Confused about the domain of characters

Question

In Stein's Fourier Analysis Ch.7 he defines characters as follows.

Let $G$ be a finite Abelian Group with operation $*$, and $S^1$ be the unit circle in $\mathbb{C}$. $e: G \to S^1$ is a character on $G$ if $\forall x,y \in G$, $e(x * y) = e(x)e(y)$.

In exercise 5 on page 237, Stein asks us to show that the only characters on $S^1$ are $e_n (x) = e^{2\pi i nx}$, where $n \in \mathbb{Z}$. This is really confusing to me: is $x\in S^1$? This seems wrong to me, because then the image of the map is no longer contained in $S^1$. Is $x \in [0,2\pi]$? This also seems wrong, since then we have $e_\xi (x) = e^{2 \pi i \xi x}$ is also a character for any $\xi \in \mathbb{R}$, by the exponent rules. Therefore, I clearly don't understand the setup of the question, and would really appreciate some help!

Answer 1

Usually, when people ask about potential errors in books, either the book is wrong or they’re wrong. In this case, you’re both a bit wrong.

The book is wrong in that it introduces the circle group with the two alternatives that you consider:

The unit circle $S_1$ in the complex plane. If we view the circle as the set of points $\{\mathrm e^{\mathrm i\theta}:\theta\in \mathbb R\}$, the group operation is the standard multiplication of complex numbers. However, if we identify points on $S_1$ with their angle $\theta$, then $S_1$ becomes $\mathbb R$ modulo $2\pi$, where the operation is addition modulo $2\pi$.

But then in the exercise you refer to (and already in Example $2$ on p. $231$) it instead identifies the circle with $\mathbb R$ modulo $1$, i.e. the interval $[0,1]$ with endpoints identified.

The part where you’re wrong is where you write “since then we have $e_\xi(x)=\mathrm e^{2\pi\mathrm i\xi x}$ is also a character for any $\xi\in\mathbb R$, by the exponent rules”. That’s not true because e.g. $\frac23+\frac23\equiv\frac13\bmod1$, so you need to have $e_\xi\left(\frac23\right)\cdot e_\xi\left(\frac23\right)=e_\xi\left(\frac13\right)$, which only holds for integer $\xi$.

By the way, there’s also the footnote in Example $2$ on p. $231$, where characters on infinite groups are first introduced:

In addition to $(2)$, the definition of a character on an infinite abelian group requires continuity.

This also requires integer $\xi$, since the character would otherwise not be continuous at the endpoints of $[0,1]$.