On Bykovsky estimates for deviations of generalized parallelepipedal grids

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

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 214-227

Résumé

This paper is devoted to obtaining estimates of the type of Bykovsky estimates for the deviation of a generalized parallelepipedal grid. It continues the studies similar to those that we previously performed to assess the quality measure and the quantitative measure of the parallelepipedal grid. The main idea used in this paper goes back to the work of V. A. Bykovsky (2002) on estimating the error of approximate integration over parallelepipedal grids and its generalization in the work of O. A. Gorkusha and N. M. Dobrovolsky (2005) for the case of a hyperbolic zeta function of an arbitrary lattice. The central place in these works is played by the Bykovsky set, consisting of local minima of the second kind, and sums over these sets. As in the work "On Bykovsky estimates for a measure of the quality of optimal coefficients  the effect was found that a multiplier with a logarithmic order of growth appears in the deviation estimates, which began to include the definition of the modified Bykovsky sum. The method of work is to combine the approaches from the work "Estimates of deviations of generalized parallelepipedal grids" (1984) with the approaches of 2005. Further ways to obtain clarification of the received estimates are outlined.

Détail

Keywords: quality function, generalized parallelepipedal grid, Bykovsky set, Bykovsky sum, local lattice minima, minimal comparison solutions.