Multiple exponential and character sums with monomials (Q2874614)

Let \(p\) be a prime number, and let \(n\geq 2\) be a fixed integer. Let \({\boldsymbol \rho}=(\rho_1,\ldots,\rho_n)\) be a vector of \(n\) complex-valued weights with NEWLINE\[NEWLINE \left|\rho_j(x)\right|\leq 1,\qquad x\in \mathbb{R}, \qquad j=1,\cdots,n, NEWLINE\]NEWLINE and let \({\mathbf e}\) be an integer vector with non-zero components NEWLINE\[NEWLINE {\mathbf e}=(e_1,\ldots,e_n)\in \mathbb{Z}^n \quad\text{and}\quad e_1\cdots e_n\neq 0. NEWLINE\]NEWLINE Let \(\lambda\in \mathbb{Z}\), and let \(\mathcal{B}\) be an \(n\)-dimensional cube NEWLINE\[NEWLINE \mathcal{B}=\left[k_1+1, k_1+h\right]\times\cdots\times\left[k_n+1, k_n+h\right] NEWLINE\]NEWLINE with \(h<p\) and some integers \(k_1,\ldots,k_n\).NEWLINENEWLINEThe multiple exponential sums with monomials are defined by NEWLINE\[NEWLINE S_p(\lambda,{\boldsymbol \rho},{\mathbf e}; \mathcal{B})=\mathop{\sum_{(x_1,\ldots,x_n)\in \mathcal{B}}}_{x_1\cdots x_n\not\equiv 0 (\bmod p)}\rho_1(x_1)\cdots\rho_n(x_n){\mathbf e}_p\left(\lambda x_1^{e_1}\cdots x_n^{e_n}\right), NEWLINE\]NEWLINE where \({\mathbf e}_p(z)=\exp(2\pi iz/p)\). For multiplicative character \(\chi\) modulo \(p\), we denote NEWLINE\[NEWLINE T_p\left(\lambda,\chi, {\boldsymbol \rho},{\mathbf e}; \mathcal{B}\right)=\mathop{\sum_{(x_1,\ldots,x_n)\in \mathcal{B}}}_{x_1\cdots x_n\not\equiv 0 (\bmod p)}\rho_1(x_1)\cdots\rho_n(x_n)\chi\left(x_1^{e_1}\cdots x_n^{e_n}+\lambda\right). NEWLINE\]NEWLINENEWLINENEWLINEThe author obtains new bounds for \(S_p(\lambda,{\boldsymbol \rho},{\mathbf e}; \mathcal{B})\) and \(T_p\left(\lambda,\chi, {\boldsymbol \rho},{\mathbf e}; \mathcal{B}\right)\), when the variables run over rather short intervals. In particular for \(T_p\left(\lambda,\chi, {\boldsymbol \rho},{\mathbf e}; \mathcal{B}\right)\), the results given in the paper break the barrier of \(p^{1/4}\) for ranges of individual variables.