Well-rounded sublattices of planar lattices
نویسندگان
چکیده
منابع مشابه
On Distribution of Well - Rounded Sublattices
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z 2 , as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of...
متن کاملSimilar Sublattices of Planar Lattices
The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar subla...
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We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s = 1 with a real pole of order 2, improving upon a result of [11]. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal N is O(N...
متن کاملOn Well-rounded Ideal Lattices
We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study ...
متن کاملOn Distribution of Well-Rounded Sublattices of Z2
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe wellrounded full-rank sublattices of Z, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of suc...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2014
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa166-4-1