TITLE: How does the bound in the large sieve depend on the norm on the lattice?
QUESTION [8 upvotes]: I've been reading about the large sieve inequality in Serre's "Lectures on the Mordell-Weil theorem", which states it in the following setting, which I've simplified a bit here:
Suppose $\Lambda \cong \Bbb Z^n$ is a lattice and we have a norm $\| \cdot \|$ on $\Lambda_{\Bbb R} = \Lambda \otimes \Bbb R \cong \Bbb R^n$.
Then there is a constant $c = c(\Lambda, \|\cdot \|)$ depending only on $\Lambda$ and the norm $\| \cdot \|$, such that for any set $X$ with the following properties:

$X$ is contained in a ball of radius $N$ for the norm  $\| \cdot \|$
some "local sieving" conditions at primes $\le Q$,

we have $|X| \le c \sup(N^n, Q^{2n})/L(Q)$, where $L(Q)$ is a function of the local densities used in the sieve.
Is there a treatment of the large sieve that spells out how the upper bound in this sieve depends on the choice of norm $\| \cdot \|$, or equiavlently on the shape of the ball bounding $X$?  I don't want explicit constants, just a qualitative sense of the dependence.
I've looked at the example of Chapter 4 of Kowalski's book on the large sieve, which gives a bound in the case where the ball is a paralellepiped aligned with the coordinate axes, but I'm interested in a more general situation (e.g. a general parallelepiped or ellipsoid).

REPLY [4 votes]: One approach is to reduce to the parallelepiped case. Let $\Lambda'$ be the dual lattice of $\Lambda$ with the dual norm $$|| y|| =  sup_{x \in \Lambda} \frac{ x \cdot y}{ || x||}.$$ Let $\lambda_1,\dots, \lambda_n$ be the successive minima of $\Lambda'$. Let $w_1,\dots, w_n$ be the associated shortest vectors, forming a basis of $\Lambda'$. Let $v_1,\dots, v_n$ be the dual basis of $\Lambda$. Then we have
$$ \left|\left|\sum_i a_i v_i  \right|\right| \leq \sup_{j\in [n]} \frac{|\sum_i a_i v_i \cdot w_j|} { \lambda_j} = \sup_{j\in [n]} \frac{ |a_j| }{\lambda_j}$$ so the ball of radius $N$ is contained in the parallelepiped $$\left \{ \sum_i a_i v_i ,  |a_i| \leq \lambda_i N \right\}$$
and then you can apply the version from Kowalski's book.