Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers?

Table of contents:

1. A sketchy outline considering cases of whether or not $G$ contains a nonroot of unity.
2. An elaboration on the claim about what happens if $G$ contains a nonroot of unity.
3. An alternative approach not depending on whether or not $G$ contains a nonroot of unity.

1.

If $a\in G$ is not a root of unity, then $\{a^n:n\in\mathbb N\}$ is dense in $S^1$. Since $G$ is closed, this implies $G=S^1$.

On the other hand, if every $a\in G$ is a root of unity, and $G$ is infinite, then the set of $n$ such that $G$ has a primitive $n$th root of unity is infinite, and the union of the sets of all powers of such roots is dense, containing $n$ equally spaced points on the circle for arbitrarily large $n$. Again, density and closedness implies $G=S^1$.

2.

Elaboration on the first claim:

Suppose $a\in G$ is not a root of unity. Take $a=e^{i\theta}$ with $0<|\theta|<\pi$. Taking powers of $a$, let $k\in \mathbb N$ be the smallest exponent greater than $1$ such that $a^k$ passes $1\in S^1$, going counterclockwise if $\theta>0$, otherwise clockwise. Then either $a^k$ or $a^{k-1}$ has less than half the arclength to $1$ as $a$, that is, we can write it as $a_2=e^{i\theta_2}$ with $0<|\theta_2|<\dfrac{|\theta|}{2}$. Continuing this with $a_2\in G$ (also not a root of unity), we get an $a_3$, a power of $a_2$, such that $a_3=e^{i\theta_3}$ and $0<|\theta_3|<\dfrac{|\theta_2|}{2}$. Continuing indefinitely we get a sequence $a_1=a, a_2,a_3,\ldots$ of powers of $a$ getting arbitrarily close (but not equal) to $1$. If $|a_n-1|<\varepsilon$, then powers of $a_n$ cover $S^1$ with consecutive points with distance less than $\varepsilon$.

3.

Because $G$ is infinite and $S^1$ is compact, $G$ has an accumulation point, and thus $G$ contains unequal but arbitrarily close elements. For all $\varepsilon>0$, there exist $a,b\in G$ such that $0<|a-b|<\varepsilon$. Then $c=ab^{-1}$ is in $G$ and $0<|c-1|<\varepsilon$. Consecutive powers of $c$ cover the circle with points less than $\varepsilon$ apart. Hence $G$ is dense, and assuming it is closed, it must be $S^1$.

3.b.

Elaborating by request on showing that density follows from the above:

To show that $G$ is dense in $S^1$ is equivalent to showing that for all $x\in S^1$, for all $\varepsilon>0$, there exists $g\in G$ such that $|x-g|<\varepsilon$. So, given such $x$ and $\varepsilon$, let $c=c(\varepsilon)$ be as above. Either $x$ is a power of $c$ or there are consecutive powers $c^n$ and $c^{n+1}$ with $x$ lying between them, hence $|x-c^n|<|c^{n+1}-c^n|<\varepsilon$.

Consider the continuous surjective homomorphism $f: \mathbb R \to S^1$ as $f(x)=e^{2\pi i x} , \forall x \in \mathbb R$ .

Then for any subgroup $H$ of $S^1$ , $f(f^{-1}(H))=H$ , where $f^{-1}(H)$ is a subgroup (as $f$ is a group homomorphism) of $(\mathbb R,+)$ . So , $f^{-1}(H)$ is either cyclic or dense in $\mathbb R$ . If $f^{-1}(H)$ is dense , then $f$ being a continuous surjection , $f(f^{-1}(H))$ is dense , i.e. $H$ is dense in $S^1$ . Otherwise , $f^{-1}(H)$ is cyclic i.e. $f^{-1}(H)=b\mathbb Z$ for some $b \in \mathbb R$ . Two cases appear .

CASE I : $ b \in \mathbb Q$ : Say $b=p/q$ , where $p \in \mathbb Z , q\in \mathbb N$ . Then for any $n \in \mathbb Z$ , $n=kq+r ; 0\le r<q$ ,

then $e^{2\pi i bn}=e^{2\pi i(p/q)(kq+r)}=e^{2\pi i r/q } ; 0\le r<q$ . So then

$H=f(f^{-1}(H))=f(b\mathbb Z)=\{e^{2\pi i bn}:n\in \mathbb Z\} \subseteq \{e^{2\pi i r/q } : 0\le r <q\} $ , hnece $H$ is finite .

CASE II : $b$ is irrational

Then $\mathbb Z+b\mathbb Z$ is dense in $\mathbb R$ and noticing $\mathbb Z \subseteq ker f$ (actually it is equal , but we don't need that ) , we get

$H=f(f^{-1}(H))=f(b\mathbb Z)=f(\mathbb Z+b\mathbb Z)$ , and since $f$ is continuous surjection , so $H=f(\mathbb Z+b \mathbb Z)$ is dense in $S^1$ .

So we see that any subgroup of $S^1$ is either finite or dense in $S^1$ , hence if it is closed and infinite , it should be whole of $S^1$ .