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The limiting curve of Jarník's polygons
Author(s):
Greg
Martin
Journal:
Trans. Amer. Math. Soc.
355
(2003),
4865-4880.
MSC (2000):
Primary 52C05;
Secondary 11H06
Posted:
July 28, 2003
MathSciNet review:
1997588
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Abstract:
In 1925, Jarník defined a sequence of convex polygons for use in constructing curves containing many lattice points relative to their curvatures. Properly scaled, these polygons converge to a certain limiting curve. In this paper we identify this limiting curve precisely, showing that it consists piecewise of arcs of parabolas, and we discuss the analogous problem for sequences of polygons arising from generalizations of Jarník's construction.
References:
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- 1.
- M. N. Huxley, Area, lattice points, and exponential sums, The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications. MR 97g:11088
- 2.
- Alex Iosevich, Curvature, combinatorics, and the Fourier transform, Notices Amer. Math. Soc. 48 (2001), no. 6, 577-583. MR 2002e:42004
- 3.
- V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Math. Zeitschrift 24 (1925), 500-518.
- 4.
- Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An introduction to the theory of numbers, fifth ed., John Wiley & Sons Inc., New York, 1991.MR 91i:11001
- 5.
- A. M. Vershik, The limit form of convex integral polygons and related problems, Funktsional. Anal. i Prilozhen. 28 (1994), no. 1, 16-25, 95; English transl., Functional Anal. Appl. 28 (1994), 13-20. MR 95i:52010
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Additional Information:
Greg
Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
Email:
gerg@math.ubc.ca
DOI:
10.1090/S0002-9947-03-03219-7
PII:
S 0002-9947(03)03219-7
Received by editor(s):
June 20, 2002
Posted:
July 28, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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