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).



How does the bound in the large sieve depend on the norm on the lattice?
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.


