Weyl sums for quadratic roots (Q2911498)

Let \(e(z)=\exp(2\pi iz)\). The paper under review deals with the sum NEWLINE\[NEWLINE \mathcal{W}_h(D)=\sum_{c\equiv 0 \pmod q}f(c)W_h(D;c), NEWLINE\]NEWLINE of the Weyl sum NEWLINE\[NEWLINE W_h(D;c)=\sum_{\substack{ b \pmod c\\ b^2\equiv D \pmod c }} e\left(\frac{hb}{c}\right),NEWLINE\]NEWLINE where \(f\) is a nice test function. Indeed, assuming \(h\geq 1\), \(q\geq 1\) and \(D\) a positive (odd) fundamental discriminant, and letting \(f(y)\) be a smooth function supported on \(Y\leq y\leq 2Y\) with \(Y\geq 1\), such that \(|f(y)|\leq 1\) and \(y^3|f'''(y)|\leq 1\), then the authors prove that NEWLINE\[NEWLINE \mathcal{W}_h(D)\ll h^{\frac14}\left(Y+h\sqrt{D}\right)^{\frac34}D^{\frac18-\frac{1}{1.331}}, NEWLINE\]NEWLINE where the implied constant is absolute. To obtain this, they utilize a bound for a weighted sum of Kloosterman sums. The authors give four arithmetic applications of the above bound, all of which are concerned with positive discriminant. The first application studies a problem in the theory of uniform distribution modulo one of sequences. The second one studies a problem concerning the distribution of integral points \((a,b,c)\) on the one-sheeted hyperboloid \(b^2-4ac=D\) reduced by \(0<b<\sqrt{D}\) and \(\sqrt{D}-b<2|a|<\sqrt{D}+b\). As third application, the authors study the asymptotic behaviour of cycle integrals of the classical modular function \(j(z)\). The forth application is the resolution of a problem concerning prime points on the sphere.NEWLINENEWLINETo prove these results, the authors apply their previous results about sums of Salié sums [\textit{W. Duke} et al., Invent. Math. 128, No. 1, 23--43 (1997; Zbl 0873.11050)]. Both the above referred paper and the paper under review contain some errors, which the authors have addressed and corrected in the erratum.