A multi-parameter variant of the Erd\H{o}s distance problem

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