Asymptotics of best-packing on rectifiable sets

Borodachov, S.; Hardin, D.; Saff, E.

doi:10.1090/S0002-9939-07-08975-7

Asymptotics of best-packing on rectifiable sets
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by S. V. Borodachov, D. P. Hardin and E. B. Saff PDF

Proc. Amer. Math. Soc. 135 (2007), 2369-2380 Request permission

Abstract:

We investigate the asymptotic behavior, as $N$ grows, of the largest minimal pairwise distance of $N$ points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz $s$-energy configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty )$ of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in $\textrm {\textbf {R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.

References

N.N. Andreev, On positions of points with minimal energy, Papers of the V.A. Steklov Math. Institute 219 (1997), 27–31.
Károly Böröczky Jr., Finite packing and covering, Cambridge Tracts in Mathematics, vol. 154, Cambridge University Press, Cambridge, 2004. MR 2078625, DOI 10.1017/CBO9780511546587
S.V. Borodachov, D.P. Hardin, E.B. Saff, Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets, Trans. Amer. Math. Soc., to appear.
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). MR 0057566
L. Fejes Tóth, Regular figures, A Pergamon Press Book, The Macmillan Company, New York, 1964. MR 0165423
Siegfried Graf and Harald Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730, Springer-Verlag, Berlin, 2000. MR 1764176, DOI 10.1007/BFb0103945
Peter M. Gruber, Optimum quantization and its applications, Adv. Math. 186 (2004), no. 2, 456–497. MR 2073915, DOI 10.1016/j.aim.2003.07.017
Thomas C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. MR 2179728, DOI 10.4007/annals.2005.162.1065
D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194. MR 2104914
D. P. Hardin and E. B. Saff, Minimal Riesz energy point configurations for rectifiable $d$-dimensional manifolds, Adv. Math. 193 (2005), no. 1, 174–204. MR 2132763, DOI 10.1016/j.aim.2004.05.006
Walter Habicht and B. L. van der Waerden, Lagerung von Punkten auf der Kugel, Math. Ann. 123 (1951), 223–234 (German). MR 42730, DOI 10.1007/BF02054950
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
A. V. Kolushov and V. A. Yudin, Extremal dispositions of points on the sphere, Anal. Math. 23 (1997), no. 1, 25–34 (English, with Russian summary). MR 1630001, DOI 10.1007/BF02789828
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov, and E. B. Saff, Asymptotics for minimal discrete Riesz energy on curves in $\Bbb R^d$, Canad. J. Math. 56 (2004), no. 3, 529–552. MR 2057285, DOI 10.4153/CJM-2004-024-1
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
C. A. Rogers, Packing and covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. MR 0172183
J.J. Thomson, On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the results to the Theory of Atomic Structure, Philosophical Magazine, Sixth Series, 7 (1904), 237–265.
B. L. van der Waerden, Punkte auf der Kugel. Drei Zusätze, Math. Ann. 125 (1952), 213–222 (German). MR 50912, DOI 10.1007/BF01343118
V. A. Yudin, Minimum potential energy of a point system of charges, Diskret. Mat. 4 (1992), no. 2, 115–121 (Russian, with Russian summary); English transl., Discrete Math. Appl. 3 (1993), no. 1, 75–81. MR 1181534, DOI 10.1515/dma.1993.3.1.75