Trigonometric sums of grids of algebraic lattices with infinitely differentiable weights

Abstract

The paper continues the authors ’ research on the evaluation of trigonometric sums of an algebraic grid with weights. The case of an arbitrary weight function of infinite order is considered.
For the parameter ⃗𝑚 of the trigonometric sum 𝑆𝑀(𝑡),⃗𝜌∞(⃗𝑚), three cases are highlighted. If ⃗𝑚 belongs to the algebraic lattice Λ(𝑡·𝑇(⃗𝑎)), then for any natural 𝑟 the asymptotic formula
is valid 𝑆𝑀(𝑡),⃗𝜌∞(𝑡(𝑚, . . . ,𝑚)) = 1 + 𝑂 (︂ ln𝑠−1 det Λ(𝑡) (detΛ(𝑡))𝑟+1)︂.
If ⃗𝑚 does not belong to the algebraic lattice Λ(𝑡·𝑇(⃗𝑎)), then two vectors are defined ⃗𝑛Λ(⃗𝑚) =(𝑛1, . . . , 𝑛𝑠) and ⃗𝑘Λ(⃗𝑚) from the conditions ⃗𝑘Λ(⃗𝑚) ∈ Λ, ⃗𝑚 = ⃗𝑛Λ( ⃗𝑀)+ ⃗𝐾𝜆(⃗𝑚) and the product𝑞(⃗𝑛𝜆(⃗𝑚)) = 𝑛1 · . . . · 𝑛𝑠 is minimal. Asymptotic estimation is proved|𝑆𝑀(𝑡),⃗𝜌∞(⃗𝑚)| 6 𝐵(𝑟,∞)(︃1 − 𝛿(⃗𝑘Λ(⃗𝑚))(𝑞(⃗𝑛Λ(⃗𝑚)))𝑟+1 + 𝑂(︂𝑞(⃗𝑛Λ(⃗𝑚))𝑟+1 ln𝑠−1 det Λ(𝑡)(det Λ(𝑡))𝑟+1)︂)︃.