» Distribution of some quadratic linear recurrence sequences modulo 1

We show that if $a$ is an even integer then for every $\xi\in \R$ the smallest limit point of the sequence $||\xi a^n||_{n=1}^{\infty}$ does not exceed $|a|/(2|a|+2)$ and this bound is best possible in the sense that for some $\xi$ this
constant cannot be improved. Similar (best possible) bound is also obtained for the smallest limit point of the sequence $||\xi x_n||_{n=1}^{\infty}$ , where $(x_n)_{n=1}^{\infty}$ satisfies the second order linear recurrence $x_n=ax_{n-1}+bx_{n-2}$ with
$a,b \in \N$ satisfying $a\!\geq\!b$ . For the Fibonacci sequence $(F_n)_{n=1}^{\infty}$ our result implies that $\sup_{\xi \in \R} \liminf_{n\to\infty}||\xi F_n||=1/5$ , and e.g., in case when $a \geq 3$ is an odd integer, $b=1$ and $\theta:=a/2+\sqrt{a^2/4+1}$ it shows that $\sup_{\xi \in \R} \liminf_{n\to \infty}||\xi\theta^n||=(a-1)/2a$ .