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A generalization of Voronoï’s Theorem to algebraic lattices

Kenji Okuda, Syouji Yano (2010)

Journal de Théorie des Nombres de Bordeaux

Let K be an algebraic number field and 𝒪 K the ring of integers of K . In this paper, we prove an analogue of Voronoï’s theorem for 𝒪 K -lattices and the finiteness of the number of similar isometry classes of perfect 𝒪 K -lattices.

A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

Christian Drouin (2014)

Journal de Théorie des Nombres de Bordeaux

A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued...

Asymptotics of variance of the lattice point count

Jiří Janáček (2008)

Czechoslovak Mathematical Journal

The variance of the number of lattice points inside the dilated bounded set r D with random position in ℝ d has asymptotics ∼ r d - 1 if the rotational average of the squared modulus of the Fourier transform of the set is O ( ρ - d - 1 ) . The asymptotics follow from Wiener’s Tauberian theorem.

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