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

Let Q ( x ) = Q ( x 1 , x 2 , … , 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 ( x ) ≡ 0 ( mod p 2 ) in small boxes of the type { x ∈ Z p 2 n | a i ⩽ x i < a i + m i , 1 ⩽ i ⩽ n } , centered about the origin, where a i , m i ∈ Z , 0 < m i ≤ p 2 , 1 ⩽ i ⩽ 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.