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Variants of the Mattila integral, measures with nonnegative Fourier transforms, and the distance set problem


Author: Bassam Shayya
Journal: Trans. Amer. Math. Soc. 368 (2016), 2623-2644
MSC (2010): Primary 42B10; Secondary 28A75
DOI: https://doi.org/10.1090/tran/6394
Published electronically: July 9, 2015
MathSciNet review: 3449251
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Abstract: Suppose $ \mu \in M(\mathbb{R}^n)$ is a measure with $ \Vert \mu \Vert > 0$, $ \sigma $ is surface measure on the unit sphere $ \mathbb{S}^{n-1} \subset \mathbb{R}^n$, and $ \phi \in L^2(\mathbb{S}^{n-1})$ is a function with $ \Vert \phi \Vert _{L^2(\mathbb{S}^{n-1})} > 0$. If $ \Vert \widehat {\mu } \Vert _{L^2(\mathbb{R}^n)} < \infty $, then $ \mbox {supp} \, \mu $ has positive Lebesgue measure. We ask the question, what can we say about $ \mbox {supp} \, \mu $ under the weaker assumption

$\displaystyle \int _0^\infty \Big \vert \int _{\mathbb{S}^{n-1}} \widehat {\mu }(r \theta ) \phi (\theta ) d\sigma (\theta ) \Big \vert^2 r^{n-1} dr < \infty ? $

We give an answer in the case $ \phi \in C^\infty (\mathbb{S}^{n-1})$ and relate our result to Falconer's distance set problem. Our line of investigation naturally leads us to the study of measures with nonnegative Fourier transforms, which we also relate to Falconer's distance set problem in both the Euclidean setting and in vector spaces over finite fields. As an application of our results, we give a new proof of the Erdös-Volkmann ring conjecture.

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Additional Information

Bassam Shayya
Affiliation: Department of Mathematics, American University of Beirut, Beirut, Lebanon
Email: bshayya@aub.edu.lb

DOI: https://doi.org/10.1090/tran/6394
Received by editor(s): February 28, 2013
Received by editor(s) in revised form: November 13, 2013, November 17, 2013, December 22, 2013, and January 26, 2014
Published electronically: July 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society