Abstract
We improve on the Peres–Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with (Forumala Presented)., we prove that for any E, F ⊂ R d , there exists a probability measure μ F on F such that for μ F -a.e. y ∈ F, (Forumala Presented). has positive Lebesgue measure. This describes dimensions of slicing subsets of E, sliced by spheres centered at y. In our proof, local smoothing estimates of Fourier integral operators plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.
| Original language | English |
|---|---|
| Pages (from-to) | 4459-4474 |
| Number of pages | 16 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 371 |
| Issue number | 6 |
| DOIs | |
| State | Published - 15 Mar 2019 |
| Externally published | Yes |
Bibliographical note
Funding Information:Received by the editors September 23, 2017, and, in revised form, July 30, 2018. 2010 Mathematics Subject Classification. Primary 28A75; Secondary 42B20. The second author would like to thank Professor Ka-Sing Lau for the financial research assistantship at Chinese University of Hong Kong. This work was partially supported by NSA Grant H98230-15-1-0319.
Funding Information:
The second author would like to thank Professor Ka-Sing Lau for the financial support of a research assistantship at Chinese University of Hong Kong. This work was partially supported by NSA Grant H98230-15-1-0319.
Publisher Copyright:
© 2018 Amerian Mathematial Soiety.