On distribution of points with algebraically conjugate coordinates in neighborhood of smooth curves

Let φ : R → R be a continuously differentiable function on an interval J ⊂ R and let α = (α1, α2) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α1, α2 is of degree ≤ n and height ≤ Q. Denote by Mn φ (Q, γ, J) the set of such points α such that |φ(α1)− α2| ≤ c1Q −γ . We show that for a real 0 < γ < 1 and any sufficiently large Q there exist positive values c2 < c3, where ci = ci(n), i = 1, 2, which are independent of Q, such that c2 · Q n+1−γ < #Mn φ (Q, γ, J) < c3 ·Q n+1−γ .