On algebraic integers in short intervals and near smooth curves

TL;DR

This paper explores the distribution of algebraic integers within short intervals and near smooth curves, applying the concept of regular systems to advance understanding in metric number theory.

Contribution

It extends the concept of regular systems to algebraic integers of bounded height in variable-length intervals, providing new insights into their distribution.

Findings

Algebraic integers form regular systems in short intervals.

Distribution results depend on the height bound Q.

Applications to metric number theory are demonstrated.

Abstract

In 1970 A. Baker and W. Schmidt introduced regular systems of numbers and vectors, showing that the set of real algebraic numbers forms a regular system on any fixed interval. This fact was used to prove several important results in the metric theory of transcendental numbers. In this paper the concept of a regular system is applied to the set of algebraic integers $\alpha$ of height $\leq Q$ in intervals of length depending on $Q$.