Intersections of sets and Fourier analysis

A classical theorem due to Mattila says that if A , B ⊂ ℝ d of Hausdorff dimension s A , s B respectively with s A + s B ≥ d , s B > ( d + 1)/2, and dim H ( A × B ) = s A + s B ≥ d , then for almost every z ∈ ℝ d , in the sense of Lebesgue measure. In this paper, we replace the Hausdorff dimensio...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2016-02, Vol.128 (1), p.159-178
Main Authors: Eswarathasan, Suresh, Iosevich, Alex, Taylor, Krystal
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Arithmetic
Dynamical Systems and Ergodic Theory
Formulas (mathematics)
Fourier analysis
Functional Analysis
Inequalities
Intersections
Mathematics
Mathematics and Statistics
Partial Differential Equations
Sharpness
Texts
Theorems
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Summary:A classical theorem due to Mattila says that if A , B ⊂ ℝ d of Hausdorff dimension s A , s B respectively with s A + s B ≥ d , s B > ( d + 1)/2, and dim H ( A × B ) = s A + s B ≥ d , then for almost every z ∈ ℝ d , in the sense of Lebesgue measure. In this paper, we replace the Hausdorff dimension on the left hand side of the first inequality above by the lower Minkowski dimension and replace the Lebesgue measure of the set of translates by a Hausdorff measure on a set of sufficiently large dimension. Interesting arithmetic issues arise in the consideration of sharpness examples.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-016-0004-1