大点配置的大扩张的近最优密度定理
Near-optimal density theorems for large dilates of large point configurations
AI总结：
研究迫使可测集包含所有足够大相似拷贝的密度阈值，证明下界为1-O((log n)/n)，匹配已知上界，解决Falconer等人提出的问题，同时研究嵌入到ℓ^p空间中的情况，得到渐近精确的上界。
摘要：
我们研究迫使可测集E⊆R^d包含所有足够大相似拷贝的密度阈值。我们证明下界形式为1-O((log n)/n)，这与已知上界在对数因子范围内匹配，从而基本解决了Falconer、Yavicoli和本文第一作者提出的问题。我们还研究了将n点配置嵌入到R^d中并配备ℓ^p范数的情况，得到渐近精确的上界1-1/n+o(1/n)，只要p∈(1,∞)\{2}。在前者估计的证明中使用了多项式序列模1的等分布性结合概率性稀释。后者估计的证明依赖于p≠2的ℓ^p空间的几何性质。
英文摘要：
We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$.
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。
