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 $$\mid \sum _{n\in {\mathbb{Z}}^k}f\text{\hspace{0.17em}⁣}\left(\frac{1}{N}n\right)\exp \left\{2\pi i((\frac{1}{2}\mathrm{\parallel }n{\mathrm{\parallel }}^2+\beta \cdot n)x+\alpha \cdot n)\right\}\mid \le C{N}^{k\mathrm{/}2}$$ 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 $N^{k\mathrm{/}2}$ agrees with an automorphic function $\mathrm{\mid }{\mathrm{\Theta }}_f\mathrm{\mid }$ 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 $\mathrm{\mid }{\mathrm{\Theta }}_f\mathrm{\mid }$ remains bounded along the entire horocycle lift.