An almost all result on q1q2 ≡ c (mod q)

Davenport [2] used Kloosterman sum estimates to show that the above question is true for all ǫ > 1/3. Using Weil’s bound on Kloosterman sums (see equation (2)), Davenport’s argument implies the truth of Question 1 for all ǫ > 1/4. Recently in [11], Shparlinski got the same result with the further restriction that q1, q2 are relatively prime to one another. When q is a prime number, Garaev [6] obtained a slight improvement that Question 1 is true for all ǫ ≥ 1/4. Question 1 seems to be hard. How about proving it for almost all c? Recently Garaev and Karatsuba [8], and Shparlinski [12] proved that the above question is true for almost all c with any ǫ > 0 when q is prime and q in general respectively. Their results are more general as one of the interval can be replaced by a sufficiently large subset of the interval and the other interval does not have to start from 1. Furthermore when q is prime, Garaev and Garcia [7] showed the above almost all result with q1, q2 in any intervals of length q 1/2+ǫ by considering solutions to q1q2 ≡ q3q4 (mod q). It used both character sum technique of [1] and exponential sum technique of [6]. Thus, in general, one does not have to restrict the ranges of q1 and q2 to start from 1. In fact, the above question should be true for q1 and q2 in any interval of length Oǫ(q ). In this paper, we will prove that this is indeed the case for almost all such pairs of intervals for q1 and q2, namely Theorem 1 For any modulus q ≥ 1 and any integers 1 ≤ L ≤ q and (c, q) = 1,