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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)
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.
Received: 18.03.2017 Accepted: 12.07.2019
Citation:
E. M. Rarova, “Trigonometric sums of nets of algebraic lattices”, Chebyshevskii Sb., 20:2 (2019), 399–405
Linking options:
https://www.mathnet.ru/eng/cheb780 https://www.mathnet.ru/eng/cheb/v20/i2/p399
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