Question

The function $f(x) = e^{2\pi ix}$ on the domain $\mathbb{R}/\mathbb{Z}$ has the property that, for every $n > 1$ and every $x$, $\displaystyle \sum_{i = 0}^{n-1} f(x + \frac{i}{n}) = 0$.

Other such functions can be found by simply postcomposing a linear function to this example (thus, for example, $x \mapsto \cos(2\pi x) : \mathbb{R}/\mathbb{Z} \to \mathbb{R}$ also has this property).

Beyond those, are there any other "natural" examples of functions with this property (on the same domain $\mathbb{R}/\mathbb{Z}$, but with any codomain)?

[Of course, one can freely construct the codomain as a monoid with generators $\mathbb{R}/\mathbb{Z}$ and all the necessary relations; slightly less trivially, one could take the codomain to be a ring, the map to be exponential, and $f(\frac{1}{n}) - 1$ to be invertible for each prime $n$, in the same freely constructed fashion. I can't quite put my finger on why, but I don't want to count these as "natural" examples (probably because the condition for each (or essentially each) distinct $n$ is handled separately, instead of flowing all at once from some underlying property)]

Answer 1

Every continuous integrable function satisfying your hypothesis is a linear combination of $\sin(2\pi x)$ and $\cos (2\pi x)$. The key fact is that if $f:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{C}$ is continuous and integrable then it is uniquely determined by its Fourier coefficients. If the Fourier series expansion of $f$ is $$f(x)=\sum_{n\in\mathbb{Z}}c(n)e(nx),$$ where $e(x)=\exp (2\pi i x)$, then for $n>1$ the Fourier expansion of $$g_n(x):=\sum_{a=1}^nf\left(x+\frac{a}{n}\right)$$ is $$g_n(x)=\sum_{m\in\mathbb{Z}}c(m)\left(\sum_{a=1}^ne(ma/n)\right)e(mx)$$ $$=n\sum_{m\in\mathbb{Z}}c(mn)e(mnx).$$ If $g_n(x)=0$ for all $n>1$ and for all $x$ then we must have that $c(mn)=0$ for all $m\in\mathbb{Z}$ and for all $n>1$, and this proves the result.