Let p be a large prime number, K,L,M,λ be integers with 1≤M≤p and gcd(λ,p)=1. The aim of our paper is to obtain
sharp upper bound estimates for the number I2(M;K,L) of solutions of the
congruence xy≡λ(modp),K+1≤x≤K+M,L+1≤y≤L+M and for the number I3(M;L) of solutions of the congruence
xyz≡λ(modp),L+1≤x,y,z≤L+M. We obtain a bound for
I2(M;K,L), which improves several recent results of Chan and Shparlinski.
For instance, we prove that if M<p1/4, then I2(M;K,L)≤Mo(1). For
I3(M;L) we prove that if M<p1/8 then I3(M;L)≤Mo(1). Our
results have applications to some other problems as well. For instance, it
follows that if I1,I2,I3 are intervals in
\F^*_p of length ∣Ii∣<p1/8, then ∣I1⋅I2⋅I3∣=(∣I1∣⋅∣I2∣⋅∣I3∣)1−o(1).Comment: 12 page