On the multiparameter Falconer distance problem

We study an extension of the Falconer distance problem in the multiparameter setting. Given $ellgeq 1$ and $mathbb{R}^{d}=mathbb{R}^{d_1}timescdots timesmathbb{R}^{d_ell}$, $d_igeq 2$. For any compact set $Esubset mathbb{R}^{d}$ with Hausdorff dimension larger than $d-frac{min(d_i)}{2}+frac{1}{4}$ if $min(d_i) $ is even, $d-frac{min(d_i)}{2}+frac{1}{4}+frac{1}{4min(d_i)}$ if $min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.