On bilinear exponential and character sums with reciprocals of polynomials (Q2827918)

Let \(\mathbb{F}_p\) denote the finite field of \(p\) elements where \(p\) is a sufficiently large prime. The sums in question have the form \(S_f(\mathcal{A}, \mathcal{B},\mathcal{C})=\sum_{(u,v)\in\mathcal{C}} \alpha_u\beta_v \mathbf{e}_p(v/f(u))\) where \(\mathcal{A} = \{\alpha_u\}, \mathcal{B} = \{\beta_v\}\) are complex weights with \(|\alpha_u|,|\beta_v| \leq 1\), \(\mathcal{C}\subseteq [1,U]\times[1,V]\) is a convex set, \(U, V\) are integers with \(1\leq U,V < p\), \(f\in\mathbb{F}_p[X]\) is a polynomial and \(\mathbf{e}_p(z)=\exp(2\pi iz/p)\). Special attention focuses on the cases \(S_f(\mathcal{A},\mathcal{C})\) in which the weights \(\beta_v=1\) for \(v\in [1,V]\) and \(K_{a,b}(\mathcal{A},\mathcal{B},\mathcal{C})= \sum_{(u,v)\in\mathcal{C}} \alpha_u\beta_v\mathbf{e}_p(v/(au+b))\) in which \(f(X)=aX+b\).NEWLINENEWLINEThere is a relatively easy general bound \(S_f(\mathcal{A},\mathcal{B},\mathcal{C})=O(\sqrt{UVp}\log p)\). The main results give improvements on this bound for certain ranges of the parameters. For example, \(|S_f(\mathcal{A},\mathcal{C})| \leq p^{d/(d+1)+o(1)}U^{d/2}+Vp^{o(1)}\) uniformly over polynomials \(f\) of degree \(d\geq 1\), and \(|K_{a,b}(\mathcal{A},\mathcal{B},\mathcal{C})| \leq V^{3/4}(U^{7/8}p^{1/8}+U^{1/2}p^{1/4})p^{o(1)}\) and \(|K_{a,b}(\mathcal{A},\mathcal{B},\mathcal{C})| \leq UV^{1/2}p^{1/3+o(1)}+U^{1/2}Vp^{o(1)}\), uniformly for \((a,b)\in\mathbb{F}_p^*\times\mathbb{F}_p\). There are also bounds for certain character sums of the shape \(\sum_{(u,v)\in\mathcal{C}}\alpha_u\beta_v\chi(v+f(u))\), where \(\chi\) is a fixed nonprincipal character of \(\mathbb{F}_p^*\), which give improvement on the Burgess bound for certain choices of the parameters.NEWLINENEWLINEThe ingredients that go into the proofs include estimates for the number of solutions of congruences with uniformly distributed sequences, the number of solutions of additive congruences with reciprocals and the number of solutions \(N_f(U,Z)\) of multiplicative congruences of the shape \(f(u)z\equiv 1\bmod p\) with \(1\leq u<U, 1\leq z<Z\). The last estimate is \(N_f(U,Z)\leq (U^{d/2}Zp^{-1/(d+1)}+1)p^{o(1)}\), uniformly over polynomials \(f\) of degree \(d\geq 1\), for \(1\leq U,Z <p\). As the author remarks, the methods used in the special cases do not seem to apply to the general sums \(S_f(\mathcal{A},\mathcal{B},\mathcal{C})\). For this, the results for \(N_f(U,z)\) would need to be extended to arbitrary rational functions. Such bounds follow from the Weil estimates for \(U\geq p^{1/2+\varepsilon} \), but the interest is in small values of \(U\) beyond the reach of the Weil bound.