Let $F_n$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$:1) ${inf}_{n\in \mathbb{N}}\left|\right|F_n\alpha \left|\right|\le \frac{\varphi -1}{\varphi +2}$,2) ${lim\; inf}_{n\to \infty }\left|\right|F_n\alpha \left|\right|\le \frac{1}{5}$,3) ${lim\; inf}_{n\to \infty }\left|\right|{\varphi }^n\alpha \left|\right|\le \frac{1}{5}$.These results are the best possible.

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. "$d$-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$-th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$-th root) is itself a recurrence. We shall also relate such...

The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$for some $b_i\in \overline{\mathbb{Q}}$ and $c_i\in \mathbb{Z}$. Let $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ be absolutely irreducible, monic and of degree at least $2$ in $y$. We consider Diophantine inequalities of the form$$\left|F(\alpha \left(n\right),y)\right|\<|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$and show that all the solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ have $y$ parametrized by some power sums in a finite set. As a consequence, we prove that the equation$$F\left(\alpha \right(n),y)=f\left(n\right),$$with $f\in \mathbb{Z}\left[x\right]$ not constant, $F$ monic in $y$ and $\alpha$ not constant, has only finitely many solutions.