Distance sets of well-distributed planar sets for polygonal norms
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- J. Bourgain,On the dimension of Kakeya sets and related maximal inequalities, Geometric and Functional Analysis9 (1999), 256–282. CrossRef
- 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.
- P. Erdős,On sets of distances of n points, The American Mathematical Monthly53 (1946), 248–250. CrossRef
- K. J. Falconer,Dimensions of intersections and distance sets for polyhedral norms, preprint, 2004.
- G. Freiman,Foundations of a Structural Theory of Set Addition (translation from Russian), Translations of Mathematical Monographs, Vol. 37, American Mathematical Society, Providence, RI, 1973.
- J. Garibaldi,Erdős Distance Problem for Convex Metrics, Ph.D. thesis, UCLA, 2004.
- A. Iosevich,Curvature, combinatorics and the Fourier transform, Notices of the American Mathematical Society46 (2001), 577–583.
- A. Iosevich and I. Łaba,Distance sets of well-distributed planar point sets, Discrete and Computational Geometry31 (2004), 243–250. CrossRef
- A. Iosevich and I. Łaba,K-distance sets, Falconer conjecture and discrete analogs, Integers: Electronic Journal of Combinatorial Number Theory, to appear.
- 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.
- 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.
- M. Nathanson,Additive Number Theory, II: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996.
- I. Ruzsa,Sum of sets in several dimensions, Combinatorica14 (1994), 485–490. CrossRef
- J. Solymosi and Cs. Tóth,Distinct distances in the plane, Discrete and Computational Geometry25 (2001), 629–634.
- 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.
- T. Wolff,Decay of circular means of Fourier transforms of measures, International Mathematics Research Notices10 (1999), 547–567. CrossRef
- Distance sets of well-distributed planar sets for polygonal norms
Israel Journal of Mathematics
Volume 152, Issue 1 , pp 157-179
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors