Pinned configuration sets and slicing theorems via Fourier integral operators
Abstract
Let X⊂ℝdX and Y⊂ℝdY be open, where dX,dY≥2 and 1≤ p≤min{dX,dY}. Let Φ:X× Y→ℝp be a smooth mapping, and suppose that DxΦ and DyΦ have rank p. For compact sets A⊂ X and B⊂ Y, we give lower bounds on dimH A+dimH B ensuring that the pinned configuration set ΔΦy(A)={Φ(x,y):x∈ A} has positive Lebesgue measure or nonempty interior for some y∈ B. The dimensional thresholds are expressed in terms of uniform L2-Sobolev estimates for the generalized Radon transforms associated with Φ. 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 Φ. Applications include distances from points to hyperplanes, spheres and lines; multiparameter distance sets; and a restricted projection family introduced by Harris.
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BibTeX
@misc{greenleaf2026pinned,
title = {{Pinned configuration sets and slicing theorems via Fourier integral operators}},
author = {Allan Greenleaf and Eyvindur Ari Palsson and Minh-Quy Pham and Georgios Psaromiligkos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32088},
eprint = {2610.32088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32088},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pinned configuration sets and slicing theorems via Fourier integral operators
AU - Allan Greenleaf
AU - Eyvindur Ari Palsson
AU - Minh-Quy Pham
AU - Georgios Psaromiligkos
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.32088
UR - https://arxiv.org/abs/2610.32088
AB - Let X⊂ℝ^d_X and Y⊂ℝ^d_Y be open, where d_X,d_Y≥2 and 1≤ p≤min{d_X,d_Y}. Let Φ:X× Y→ℝ^p be a smooth mapping, and suppose that D_xΦ and D_yΦ have rank p. For compact sets A⊂ X and B⊂ 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∈ A} has positive Lebesgue measure or nonempty interior for some y∈ B. The dimensional thresholds are expressed in terms of uniform L^2-Sobolev estimates for the generalized Radon transforms associated with Φ. 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 Φ. Applications include distances from points to hyperplanes, spheres and lines; multiparameter distance sets; and a restricted projection family introduced by Harris.
ER -