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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
MathSciNet review: 3917228
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Abstract:

We improve on the Peres–Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with \[ \Delta ^y(E) = \{|x-y|:x\in E\},\] 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. $y\in F$, \[ \left \{t\in {\mathbb {R}} : {\dim _{{\mathcal H}}}(\{x\in E:|x-y|=t\}) \geq {\dim _{{\mathcal H}}}(E)+\frac {d+1}{d-1}{\dim _{{\mathcal H}}}(F)-d \right \} \] 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
    MR Author ID: 356191
    Email: iosevich@math.rochester.edu

    Bochen Liu
    Affiliation: Department of Mathematics, University of Rochester, Rochester, New York
    MR Author ID: 1066951
    Email: bochen.liu@rochester.edu

    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