Abstract
A classical theorem due to Mattila says that if A, B ⊂ ℝd 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 ∈ ℝd, 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.
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The first listed author was supported by a CRM-ISM Postdoctoral Fellowship and McGill University during the first part of the writing of this article, with the second part written while as a resident at the Institute des Hautes Études Scientifiques.
The work of the second listed author was partially supported by the NSF Grant DMS10-45404.
The third listed author was supported by the Technion Technical Institute during the first part of the writing of this article, with second part written while at the Institute for Mathematics and its Applications.
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Eswarathasan, S., Iosevich, A. & Taylor, K. Intersections of sets and Fourier analysis. JAMA 128, 159–178 (2016). https://doi.org/10.1007/s11854-016-0004-1
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DOI: https://doi.org/10.1007/s11854-016-0004-1