Let p>3 bc a prime, k bc any fixed positlvc integer. For any polynomiM f(x, y) with rational integer coefficients, we define set A=A(f,k,δ)={(a,b):f(a,b)=O (modp), {ak-p}-{bk-p}<δ} whcrc 0<δ≤1, [x] denotes the grcatest intcger not exceeding x and {x}=x-[x]. The main purpose of this paper is to study the asymptotic behaviour of M(p,A,K)=Σpa=1 Σpb=1(a,b)4 and give an interesting asymptotic formula for M (p,A,k) for some special polynomials f (x, y).