论文标题
在飞机上的埃尔德链上
On Erdős Chains in the Plane
论文作者
论文摘要
令$ p $为$ \ mathbb {r}^2 $的有限点设置，距离$ n $ chains定义为$$Δ_n（p）= \ {（| p_1-p_2 |，| p_2-p_2-p_3 |，| p_n-p_n-p_-p_n-p_-p_ {n+1} | p_i p_i p_i p_i p_i we we we n = o_ {| p |}（1）$我们有$ |Δ_n（p）| \ gtrsim \ frac {| p |^{n}} {\ log^{\ log^{\ frac {13} {2} {2} {2}（n-1）（n-1）} | P |}。 Rudnev最近的铰链纸可以在高度相交的Guth-Katz系列嵌套子集上有效迭代。让$ g $是$ m = o（1）$ m \ geq 2 $的简单连接图。将图形距离设置$Δ_g（p）$为$$Δ_g（p）= \ {（| p_ {i} -p_ {j} |）_ {\ {\ {i，j \} \ in E（g）} 结合Guth和Katz和Rudnev的结果与上述结果，如果$ G $有Hamiltonian路径，我们有$ |Δ_G（P）| \ gtrsim \ frac {| p |^{m-1}}} {\ text {polylog} | p |}。 $$ \ end {摘要}
Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ Δ_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|Δ_n(P)|\gtrsim \frac{|P|^{n}}{\log^{\frac{13}{2}(n-1)}|P|}.$$ Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in Rudnev's recent hinge paper which allows one to iterate efficiently on highly intersecting nested subsets of Guth-Katz lines. Let $G$ is a simple connected graph on $m=O(1)$ vertices with $m\geq 2$. Define the graph-distance set $Δ_G(P)$ as $$ Δ_G(P) = \{ (|p_{i}-p_{j}|)_{\{i,j\}\in E(G)} : p_i,p_j \in P\}.$$ Combining with results of Guth and Katz and Rudnev with the above, if $G$ has a Hamiltonian path we have $$ |Δ_G(P)| \gtrsim \frac{|P|^{m-1}}{\text{polylog}|P|}. $$ \end{abstract}
