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

Abstract

Let $φ:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbolα=(α_1,α_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $α_1,α_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_φ(Q,γ, J)$ the set of such points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. We show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, where $c_i=c_i(n)$, $i=1,2$, which are independent of $Q$, such that $c_2 Q^{n+1-γ}<\# M^n_φ(Q,γ, J)< c_3 Q^{n+1-γ}$.