# Pointwise bounds for Dirichlet kernel over truncated lattice

In 1 dimension, a "one-sided" Dirichlet kernel $D_N(x)=\sum_{k=0}^{k=N}e^{\frac{2\pi}{N}ikx}$ has its module sharply peaked around points corresponding, roughly, to the "dual lattice" $N\cdot\mathbb{Z}$ (which area really integer points in this case).

Now consider a lattice $\Lambda \subseteq \mathbb{Z}^2$. Let $\Lambda_N = \Lambda \cap [0,N]^2$ and a Dirichlet kernel restricted to this truncated lattice: $D_{\Lambda_N}(x)=\sum_{v \in \Lambda_N}e^{i\langle x,v\rangle}$.

We would expect that $|D_{\Lambda_N}|$ will be peaked around points from $N\cdot\Lambda^{\ast}$, where $\Lambda^{\ast}$ is the dual lattice. However, there is no direct way to perform the summation and obtain a compact formula as in 1D case. Is there any way to estimate $|D_{\Lambda_N}(w)|$ from below, for $w \approx N\cdot\Lambda^{\ast}$?

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