I am trying find an upper bound for the following integral, I know the expression is less than $1$, and I would prefer to find a bound that looks like $\frac{1}{2^{f(n)}}$. Where $f(n)$ is any function in $n$.
The integral is: $$\frac{1}{r^n} \int_{||\vec{u}|| \le n 2^n C} e^{-\pi ||(\vec{x} + \vec{u})/r||^2} d \vec{u}$$
We also know the following:
$n \in \mathbb{N}$
$\vec{x}$ and $\vec{u}$ are vectors in $\mathbb{R^n}$.
$C$ and $r$ are just constants.
$||\vec{x}|| \ge \sqrt{n} r$
$r > 2^{2n} C$
I am a bit stumped so any hints or tips would he gladly appreciated.
I've tried doing the substitution $\vec{y} = \vec{u} + \vec{x}$ and integrate over $\mathbb{R^n}$ instead of the ball, but the bound is $1$ in that case which is too high.
I've also tried using the reverse triangle inequality $-||\vec{x}+\vec{u}||^2 \leq -(||\vec{x} - \vec{u}||)^2$
As I indicated to you in chat, the most productive approach seems to be to make the change of variables $\vec y = \frac 1r (\vec x + \vec u)$. Setting $\delta = n2^nC/r<n/2^n$, you'll get $$\frac 1{r^n}\int_{\|\vec u\|\le n2^n} e^{-\pi\|(\vec x+\vec u)/r\|^2} d\vec u = \int_{\|\vec y-\vec x\|\le\delta} e^{-\pi\|\vec y\|^2}d\vec y < ke^{-\pi(\|\vec x\|-\delta)^2}\delta^n,$$ where the constant $k$ is the volume of the unit ball in $\Bbb R^n$. You should be able to get good estimates from this.