Approximation of quadratic algebraic lattices and nets by integer lattices and rational nets

A. V. Mikhlyaeva

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 241-256

Résumé

This paper is devoted to the approximation of quadratic algebraic lattices and grids by integer lattices and rational grids.A General formulation of the problem of approximation of algebraic lattices and corresponding meshes by integer lattices and rational meshes is given.In the case of a simple $p$ of the form $p=4k+3$ or $p=2$, we consider an integer lattice given $m$by a suitable fraction to the number $\sqrt{p}$. The corresponding algebraic lattice and the generalized parallelepipedal grid are written out explicitly.To determine the quality of the corresponding generalized parallelepipedal grid, a quality function is defined, which requires $O(N)$ arithmetic operations for its calculation, where $N$ — is the number of grid points. The Central result is an algorithm for computing a quality function for $O\left(\sqrt{N}\right)$ arithmetic operations.We hypothesize the existence of an algorithm that requires $O\left(\ln{N}\right)$ arithmetic operations. An approach for calculating sums with integral parts of linear functions is outlined.

Détail

Keywords: quadratic fields, approximation of algebraic grids, quality function, generalized parallelepipedal grid.