On the multiparameter Falconer distance problem

Abstract

We study an extension of the Falconer distance problem in the multiparameter setting. Given ℓ ≥ 1 \ell \geq 1 and R d = R d 1 × ⋯ × R d ℓ \mathbb {R}^{d}=\mathbb {R}^{d_1}\times \cdots \times \mathbb {R}^{d_\ell } , d i ≥ 2 d_i\geq 2 . For any compact set E ⊂ R d E\subset \mathbb {R}^{d} with Hausdorff dimension larger than d − min ⁡ ( d i ) 2 + 1 4 d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4} if min ⁡ ( d i ) \operatorname {min}(d_i) is even, d − min ⁡ ( d i ) 2 + 1 4 + 1 4 min ⁡ ( d i ) d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4} +\frac {1}{4\operatorname {min}(d_i)} if min ⁡ ( d i ) \operatorname {min}(d_i) is odd, we prove that the multiparameter distance set of E E has positive ℓ \ell -dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.