On distribution of points with conjugate algebraic integer coordinates close to planar 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 algebraic conjugate integer coordinates of degree $\leq n$ and of height $\leq Q$. Denote by $\tilde{M}^n_\varphi(Q,\gamma, J)$ the set of points $\boldsymbol{\alpha}$ such that $|\varphi(\alpha_1)-\alpha_2|\leq c_1 Q^{-\gamma}$. In this paper we show that for a real $0<\gamma<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, which are independent of $Q$, such that $c_2\cdot Q^{n-\gamma}<# \tilde{M}^n_\varphi(Q,\gamma, J)< c_3\cdot Q^{n-\gamma}$.