Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 37, 2012, 557-562
University of Rochester, Department of Mathematics
Rochester, NY 14627, U.S.A.; iosevich 'at' math.rochester.edu
UMR 8628 Université Paris-Sud 11-CNRS, Département de Mathématiques
Université Paris-Sud 11, F-91405 Orsay Cedex, France;
mihalis.mourgoglou 'at' math.u-psud.fr
University of Rochester, Department of Mathematics
Rochester, NY 14627, U.S.A.; taylor 'at' math.rochester.edu
Abstract. We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set E \subset Rd, d \ge 2, is greater than (d + 1) / 2, then the distance set \Delta(E) = {|x - y| : x,y \in E} contains an interval. We prove this result for distance sets \DeltaB(E) = {|x - y| : x,y \in E}, where {|.|}B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.
2010 Mathematics Subject Classification: Primary 28A75, 42B20, 52C10.
Key words: Falconer distance problem, Erdos problems, bilinear operators, distribution of angles, arithmetic of the lattice.
Reference to this article: A. Iosevich, M. Mourgoglou and K. Taylor: On the Mattila-Sjölin theorem for distance sets. Ann. Acad. Sci. Fenn. Math. 37 (2012), 557-562.
doi:10.5186/aasfm.2012.3732
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