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

TL;DR

This paper studies the distribution of algebraically conjugate points near smooth curves, providing bounds on their number based on polynomial degree, height, and proximity parameters.

Contribution

It establishes precise asymptotic bounds for the count of algebraically conjugate points near smooth curves, extending understanding of their distribution in Diophantine approximation.

Findings

Number of such points grows like Q^{n+1-γ} for large Q.

Bounds are independent of Q, depending only on degree and function parameters.

Results hold for any smooth curve with specified regularity.

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 $c_2 Q^{n+1-\gamma}<\# M^n_{\varphi}(Q,\gamma, J)< c_3 Q^{n+1-\gamma}$.