Bounds for Smooth Theta Sums with Rational Parameters

Abstract

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.