Pinned configuration sets and slicing theorems via Fourier integral operators

Abstract

Let $X\subset {\mathbb{R}}^{d_X}$ and $Y\subset {\mathbb{R}}^{d_Y}$ be open, where $d_X,d_Y\ge 2$ and $1\le p\le \min \{d_X,d_Y\}$. Let $\Phi :X\times Y\to {\mathbb{R}}^p$ be a smooth mapping, and suppose that $D_x\Phi$ and $D_y\Phi$ have rank $p$. For compact sets $A\subset X$ and $B\subset Y$, we give lower bounds on ${\dim }_{H}A+{\dim }_{H}B$ ensuring that the pinned configuration set \[Δ_Φ^y(A)=\{Φ(x,y):x\in A\}\] has positive Lebesgue measure or nonempty interior for some $y\in B$. The dimensional thresholds are expressed in terms of uniform $L^2$-Sobolev estimates for the generalized Radon transforms associated with $\Phi$. Our results extend the Fourier integral operator approach developed by Greenleaf, Iosevich, and Taylor for unpinned $2$-point configuration problems to the pinned setting. We also prove a nonlinear projection theorem that remains valid even when the Peres-Schlag transversality condition fails. Under the smoothing hypotheses used for the pinned configuration theorem, we obtain lower bounds for the Hausdorff dimensions of the corresponding slices of $A$ by the level sets of $\Phi$. Applications include distances from points to hyperplanes, spheres and lines; multiparameter distance sets; and a restricted projection family introduced by Harris.

arXiv comment35 pages