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 $W={e^{it}:|t|<pi e}$ 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.