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 Chebyshevskii Sb., 2020, Volume 21, Issue 3, Pages 232–240 (Mi cheb938)

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Asymptotic estimation for trigonometric sums of algebraic grids

E. M. Rarovaa, N. N. Dobrovol'skiiab, I. Yu. Rebrovaa

a Tula State Lev Tolstoy Pedagogical University (Tula)
b Tula State University (Tula)

Abstract: 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
$$S_{M(t),\vec\rho}(t(m,\ldots, m))=1+O(\frac{\ln^{s-1}\det \Lambda(t)} { (\det\Lambda(t))^{r+1}}).$$

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
$$|S_{M(t),\vec\rho}(\vec{m})|\le B_r(\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{(q(\vec{n}_\Lambda(\vec{m})))^{r+1}}+O(\frac{q(\vec{n}_\Lambda(\vec{m}))^{r+1}\ln^{s-1}\det \Lambda(t)}{ (\det\Lambda(t))^{r+1}})).$$

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_ð_à The work has been prepared by the RFBR grant №19-41-710004_ð_à.

DOI: https://doi.org/10.22405/2226-8383-2018-21-3-232-240

Full text: PDF file (752 kB)

UDC: 511.3
Accepted:22.10.2020

Citation: E. M. Rarova, N. N. Dobrovol'skii, I. Yu. Rebrova, “Asymptotic estimation for trigonometric sums of algebraic grids”, Chebyshevskii Sb., 21:3 (2020), 232–240

Citation in format AMSBIB
\Bibitem{RarDobReb20} \by E.~M.~Rarova, N.~N.~Dobrovol'skii, I.~Yu.~Rebrova \paper Asymptotic estimation for trigonometric sums of algebraic grids \jour Chebyshevskii Sb. \yr 2020 \vol 21 \issue 3 \pages 232--240 \mathnet{http://mi.mathnet.ru/cheb938} \crossref{https://doi.org/10.22405/2226-8383-2018-21-3-232-240}