On the hyperbolic parameter of a two-dimensional lattice of comparisons

A. N. Kormacheva ; N. N. Dobrovol'skii ; I. Yu. Rebrova ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 168-182

Résumé

This paper is devoted to the refinement of the results of V. A. Bykovsky on the estimation of the error of approximate integration on the Korobov class $E_s^\alpha$ for two-dimensional parallelepipedal grids. The necessary information from the theory of continued fractions and Euler brackets is given. With the help of the theory of best approximations of the second kind, the Bykovsky set consisting of local minima of the lattice of Dirichlet approximations for a rational number is described. The Bykovsky set for a two-dimensional lattice of linear comparison solutions is explicitly described. A formula is obtained expressing the hyperbolic parameter of this lattice in terms of denominators of suitable fractions and Euler brackets and allowing it to be calculated in $O(N)$ arithmetic operations. Estimates of the hyperbolic zeta function of a two-dimensional lattice of linear comparison solutions are obtained in terms of the Bykovsky sum, which is a partial sum of the zeta series for the hyperbolic zeta function of the lattice. The partial sum is taken by the Bykovsky set. For the Bykovsky sum, estimates are obtained from above and from below, from which it follows that the main term for these sums is the sum of the $\alpha$-th degrees of the elements of the continued fraction for $\frac{a}{N}$ divided by $N^\alpha$. In conclusion, the current directions of research on this topic are noted.

Détail

Keywords: quadratic fields, approximation of algebraic grids, quality function, generalized parallelepipedal grid, Bykovsky set, Bykovsky sum, local lattice minima, minimal comparison solutions.