特曼测度明确定义。对于\(1\leq k < n\)，得到阈值\(\dim(E)>\alpha_{+}(n,k):=\frac{n^2 + nk + k}{2n + 1}=\frac{n + k - 1}{2}+\frac{1}{4}+\frac{2k + 1}{4(2n + 1)}\)。利用相关的图构建机制，k星的正测度结果可用于具有规定固定点的有限距离图配置。结果改进了\(n\geq3\)时固定k单形和\(n\geq3\)时项链图（环）的最佳已知正测度阈值。还证明了k星的非空内部结果。在\(k = 1\)的特殊情况下，对应距离集\(\Delta_{x}(E)=\{|x - y|\colon y\in E\}\)的固定非空内部，使用更精确的论证改进了\(n\geq4\)时的固定非空内部阈值。
英文摘要
In this paper, we study pinned $k$-star distance sets associated to compact subsets of $\mathbb{R}^n$, $n\geq 2$. For pins $x_1,\dots,x_k\in E$, the pinned $k$-star distance set is \[ Δ_{x_1,\dots,x_k}^{k\text{-star}}(E) = \{(|x_1-x|,\dots,|x_k-x|):x\in E\}\subset\mathbb{R}^k. \] We obtain improved Hausdorff-dimension thresholds on $E$ guaranteeing that pinned $k$-star distance sets have positive $k$-dimensional Lebesgue measure. The main analytic input is a reformulation of the connection, first observed in \cite{IPPS22}, between $k$-stars in $\mathbb{R}^n$ and pinned dot products on the paraboloid in $\mathbb{R}^{n+1}$. In our framework, $L^2(\mathbb{R}^k)$ estimates for the densities of pinned $k$-star distance measures are reduced to a weighted Fourier extension estimate for the paraboloid whose weight is defined explicitly in terms of Frostman measures on $E$. For $1\leq k<n$, this yields the threshold \[\dim(E)>α_{+}(n,k):=\frac{n^2+nk+k}{2n+1}=\frac{n+k-1}{2}+\frac14 +\frac{2k+1}{4(2n+1)}.\] Using the graph-building machinery of \cite{BFOPR2026}, our positive-measure results for $k$-stars can be used as building blocks for finite distance graph configurations with prescribed pins. As a consequence, we improve the best-known positive-measure thresholds for pinned $k$-simplices in every dimension $n\geq 3$ and for necklace graphs (cycles) in every dimension $n\geq 3$. We further prove nonempty interior results for $k$-stars. In the special case $k=1$, corresponding to the pinned nonempty interior of the distance set $Δ_{x}(E)=\{|x-y|\colon y\in E\}$, we use a sharper argument to improve the pinned nonempty-interior thresholds of \cite{BFOP2026} in all dimensions $n\geq 4$.
