Let $a\in\mathbb{C}, |a|=1$ and $c$ an irrational real number. Prove: $a^c$ is dense in the unit circle.

Question

Let $a\in\mathbb{C}$ a complex number such that $|a|=1$ and $c$ an irrational real number.

Prove: The set $a^c$ is dense in the unit circle.

The problem is taken from Notes on Complex Function Theory by Donald Sarason.

Would appreciate any help.

Answer 1

Let $c$ and $a$ be fixed as in the question. For any complex number $w$ with $|w|=1$ define $$ S_w:=\{w\exp(2\pi i nc):n\in\mathbb{N}\} $$

According to your specifications if $s=Arg(a)$ then the set you are asking about is precisely $S_w$ where we choose $w=\exp(ics)$.

However we can show $S_w$ is dense for any choice of $w$ on the unit circle. There are two steps.

1. Show $S_1$ is dense (details below).
2. Given arbitrary $w$, write $S_w=w\cdot S_1$. So $S_w$ is a uniform rotation of the dense set $S_1$ and hence is dense itself.

To do step 1, this is the usual "irrational rotation". There are many sources with this proof. Here are two from the network

Dense set in the unit circle- reference needed

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

Added later: In light of the comments below the question, I can help clarify. While $a$ and $c$ are fixed, the intention seems to be to not fix a branch of logarithm, leading to a set of values for $a^c=\exp(c\log(a))$, in which $\log(a)$ takes any value $i(Arg(a)+2\pi n)$ as $n$ ranges over natural numbers. Since $c$ is irrational we obtain a countable set of values, and this is the set of values that one wants to show is dense. In other words

If $z$ is an irrational number, then the countably infinitely many choices of $\log w$ lead to infinitely many distinct values for $w^z$.

Source: https://en.wikipedia.org/wiki/Exponentiation#Complex_exponents_with_complex_bases

Note that if $a=1$ then formally one can still obtain a set of values in the same way, which would be the set $S_1$ above. Though this is abusing the notation of complex exponentiation.