Bounds for Smooth Theta Sums with Rational Parameters

We provide an explicit family of pairs \((\alpha, \beta) \in \mathbb{R}^k \times \mathbb{R}^k\) such that for sufficiently regular \(f\), there is a constant \(C>0\) for which the theta sum bound $$\left|\sum_{n\in\mathbb{Z}^k}f\!\left(\tfrac{1}{N}n\right)\exp\left\{2\pi i\left(\left(\tfrac{1}{2}...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Authors: Cellarosi, Francesco, Osman, Tariq
Format: Article
Language:English
Online Access:Get full text
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Summary:We provide an explicit family of pairs \((\alpha, \beta) \in \mathbb{R}^k \times \mathbb{R}^k\) such that for sufficiently regular \(f\), there is a constant \(C>0\) for which the theta sum bound $$\left|\sum_{n\in\mathbb{Z}^k}f\!\left(\tfrac{1}{N}n\right)\exp\left\{2\pi i\left(\left(\tfrac{1}{2}\|n\|^2+\beta\cdot n\right)x+\alpha\cdot n\right)\right\}\right|\leq C N^{k/2}$$ holds for every \(x \in \mathbb{R}\) and every \(N \in \mathbb{N}\). Central to the proof is realising that, for fixed \(N\), the theta sum normalised by \(N^{k/2}\) agrees with an automorphic function \(|\Theta_f|\) evaluated along a special curve known as a horocycle lift. The lift depends on the pair \((\alpha,\beta)\), and so the bound follows from showing that there are pairs such that \(|\Theta_f|\) remains bounded along the entire horocycle lift.
ISSN:2331-8422