Lipschitz class, narrow class, and counting lattice points

Widmer, Martin

doi:10.1090/S0002-9939-2011-10926-2

Lipschitz class, narrow class, and counting lattice points
HTML articles powered by AMS MathViewer

by Martin Widmer

Proc. Amer. Math. Soc. 140 (2012), 677-689

DOI: https://doi.org/10.1090/S0002-9939-2011-10926-2

Published electronically: June 9, 2011

PDF | Request permission

Abstract:

A well-known principle says that the number of lattice points in a bounded subset $S$ of Euclidean space is about the ratio of the volume and the lattice determinant, subject to some relatively mild conditions on $S$. In the literature one finds two different types of such conditions: one asserts the Lipschitz parameterizability of the boundary $\partial S$, and the other one is based on intersection properties of lines with $S$ and its projections to linear subspaces. We compare these conditions and address a question, which we answer in some special cases. Then we give some simple upper bounds on the number of lattice points in a convex set, and finally, we apply these results to obtain estimates for the number of rational points of bounded height on certain projective varieties.

References

Keith Ball, Ellipsoids of maximal volume in convex bodies, Geom. Dedicata 41 (1992), no. 2, 241–250. MR 1153987, DOI 10.1007/BF00182424
H. F. Blichfeldt, The April meeting of the San Francisco section of the AMS, The American Math. Monthly 28, no. 6/7 (1920/21), 285–292.
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179–183. MR 43821, DOI 10.1112/jlms/s1-26.3.179
X. Gao, On Northcott’s Theorem, Ph.D. Thesis, University of Colorado (1995).
Martin Henk and Jörg M. Wills, A Blichfeldt-type inequality for the surface area, Monatsh. Math. 154 (2008), no. 2, 135–144. MR 2419059, DOI 10.1007/s00605-008-0530-8
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 187–204. MR 0030135
Wilhelm Maak, Schnittpunktanzahl rektifizierbarer und nichtrektifizierbarer Kurven, Math. Ann. 118 (1942), 299–304 (German). MR 8463, DOI 10.1007/BF01487371
David Masser and Jeffrey D. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc. 359 (2007), no. 1, 427–445. MR 2247898, DOI 10.1090/S0002-9947-06-04115-8
J. Pila, Density of integral and rational points on varieties, Astérisque 228 (1995), 4, 183–187. Columbia University Number Theory Seminar (New York, 1992). MR 1330933
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
Wolfgang M. Schmidt, Northcott’s theorem on heights. II. The quadratic case, Acta Arith. 70 (1995), no. 4, 343–375. MR 1330740, DOI 10.4064/aa-70-4-343-375
Martin Widmer, Counting points of fixed degree and bounded height, Acta Arith. 140 (2009), no. 2, 145–168. MR 2558450, DOI 10.4064/aa140-2-4
Martin Widmer, Counting points of fixed degree and bounded height on linear varieties, J. Number Theory 130 (2010), no. 8, 1763–1784. MR 2651154, DOI 10.1016/j.jnt.2010.03.001
Martin Widmer, Counting primitive points of bounded height, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4793–4829. MR 2645051, DOI 10.1090/S0002-9947-10-05173-1
—, On number fields with nontrivial subfields, to appear in Int. J. Number Theory (2010).
I. M. Jaglom and V. G. Boltjanskiĭ, Convex figures, Holt, Rinehart and Winston, New York, 1960. Translated by Paul J. Kelly and Lewis F. Walton. MR 0123962