# What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).

In particular, I am trying to figure out if there exists a non-constant almost periodic function $$f$$ on $$\mathbb{C}$$ such that $$f$$ is invariant under rotations i.e. $$f(tz) = f(z)$$ for all $$t\in \mathbb{T}$$, $$z\in \mathbb{C}$$.

Any help is much appreciated.

Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $$\mathbb{R}^2$$ these are the functions $$e^{i(ax + by)}$$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $$f$$ is a rotationally invariant nonconstant almost periodic function. WLOG $$f(0,0) =0$$ and $$f(1,0) = 1$$. So $$f$$ is constantly $$1$$ on the unit circle. Now find $$t > 1$$ such that $$f$$ and its shift by $$(t, 0)$$ are uniformly at most $$1/3$$ apart. Then $$f(t,0)$$ is within $$1/3$$ of $$0$$, so the same must be true at any point on the circle of radius $$t$$ about the origin. But at the same time, $$f$$ must be within $$1/3$$ of $$1$$ on the circle of radius $$1$$ about $$(t,0)$$, and that is contradictory.