On the Distribution of Points with Algebraically Conjugate Coordinates in a Neighborhood of Smooth Curves

Abstract

Let φ : ℝ → ℝ be a continuously differentiable function on a finite 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 \( {M}_{\varphi}^n\left(Q,\gamma, J\right) \) the set of points α such that |φ(α₁) − α₂| ≤  c₁Q^−γ. We show that for 0 < γ < 1 and any sufficiently large Q there exist positive values c₂ < c₃, where c_i = c_i(n), i = 1, 2, that are independent of Q and such that \( {c}_2\cdot {Q}^{n+1-\upgamma}<\#{M}_{\varphi}^n\left(Q,\upgamma, J\right)<{c}_3\cdot {Q}^{n+1-\upgamma}. \) Bibliography: 17 titles.