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Intersections of sets and Fourier analysis

Eswarathasan, Suresh, Iosevich, Alex and Taylor, Krystal 2016. Intersections of sets and Fourier analysis. Journal d'Analyse Mathématique 128 , pp. 159-178. 10.1007/s11854-016-0004-1

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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 dimH(A∩(z+B)⩽sA+sB−d dimH(A∩(z+B)⩽sA+sB−d 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Additional Information: Pdf uploaded in accordance with publisher's policy at (accessed 22/06/2016)
Publisher: Springer Verlag (Germany)
ISSN: 0021-7670
Date of First Compliant Deposit: 30 March 2016
Last Modified: 10 Mar 2020 19:25

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