Is there any known characterization of the functions $\mathbb{R \to R}$ that can be written as a sum of (a finite family of) periodic functions? Not assuming any regularity condition (not even Lebesgue-measurable).


It is a well-known puzzle that assuming the axiom of choice, the identity function on $\mathbb{R}$ is a sum of two periodic functions. (Proof: decompose $\mathbb{R}$ into a direct sum $V \oplus W$ of two vector spaces over $\mathbb{Q}$; the sum of the two projections is the identity, the projection on $V$ has $W$ as its set of periods, and vice versa.)
Similarly, every polynomial function of degree $d$ is a sum of $d+1$ periodic functions. Such functions are also dense for the topology of pointwise convergence according to https://doi.org/10.1515/gmj-2019-2076 but I couldn't find a characterization in the literature.