68 research outputs found

    Bounds for solid angles of lattices of rank three

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    We find sharp absolute constants C1C_1 and C2C_2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C1,C2][C_1,C_2]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN\mathbb R^N with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R3\mathbb R^3. Such spherical configurations come up in connection with the kissing number problem.Comment: 12 pages; to appear in the Journal of Combinatorial Theory

    Frobenius problem and the covering radius of a lattice

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    Let Nβ‰₯2N \geq2 and let 1<a1<...<aN1 < a_1 < ... < a_N be relatively prime integers. Frobenius number of this NN-tuple is defined to be the largest positive integer that cannot be expressed as βˆ‘i=1Naixi\sum_{i=1}^N a_i x_i where x1,...,xNx_1,...,x_N are non-negative integers. The condition that gcd(a1,...,aN)=1gcd(a_1,...,a_N)=1 implies that such number exists. The general problem of determining the Frobenius number given NN and a1,...,aNa_1,...,a_N is NP-hard, but there has 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 Frobenius number to the covering radius of the null-lattice of this NN-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.Comment: 12 pages; minor revisions; to appear in Discrete and Computational Geometr

    Hecke operators on rational functions

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    We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L, giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m. We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.Comment: 35 pages, LaTe

    The integer point transform as a complete invariant

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    The integer point transform ΟƒP\sigma_P is an important invariant of a rational polytope PP, and here we prove that it is a complete invariant. We prove that it is only necessary to evaluate ΟƒP\sigma_P at one algebraic point in order to uniquely determine PP. Similarly, we prove that it is only necessary to evaluate the continuous Fourier transform of a rational polytope PP at a single algebraic point, in order to uniquely determine PP. By relating the integer point transform to finite Fourier transforms, we show that a finite number of integer point evaluations of ΟƒP\sigma_P suffice in order to uniquely determine PP. In addition, we give an equivalent condition for central symmetry in terms of the integer point transform, and some facts about its local maxima. We prove many of these results for arbitrary finite sets of integer points in Rd\mathbb R^d.Comment: 12 pages, 3 figure
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