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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Group actions, the Mattila integral and applications


Author: Bochen Liu
Journal: Proc. Amer. Math. Soc. 147 (2019), 2503-2516
MSC (2010): Primary 28A75, 42B20
DOI: https://doi.org/10.1090/proc/14406
Published electronically: February 14, 2019
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Abstract: The Mattila integral,

$\displaystyle {\mathcal M}(\mu )=\int {\left ( \int _{S^{d-1}} {\vert\widehat {\mu }(r \omega )\vert}^2 d\omega \right )}^2 r^{d-1} dr,$    

developed by Mattila, is the main tool in the study of the Falconer distance conjecture. In this paper we develop a generalized version of the Mattila integral that works on more general Falconer-type problems. As applications, we consider when the product of distance set

$\displaystyle (\Delta (E))^k= \left \{\prod _{j=1}^k \vert x^j-y^j\vert: x^j, y^j\in E\subset \mathbb{R}^d\right \}$    

has positive Lebesgue measure and when the sum-product set

$\displaystyle E\cdot (F+H)=\{x\cdot (y+z): x\in E\subset \mathbb{R}^2, y\in F\subset \mathbb{R}^2, z\in H\subset \mathbb{R}^2\},$    

has positive Lebesgue.

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Additional Information

Bochen Liu
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
Email: bochen.liu1989@gmail.com

DOI: https://doi.org/10.1090/proc/14406
Received by editor(s): January 17, 2018
Received by editor(s) in revised form: August 19, 2018
Published electronically: February 14, 2019
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society