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

Abstract

Let $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $\alpha_1,\alpha_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_\varphi(Q,\gamma, J)$ the set of such points $\boldsymbol{\alpha}$ such that $|\varphi(\alpha_1)-\alpha_2|\leq c_1 Q^{-\gamma}$. We show that for a real $0<\gamma<1$ 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 $c_2 Q^{n+1-\gamma}<\# M^n_{\varphi}(Q,\gamma, J)< c_3 Q^{n+1-\gamma}$.