부산대학교 도서관

Let $Q({\mathbf{x}}) = Q(x_{1} ,x_{2} ,\ldots,x_{n} )$ be a nonsingular quadratic form with integer coefficients, n be even and p be an odd prime. In Hakami (J. Inequal. Appl. 2014:290, 2014, doi:) we obtained an upper bound on the number of integer solutions of the congruence $Q({\mathbf{x}}) \equiv 0\ (\operatorname{mod} p^{2} )$ in small boxes of the type $\{ { {{\mathbf{x}} \in \mathbb{Z}_{p^{2} }^{n} | {a_{i} \leqslant x_{i} < a_{i} + m_{i} , 1 \leqslant i \leqslant n} } }\} $, centered about the origin, where $a_{i} ,m_{i} \in\mathbb{Z}$, $0< m_{i}\le p^{2}$, $1 \leqslant i \leqslant n$. In this paper, we shall drop the hypothesis of 'centered about the origin' and generalize the result of paper Hakami (J. Inequal. Appl. 2014:290, 2014, doi:) to boxes of arbitrary size and position. [ABSTRACT FROM AUTHOR]