Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ according to the discrete gaussian distribution with parameter $s$ defined as follows. For any real $s>0$, define the (spherical) Gaussian function $\rho_{s} : \mathbb{R}^n \rightarrow (0,1]$ with parameter $s$ as:
$\forall \vec{x} \in \mathbb{R}^n, \rho_{s}(\vec{x})=\exp(-\pi \langle\vec{x},\vec{x}\rangle/s^2) = \exp(-\pi \|\vec{x}\|^2/s^2).$
For any real $s > 0$, define the (spherical) discrete Gaussian distribution over $\mathbb{Z}^n$ as:
$\forall \vec{x} \in \mathbb{Z}^n, D_{Z^n,s}(\vec{x})= \frac{\rho_{s}(\vec{x})}{\rho_{s}(\mathbb{Z}^n)},$
where $\rho_s(\mathbb{Z}^n)$ is equal to $\sum_{\vec{x} \in \mathbb{Z}^n} \rho_s(\vec{x})$.
This vector $g$ is then viewed as a polynomial where the ith component of the vector is seen as the coefficient of the term $x^{i-1}$. Now viewing this g as a polynomial if we evaluate this polynomial at a root of unity then this value will be complex.
My question is what is the probability that $|g(\omega)| < \epsilon$ where $\omega$ is a root of the polynomial $x^n +1$ (that is $\omega$ is a primitive $2n$th root of unity) for some $\epsilon < 1$.