On small solutions to quadratic congruences

Abstract

We estimate the deviation of the number of solutions of the congruence m 2 − n 2 ≡ c ( mod q ) , 1 ⩽ m ⩽ M , 1 ⩽ n ⩽ N , from its expected value on average over c = 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 α k 2 , k = 1 , 2 , … with a real α.