Falconer's problem for dot product on paraboloids

Abstract: We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in $\mathbb{R}^3$, and $ \frac{d}{2} - \frac{1}{4} - \frac{1}{8d - 4} $ in $\mathbb{R}^d$ for $d\geq 4$, its dot product set has positive Lebesgue measure. This result demonstrates that if a compact set in $ \mathbb{R}^d $ exhibits a paraboloidal structure, then the usual dimensional barrier of $ \frac{d}{2} $ for dot product sets can be lowered for $ d \geq 3 $. Our work serves as the continuous counterpart of a paper by Che-Jui Chang, Ali Mohammadi, Thang Pham, and Chun-Yen Shen, which examines the finite field setting with partial reliance on the extension conjecture. The key idea, closely following their paper, is to reformulate the dot product set on the paraboloid as a variant of a distance set. This reformulation allows us to leverage state-of-the-art results from the pinned distance problem, as established by Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang for $ d = 2 $, and by Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang for higher dimensions. Finally, we present explicit constructions and existence proofs that highlight the sharpness of our results.