Rarova, Elena Mikhaĭlovna Trigonometric sums of nets of algebraic lattices. (Russian. English summary) Zbl 1434.11161 Chebyshevskiĭ Sb. 20, No. 2(70), 399-405 (2019). Summary: 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 \[S_{M(t),\vec\rho_1}(t(m,\ldots,m))=\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{q(\vec{n}_\Lambda(\vec{m}))^2}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^2\ln^{s-1}\det \Lambda (t)}{ (\det\Lambda(t))^2}\right). \] MSC: 11L03 Trigonometric and exponential sums (general theory) 11H06 Lattices and convex bodies (number-theoretic aspects) Keywords:algebraic lattices; algebraic net; trigonometric sums of algebraic net with weights; weight functions × Cite Format Result Cite Review PDF Full Text: MNR