On Erdős Chains in the Plane

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}

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