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 $p = (p_1, \ldots, p_q)$ with $p_1+ \cdots + p_q = d$ is an increasing partition of $d$ define $$B_p(E,F)=\{(|x_1-y_1|, \ldots, |x_q-y_q|): x \in E, y \in F \},$$ where $x=(x_1, \ldots, x_q)$ with $x_i$ in $\mathbb{R}^{p_i}$. For $p_1 \geq 2$ 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 $|B_p(E,E)| \geq C{|E|}^{\frac{\gamma_q}{q}}$. 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 $$|B_{2,2}(E)| \gtrapprox |E|.$$ 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 Erd\H{o}s-distance problem in $\mathbb{R}^m$ for $k \neq l$ we gain a further optimal result of, $$|B_{k,l}(E)| \gtrapprox |E|^{\gamma_l}.$$ 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 $$|B_{k,k}(E)| \gtrapprox |E|^{\frac{13}{14}\gamma_k}.$$ 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 $$B_p(E) \gtrapprox |E|^\tau \hspace{0.5cm} \text{where} \hspace{0.5cm} \tau = \gamma_q\left(\frac{\gamma_1+\eta_1}{\gamma_q+(q-1)(\gamma_1+\eta_1)}\right).$$ 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 $\tau \sim \gamma_{q}\left(\frac{1}{q}+\frac{1}{q^2}\right)$.