Asymptotic estimation for trigonometric sums of algebraic grids

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

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