  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.
