Paper detail

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

After defining a notion of $ε$-density, we provide for any real algebraic number $α$ an estimate of the smallest $ε$ such that for each $m>1$ the set of vectors of the form $(t,tα,...,tα^{m-1})$ for $t\in\R$ is $ε$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $α$, and independently of $m$. In particular, we show that if $α$ has degree $d$ it is possible to take $ε= 2^{[d/2]}/M(A(x))$. On the other hand using asymptotic estimates for Toeplitz determinants we show that for sufficiently large $m$ we cannot have $ε$-density if $ε$ 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.