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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Pinned distance problem, slicing measures, and local smoothing estimates
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by Alex Iosevich and Bochen Liu PDF
Trans. Amer. Math. Soc. 371 (2019), 4459-4474 Request permission

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
    • © Copyright 2018 American Mathematical Society
    • Journal: Trans. Amer. Math. Soc. 371 (2019), 4459-4474
    • MSC (2010): Primary 28A75; Secondary 42B20
    • DOI: https://doi.org/10.1090/tran/7693
    • MathSciNet review: 3917228