Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
\Phi[f](P) = \sum\_{r=0}^\infty \sum\_{s=0}^\infty (-1)^r f\left(P+\beta r+\frac{s}{\beta}\right)
$$
Numerical experiments, and the study of its behaviour under shifts by $\beta$ and $\frac{1}{\beta}$, show that the function $\Phi[f](P)$ is bounded and quasiperiodic when $P\to -\infty$, with the quasiperiod $\frac{1}{\beta}$ and average value $\frac{\beta}{2}\int\_{-\infty}^\infty f$.


**Conjecture:** For any $P\_0\in\mathbb{R}$, and any sequence $((r\_i,s\_i))\_i$ of pairs of positive integers such that $r\_i$ is even and 
$$\lim\_i \left(\beta r\_i-\frac{s\_i}{\beta}\right) = P\_0
$$
the following limit exists:
$$
L[f](P\_0) = \lim\_i \Phi[f](-\beta r\_i) = \lim\_i \Phi[f]\left(-P\_0 -\frac{s\_i}{\beta}\right)
$$


**Question:** Prove (or disprove) the conjecture. If the limit exists, compute it. 


**Remarks:** 


* The limit $\lim\_{s\to\infty} \Phi[f](-P\_0-\frac{s}{\beta})$ does not exist, because $\Phi[f](P)$ becomes quasiperiodic, not periodic, for $P\to-\infty$. For the limit to exist we need the integer $s$ to take rather sparse values.
* This conjecture is suggested by considerations on limits of non-diagonal two-dimensional conformal field theory. The parameter $\beta$ is related to the central charge, the integers $r,s$ are indices of degenerate fields, and the function $f$ is a conformal block.
* Numerical tests suggest that the conjecture is true.



Limits of a quasiperiodic function with two pseudoperiods
$$
L[f](P\_0) = \left( \sum\_{r,s\in \mathbb{N}} - \sum\_{r,s\in -\mathbb{N}^\*}\right) (-1)^r f(\beta r +\beta^{-1}s) 
\\
-2\beta \sum\_{n\in\mathbb{N}^\*} \frac{\sin \pi n\beta P\_0}{\cos \pi n\beta^2} \int\_{\mathbb{R}} f(P') \sin\pi \beta n(P\_0+2P'+\beta) dP'
$$
Idea of the proof: complete the sum on $s$ to a sum over $\mathbb{Z}$, then Fourier transform the resulting periodic function and perform the sum over $r$, which has become a geometric series.


