Suppose we have $\Gamma$ which is a lattice of $\mathbb{R}^d$ ($rk(\Gamma) = d$), and $| \cdot |$ a norm on $\mathbb{R}^d$. We consider as well $c_1, c_2 > 0$, and the annulus $S = \{ P \in \Gamma \, | \, c_1 \leq |P| \leq c_2 \}$. How one could find a bound $\#S \leq c(c_1, c_2, \Gamma)$, if it's possible to give a "closed form" for $c(c_1, c_2, \Gamma)$ ?
Thank you !
Well, the annulus is contained in the ball of radius $c_2,$ so the asymptotic bound is the volume of the ball of radius $c_1$ divided by the covolume. This will be (possibly) way off for small radii, so if $\ell$ is the shortest vector in $\Gamma,$ then the volume of the ball divided by $\ell^d$ will work.