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On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at Institut de Mathématiques in Bordeaux, France, June 2008.
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
CMC Faculty Publications and Research
The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current …
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Illinois Number Theory Fest, May 2007.
Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins
Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins
CMC Faculty Publications and Research
Abstract. Let N ≥ 2 and let 1 < a(1) < ... < a(N) be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as Sigma(N)(i=1) a(i) x(i) where x(1),..., x(N) are non-negative integers. The condition that gcd(a(1),..., a(N)) = 1 implies that such a number exists. The general problem of determining the Frobenius number given N and a(1),..., a(N) is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
CMC Faculty Publications and Research
Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.
Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky
Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005.
Problems From The Cottonwood Room, Matthias Beck, Beifang Chen, Lenny Fukshansky, Christian Haase, Allen Knutson, Bruce Reznick, Sinai Robins, Achill Schürmann
Problems From The Cottonwood Room, Matthias Beck, Beifang Chen, Lenny Fukshansky, Christian Haase, Allen Knutson, Bruce Reznick, Sinai Robins, Achill Schürmann
CMC Faculty Publications and Research
This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in polyhedra. Lattice points in homogeneously expanding compact domains. Presented by Lenny Fukshansky (Texas A&M University).