Fourier-type series representation of "quasi-periodic" functions

Question

By a quasi-$T$-periodic function on some set $S \subseteq \mathbb{R}$, I mean a function $f : \mathbb{R} \to \mathbb{C}$ such that $f(x) = f(x + T) + o(1)$ for all $x \in S$, where $o(1)$ denotes a quantity that tends to zero as $x \to \infty$. For example, the function $\log(x)$ is quasi-1-periodic for $x$ large enough, since $\log(x + 1) = \log(x) + O(1/x)$ for $x > 1$. Thus as $x$ grows large, $\log(x)$ behaves more and more like a $1$-periodic function. I am wondering if there's a theory of Fourier-type series representation for such functions. For example, given a quasi-$T$-periodic function $f$ satisfying $f(x + T) = f(x) + O(1/x)$, can we find a representation of the form $$ \sum_{n \in \mathbb{Z}}c_n e^{2\pi i (n + \epsilon(n))x/T} $$ for $f(x)$ converging almost everywhere, where the real numbers $\epsilon(n)$, $n \in \mathbb{Z}$, vanish at infinity? Thanks.