Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra.

I want to understand the following theorem from the book "Symmetric Bilinear Forms" by J. Milnor and D. Husemoller.

Theorem (9.5, pp 46) For any dimension n there exists a positive definite inner product space $X$ of type $1$ and rank $n$ with $\min_{x \in X \setminus {0}} x.x \geq n ($n \rightarrow \infty$)$.

In particular, the theorem implies the existence of a self-dual lattice with shortest vector \geq \sqrt{n}.

The proof of this theorem is a byproduct of Siegel's theorem, which can be seen as a bound on the "average" number of solutions to a quadratic equation. The equation of interest for me is x.x = k i.e., the number of vectors of a particular length $k$ from an inner-product space $X$.

The above mentioned book does not give a proof for the Siegel's theorem. Additionally, the proof of Theorem 9.5 uses some results over p-adic integers and others on genus of bilinear form spaces to show that the number of solutions of the equation x.x=k, summed for k \in {1, \dots, n} is <2 if averaged over all distinct linear product space in the genus $I_n$.

My aim is to understand this proof.

What are the books which I should start with to understand the proof ? The Milnor book is too short and seems to be aimed at experts. Note that I do not understand the p-adic integers and so it is very difficult for me to make sense of the proof.