Bounds for smooth theta sums with rational parameters

We provide explicit families of pairs $(\text{α},\text{β})\in R^k\times R^k$ such that for sufficiently regular f, there is a constant C for which the theta sum bound$|\sum _{\text{n}\in Z^k}f\left(\frac{1}{N}n\right)\exp \{2\pi i\left(\right(\frac{1}{2}{\Vert n\Vert }^2+\beta \cdot n)x+\alpha \cdot n)\left\}\right|\le C{N}^{k/2},$ holds for every $x\in R$ and every $N\in N$. 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 $(\text{α},\text{β})$, and so the bound follows from showing that there are pairs such that $\left|{\Theta }_f\right|$ remains bounded along the entire horocycle lift.