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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Pinned distance problem, slicing measures, and local smoothing estimates


Authors: Alex Iosevich and Bochen Liu
Journal: Trans. Amer. Math. Soc. 371 (2019), 4459-4474
MSC (2010): Primary 28A75; Secondary 42B20
DOI: https://doi.org/10.1090/tran/7693
Published electronically: November 19, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: We improve on the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with

$\displaystyle \Delta ^y(E) = \{\vert x-y\vert:x\in E\},$

we prove that for any $ E, F\subset {\mathbb{R}}^d$, there exists a probability measure $ \mu _F$ on $ F$ such that for $ \mu _F$-a.e. $ y\in F$,

  • $ {\dim _{{\mathcal H}}}(\Delta ^y(E))\geq \beta $ if $ {\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d-1+\beta $,
  • $ \Delta ^y(E)$ has positive Lebesgue measure if $ {\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d$,
  • $ \Delta ^y(E)$ has nonempty interior if $ {\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d+1$.

We also show that in the case in which $ {\dim _{{\mathcal H}}}(E)+\frac {d-1}{d+1}{\dim _{{\mathcal H}}}(F)>d$, for $ \mu _F$-a.e. $ y\in F$,

$\displaystyle \left \{t\in {\mathbb{R}} : {\dim _{{\mathcal H}}}(\{x\in E:\vert... ...\dim _{{\mathcal H}}}(E)+\frac {d+1}{d-1}{\dim _{{\mathcal H}}}(F)-d \right \} $

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.


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Additional Information

Alex Iosevich
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York
Email: iosevich@math.rochester.edu

Bochen Liu
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York
Email: bochen.liu@rochester.edu

DOI: https://doi.org/10.1090/tran/7693
Received by editor(s): September 23, 2017
Received by editor(s) in revised form: July 30, 2018
Published electronically: November 19, 2018
Additional Notes: 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
Article copyright: © Copyright 2018 American Mathematical Society