Distribution of some quadratic linear recurrence sequences modulo 1

Dubickas, Artūras



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.

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Dubickas, Artūras