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Chebyshevskii Sbornik, 2019, Volume 20, Issue 2, Pages 399–405
DOI: https://doi.org/10.22405/2226-8383-2018-20-2-399-405
(Mi cheb780)
 

This article is cited in 2 scientific papers (total in 2 papers)

BRIEF MESSAGE

Trigonometric sums of nets of algebraic lattices

E. M. Rarova

Tula State L. N. Tolstoy Pedagogical University (Tula)
Full-text PDF (711 kB) Citations (2)
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Abstract: 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.
Funding agency Grant number
Russian Foundation for Basic Research 19-41-710004_р_а
The study was carried out under the RFBR grant №~19-41-710004_r_а.
Received: 18.03.2017
Accepted: 12.07.2019
Document Type: Article
UDC: 511.3
Language: Russian
Citation: E. M. Rarova, “Trigonometric sums of nets of algebraic lattices”, Chebyshevskii Sb., 20:2 (2019), 399–405
Citation in format AMSBIB
\Bibitem{Rar19}
\by E.~M.~Rarova
\paper Trigonometric sums of nets of algebraic lattices
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 2
\pages 399--405
\mathnet{http://mi.mathnet.ru/cheb780}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-2-399-405}
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  • https://www.mathnet.ru/eng/cheb/v20/i2/p399
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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