Pinned distance problem, slicing measures, and local smoothing estimates

Alex Iosevich, Bochen Liu

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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 languageEnglish
Pages (from-to)4459-4474
Number of pages16
JournalTransactions of the American Mathematical Society
Volume371
Issue number6
DOIs
StatePublished - 15 Mar 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 Amerian Mathematial Soiety.

Funding

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. 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.

FundersFunder number
National Security Agency
Chinese University of Hong KongH98230-15-1-0319

    Fingerprint

    Dive into the research topics of 'Pinned distance problem, slicing measures, and local smoothing estimates'. Together they form a unique fingerprint.

    Cite this