Asymptotic estimation for trigonometric sums of algebraic grids

E. M. Rarova ; N. N. Dobrovol'skii ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 232-240

Résumé

The paper continues the author's research on the evaluation of trigonometric sums of an algebraic net with weights with the arbitrary weight function of the $r+1$ order. For the parameter $\vec{m}$ of the trigonometric sum $S_{M(t),\vec\rho} (\vec m)$, three cases are highlighted. If $\vec{m}$ belongs to the algebraic lattice $\Lambda (t \cdot T(\vec a))$, then the asymptotic formula is valid $$ S_{M(t),\vec\rho}(t(m,\ldots, m))=1+O\left(\frac{\ln^{s-1}\det \Lambda(t)} { (\det\Lambda(t))^{r+1}}\right). $$ If $\vec{m}$ does not belong to the algebraic lattice $\Lambda(t\cdot T(\vec a))$, then two vectors are defined $\vec{n}_\Lambda(\vec{m})=(n_1,\ldots,n_s)$ and $\vec{k}_\Lambda(\vec{m})$ from the conditions $\vec{k}_\Lambda(\vec{m})\in\Lambda$, $\vec{m}=\vec{n}_\Lambda(\vec{M})+\vec{K}_\lambda(\vec{m})$ and the product $q(\vec{n}_\lambda(\vec{m}))=\overline{n_1}\cdot\ldots\cdot\overline{n_s}$ is minimal. Asymptotic estimation is proved $$ |S_{M(t),\vec\rho}(\vec{m})|\le B_r\left(\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{(q(\vec{n}_\Lambda(\vec{m})))^{r+1}}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^{r+1}\ln^{s-1}\det \Lambda(t)}{ (\det\Lambda(t))^{r+1}}\right)\right). $$

Détail

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