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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.

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$$ L[f](P_0) = \left( \sum_{r,s\in \mathbb{N}} - \sum_{r,s\in -\mathbb{N}^*}\right) (-1)^r f(\beta r +\beta^{-1}s) \\ -2\beta \sum_{n\in\mathbb{N}^*} \frac{\sin \pi n\beta P_0}{\cos \pi n\beta^2} \int_{\mathbb{R}} f(P') \sin\pi \beta n(P_0+2P'+\beta) dP' $$ 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.

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