Geodesic
Trigonometric sums of nets of algebraic lattices
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 399-405 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, trigonometric sums of algebraic net with weights, weight functions.
Mots-clés : algebraic net 
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     title = {Trigonometric sums of nets of algebraic lattices},
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     url = {http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a31/}
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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/

[1] E. M. Rarova, “Decomposition of the trigonometric sum of a grid with weights in a series by lattice points”, Proceedings of Tula state University. Natural science, 2014, no. 1-1, 37–49

[2] E. M. Rarova, “Trigonometric grid sums with weights for integer lattice”, Proceedings of Tula state University. Natural science, 2014, no. 3, 34–39

[3] E. M. Rarova, “Trigonometric sums of algebraic nets”, Algebra, number theory and discrete geometry: modern problems and applications, Proceedings of the XIII International conference dedicated to the eighty-fifth anniversary of the birth of Professor Sergei Sergeevich Ryshkov, Tula state pedagogical University L. N. Tolstoy, 2015, 356–359

[4] E. M. Rarova, “Weighted number of points of algebraic net”, Chebyshevskii sbornik, 19:1 (2018), 200–219 | MR | Zbl

[5] I. Yu. Rebrova, N. M. Dobrovolsky, N. N. Dobrovolsky, I. N. Balaba, A. R. Yesayan, Yu. A. Basalov, Numerical Theoretic method in approximate analysis and its implementation in POIVS “TMK”, Monogr. In Under 2 hours. ed., v. I, Publishing house of Tula. state PED. UN-TA im. L. N. Tolstoy, Tula, 2016, 232 pp. | MR