Weighted number of points of algebraic net

E. M. Rarova

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 200-219.

Résumé

The paper is devoted to the study of trigonometric sums of algebraic grids with weights, which play a Central role in the modification of K. K. Frolov's method proposed by N. M. Dobrovolsky in 1984. The trigonometric sum of the algebraic grid with weights for the vector $\vec{m}=\vec{0}$ is naturally called the weighted number of points of the algebraic grid. In the introduction of this paper, the justification of the relevance of the research topic is proposed, the necessary definitions and facts from the modern theory of K. K. Frolov's method are given, an important theorem on the decomposition of the trigonometric sum of an algebraic grid with weights in a row by points of an algebraic grid is proved. In the section "Auxiliary lemmas" the necessary facts from the theory of weight functions of a special kind which play a principal role in modification of H. M. Dobrovolsky are given without proof. method K. K. Frolov. Using a theorem on the decomposition of the trigonometric sum of an algebraic grid with weights in a row by points of an algebraic grid and a Lemma on the value of a trigonometric integral of the weight function, we derive an asymptotic formula for the weighted number of points of an algebraic grid with a special weight function of order $2$, which States that such a number tends to unity. Similarly, it is shown that when the determinant of an algebraic lattice grows for any vector $\vec{m}\neq\vec{0}$, the trigonometric sum of algebraic grids with weights given by the special weight function tends to $0$. For simplicity, only the case of the simplest weight function of order $2$ is considered in the main text of the article. In conclusion, we formulate without proof similar statements about the values of trigonometric sums of algebraic grids with special weight functions of the order $r+1$ for any natural $R$. Namely, it is argued that for the weighted number of points of algebraic nets with a special weight function $r$ is true desire-to-$1$ with the residual member of the order $s-1$ of the logarithm of the determinant is an algebraic lattice, divided by $r+1$ the degree of the determinant is an algebraic lattice. A similar statement is true about the tendency to zero the trigonometric sum of an algebraic grid with weights given by a special weight function of the order $r+1$.

Détail

Keywords: algebraic lattices, algebraic net, trigonometric sums of algebraic net with weights, weight functions.

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