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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: 10.1186/1029-242X-2014-290) 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: 10.1186/1029-242X-2014-290) to boxes of arbitrary size and position.

Estimates for lattice points of quadratic forms with integral coefficients modulo a prime number square (II)