A multi-parameter variant of the Erdős distance problem

Abstract

We study the following variant of the Erdős distance problem. Given $E$ and $F$ a point sets in ${\mathbb{R}}^d$ and $p=(p_1,\dots ,p_q)$ with $p_1+\cdots +p_q=d$ is an increasing partition of $d$ define $$B_p(E,F)=\{(|x_1-y_1|,\dots ,|x_q-y_q|):x\in E,y\in F\},$$ where $x=(x_1,\dots ,x_q)$ with $x_i$ in ${\mathbb{R}}^{p_i}$. For $p_1\ge 2$ it is not difficult to construct $E$ and $F$ such that $|B_p(E,F)|=1$. On the other hand, it is easy to see that if ${\gamma }_q$ is the best know exponent for the distance problem in ${\mathbb{R}}^{p_i}$ that $|B_p(E,E)|\ge C{|E|}^{\frac{{\gamma }_q}{q}}$. The question we study is whether we can improve the exponent $\frac{{\gamma }_q}{q}$. We first study partitions of length two in detail and prove the optimal result (up to logarithms) that $$|B_{2,2}(E)|⪆|E|.$$ In the generalised two dimensional case for $B_{k,l}$ we need the stronger condition that $E$ is $s$-adaptable for $s<\frac{k}{2}+\frac{1}{3}$, letting ${\gamma }_m$ be the best known exponent for the Erdős-distance problem in ${\mathbb{R}}^m$ for $k\ne l$ we gain a further optimal result of, $$|B_{k,l}(E)|⪆|E{|}^{{\gamma }_l}.$$ When $k=l$ we use the explicit ${\gamma }_m=\frac{m}{2}-\frac{2}{m(m+2)}$ result due to Solymosi and Vu to gain $$|B_{k,k}(E)|⪆|E{|}^{\frac{13}{14}{\gamma }_k}.$$ For a general partition, let ${\gamma }_i=\frac{2}{p_i}-\frac{2}{p_i(p_i+2)}$ and ${\eta }_i=\frac{2}{2d-(p_i-1)}$. Then if $E$ is $s$-adaptable with $s>d-\frac{p_1}{2}+\frac{1}{3}$ we have $$B_p(E)⪆|E{|}^{\tau }\text{where}\tau ={\gamma }_q\left(\frac{{\gamma }_1+{\eta }_1}{{\gamma }_q+(q-1)({\gamma }_1+{\eta }_1)}\right).$$ Where $p_i\sim \frac{d}{q}$ implies $\tau \sim {\gamma }_q\left(\frac{1}{q}+\frac{1}{dq}\right)$ and $p_q\sim d$ (with $q<<d$) implies $\tau \sim {\gamma }_q\left(\frac{1}{q}+\frac{1}{q^2}\right)$.

19 Pages