An equivalent of Kronecker's Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length

TL;DR

This paper extends Kronecker's theorem to algebraic numbers by estimating the density of certain vector sets modulo 1, linking it to Mahler measure, and explores the structure of linear recurrences of fixed length.

Contribution

It provides a new estimate for the smallest epsilon ensuring epsilon-density of vector sets related to algebraic numbers, independent of the vector length, and analyzes the structure of linear recurrences.

Findings

Epsilon-density bounds depend on Mahler measure of the minimal polynomial.

For large vector lengths, epsilon-density cannot be achieved below a threshold related to Mahler measure.

Structural results on the module of linear recurrences of fixed length are obtained.

Abstract

After defining a notion of $\epsilon$-density, we provide for any real algebraic number $\alpha$ an estimate of the smallest $\epsilon$ such that for each $m>1$ the set of vectors of the form $(t,t\alpha,...,t\alpha^{m-1})$ for $t\in\R$ is $\epsilon$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $\alpha$, and independently of $m$. In particular, we show that if $\alpha$ has degree $d$ it is possible to take $\epsilon = 2^{[d/2]}/M(A(x))$. On the other hand using asymptotic estimates for Toeplitz determinants we show that for sufficiently large $m$ we cannot have $\epsilon$-density if $\epsilon$ is a fixed number strictly smaller than $1/M(A(x))$. As a byproduct of the proof we obtain a result of independent interest about the structure of the $\Z$-module of integral linear recurrences of fixed length determined by a non-monic polynomial.