Geodesic
Trigonometric sums of nets of algebraic lattices
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 399-405.

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The paper continues the author's research on the evaluation of trigonometric sums of an algebraic net with weights with the simplest weight function of the second order. For the parameter $\vec{m}$ of the trigonometric sum $S_{M(t),\vec\rho_1} (\vec m)$, three cases are highlighted. If $\vec{m}$ belongs to the algebraic lattice $\Lambda (t \cdot T(\vec a))$, then the asymptotic formula is valid $$ S_{M(t),\vec\rho_1}(t(m,\ldots, m))=1+O\left(\frac{\ln^{s-1}\det \Lambda(t)} { (\det\Lambda(t))^2}\right). $$ 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 $$ S_{M(t),\vec\rho_1}(t(m,\ldots,m))=\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{q(\vec{n}_\Lambda(\vec{m}))^2}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^2\ln^{s-1}\det \Lambda (t)}{ (\det\Lambda(t))^2}\right). $$
Keywords: algebraic lattices, algebraic net, trigonometric sums of algebraic net with weights, weight functions. 
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     author = {E. M. Rarova},
     title = {Trigonometric sums of nets of algebraic lattices},
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E. M. Rarova. Trigonometric sums of nets of algebraic lattices. Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 399-405. http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a31/

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