منابع مشابه
Primitive Lattice Points in Starlike
This article is concerned with the number B D (x) of integer points with relative prime coordinates in p x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for B D (x). Applications to certain special geometric and arithmetic problems are discussed .
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This is one form of Kronecker’s theorem [4-j aura, since N can be chosen arbitrarily large, it follows that there are an infinity of integer sets (x, yl, -. . , y,) with x > 0 satisfying (A). For n > 2, it is not possible to strengthen this result by replacing the & in (A), throughout, by any function y/(x) which tends to zero as x-m (see, e.g., 15) , Kap VII,37, Satz 6). But, in the case n = 1...
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We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩lZ (provided it is non-empty) contains a point whose coefficient ofasymmetry with respect to P is at most 8d · (8l+7)2d+1. If, moreover,P is a simplex, then this bound can be improved to 8 · (8l+ 7)d+1.As an application, we deduce new upper bounds on the volume ofa lattice polytope, given its ...
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We study the problem of extending a primitive set of size k < n, in a lattice of rank n, to a basis of the lattice. Our approach relies on solving the matrix completion problem, and we show that the relevant computed matrices have nicely bounded entries. We present formulas, for the cases k = 1, n − 1 for an extended basis, and analyze the computational complexity for each of these two cases. T...
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We present algorithms for computing ranks and order statistics in the Farey sequence, taking time e O(n). This improves on the recent algorithms of Pawlewicz [Paw07], running in time e O(n). We also initiate the study of a more general algorithmic problem: counting primitive lattice points in planar shapes. Since the publication of this technical report, this work has been extended and merged w...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2010
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa142-3-4