On small solutions to quadratic congruences

We estimate the deviation of the number of solutions of the congruencem2−n2≡c(modq),1⩽m⩽M,1⩽n⩽N, from its expected value on average over c=1,…,qc=1,…,q. This estimate is motivated by the connection, recently established by D.R. Heath-Brown, between the distribution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function αk2αk2, k=1,2,…k=1,2,… with a real α.