# ds.dynamical systems – A property of rapid sequences of natural numbers

$$newcommand{IR}{mathbb R}$$
$$newcommand{IT}{mathbb T}$$
$$newcommand{w}{omega}$$
$$newcommand{e}{varepsilon}$$

Taras Banakh and me proceed a long quest answering a question of ougao at Mathematics.SE. Recently we encountered a notion of a remote sequence. We are interested whether it was studied before and in an answer to a question below.

Recall that a circle $$mathbb T={zinmathbb C:|z|=1}$$, endowed with the operation of multiplication of complex numbers and the topology inherited from $$mathbb C$$ is a topological group. Let $$(r_n)_{ninw}$$ be an increasing sequence of natural numbers. The sequence $$(r_n)_{ninw}$$ is called remote if there exists $$zinIT$$ such that $$inf_{ninw}|z^{r_n}-1|>0$$. A minimum growth rate of $$(r_n)_{ninw}$$ is a number $$inf_{ninw} r_{n+1}/r_{n}$$. We want to determine a
set $$M$$ such that if the minimum growth rate of $$(r_n)_{ninw}$$ belongs to $$M$$ then $$(r_n)_{ninw}$$ is remote.

Our try. It is easy to see that $$Msubseteq (1,infty)$$. We can show that $$(2,infty)subseteq M$$ by rather straightforward arguments.

Namely, suppose that the minimum growth rate $$m’$$ of $$(r_n)_{ninw}$$ is bigger than $$2$$ and find $$e>0$$ such that $$2+frac{e}{1-e}le m’$$. Then $$frac{r_{n+1}}{r_n}ge 2+frace{1-e}=frac{2-e}{1-e}$$ and hence $$frac{r_n}{r_{n+1}}cdotfrac{2-e}{1-e}le 1$$ for every $$ninw$$. Let $$W={e^{it}:|t| and for every $$ninw$$ consider the set $$U_n={zinIT:z^{r_n}in W}$$. Consider the exponential map $$exp:IRto IT$$, $$exp:tmapsto e^{2pi t i}$$, and observe that for every $$ninw$$ any connected component of the set $$exp^{-1}(U_n)$$ is an open interval of length $$frac{e}{r_n}$$ and every connected component of the set $$IRsetminus exp^{-1}(U_n)$$ is a closed interval of length $$frac{1-e}{r_n}$$. Since $$frac{r_n}{r_{n+1}}cdotfrac{2-e}{1-e}le 1$$ and $$frac{1-e}{r_{n+1}}+frace{r_{n+1}}+frac{1-e}{r_{n+1}}=frac{2-e}{r_{n+1}}=frac{1-e}{r_n}cdotfrac{r_n}{r_{n+1}}cdotfrac{2-e}{1-e}le frac{1-e}{r_n},$$every connected component of the set $$IRsetminus exp^{-1}(U_n)$$ contains a connected component of the set $$IRsetminusexp^{-1}(U_{n+1})$$. Then for every $$ninw$$ we can choose a connected component $$I_n$$ of the set $$IRsetminusexp^{-1}(U_n)$$ such that $$I_{n+1}subseteq I_n$$. By the compactness of the set $$I_0$$, the intersection $$bigcap_{ninw}I_n$$ contains some real number $$t$$. Then the point $$z=exp(t)$$ does not belong to $$bigcup_{ninw}U_n$$, which implies that $$z^{r_n}notin W$$ for every $$ninw$$. The definition of the neighborhood $$W$$ ensures that $$inf_{ninw}|z^{r_n}-1|>0$$, that is the
sequence $$(r_n)_{ninw}$$ is remote.

Thanks.