A problem about Gaussian distribution in paper GPV08
The Gaussian function $\rho(x)=\exp(-\pi \| x \|^2)$ satisfies the property that if $x,y$ are orthogonal vectors, then $\rho(x+y)=\rho(x) \cdot \rho(y)$. This follows directly from the definition and the Pythagorean theorem. So, we can cancel out the contribution due to $c-c’$, which is orthogonal to every element in $\Lambda$, from both the numerator and the denominator.