Estimates of Weyl's exponential sums for very small values of the variable (Q704964)

Let \(k, N\) be integers, \(k\geq 3\) and let \[ p_k (z) = a_1 z + \dfrac{a_2} 2 z^2 + \cdots + \dfrac{a_k} k z^k \] be a polynomial of degree \(k\) with coefficients \(a_i \in \mathbb R\), and \(a_k > 0\). The author considers in this article the general Weyl sum \[ S= \sum\limits^N_{n=1} e^{2\pi i \vartheta p_k (n)} \] where, as usual, \(0 < \vartheta <1 , \quad \vartheta N^k > 1\). This is an extension of previous works of the author for the case \(p_k (z) = z^k\). The main result obtained is a transformation formula of the form \(S=F + E + {\mathcal O} (\sqrt N)\) where \(F\) is the main term and \(E\) is another exponential sum. Then by using van der Corput's method of estimating exponential sums, \(E\) is bounded satisfactorily over the range \(1/p'_k (N) < \theta < N^{-k+ 2 + \delta}\) for a positive number \(\delta \leq 1\). The main feature here is that the range for \(\vartheta\) extends beyond \(N^{2-k}\). The detailed results are too complicated to be reproduced here.