Distance sets of well-distributed planar sets for polygonal norms
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LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN 3/2-ε. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.
KeywordsHausdorff Dimension Geometric Graph Positive Lebesgue Measure Affine Subspace Finite Nonempty Subset
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- [EE94]G. Elekes and P. Erdős,Similar configurations and pseudo grids, inIntuitive Geometry (Szeged, 1991), Colloquia Mathematica Societatis János Bolyai, 63, North-Holland, Amsterdam, 1994, pp. 85–104.Google Scholar
- [Fa04]K. J. Falconer,Dimensions of intersections and distance sets for polyhedral norms, preprint, 2004.Google Scholar
- [G04]J. Garibaldi,Erdős Distance Problem for Convex Metrics, Ph.D. thesis, UCLA, 2004.Google Scholar
- [IŁ04]A. Iosevich and I. Łaba,K-distance sets, Falconer conjecture and discrete analogs, Integers: Electronic Journal of Combinatorial Number Theory, to appear.Google Scholar
- [KT04]N. H. Katz and G. Tardos,A new entropy inequality for the Erdős distance problem, inTowards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, American Mathematical Society, Providence, RI, 2004.Google Scholar
- [LR96]M. Laczkovich and I. Z. Ruzsa,The number of homothetic subsets, inThe Mathematics of Paul Erdős, II, Algorithms and Combinatorics, 14, Springer, Berlin, 1997, pp. 294–302.Google Scholar
- [N96]M. Nathanson,Additive Number Theory, II: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996.Google Scholar
- [SV04]J. Solymosi and V. Vu,Distinct distances in high dimensional homogeneous sets, inTowards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, American Mathematical Society, Providence, RI, 2004.Google Scholar