Interpreting Figure 1 in the "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography"

Question

I'm reading "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography" by O. Regev.

I'm having trouble with understanding graphs in Figure 1.

By the definition of $\overline{\psi}_{\alpha}$, its support is $[0,1)$.

However, graphs in Figure 1 use support as $[-1,1] \times [-1,1]$.

Can anyone explain me about this?

Answer 1

The points of $\mathbb Z/q\mathbb Z$ are mapped to $\mathbb R/\mathbb Z$ and plotted in a circle. The plot shows \begin{align*} x &= \sin (2\pi i/q), \\ y &= \cos (2\pi i/q), \\ p &= \Psi_\alpha(i/q). \end{align*} for each $i$ in the least nonnegative residues modulo $q$.

The following gnuplot script gives a similar graph, for $q = 31$:

set parametric
set samples 31
psi(alpha, r) = exp(-pi*(r/alpha)**2) + exp(-pi*((r - 1)/alpha)**2)
splot [i=0:30] sin(2*pi*i/31), cos(2*pi*i/31), psi(0.1, i/31) with points