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.



Inner product spaces, Siegel's theorem and lattices: book suggestion 
Hi there, I think we had better give you a start here...


You have combined together a few ideas that come from very different areas of inquiry. 


In one direction, kissing numbers and Minkowski-Hlawka (Milnor and Husemoller, page 31) see Table 1.3 on pages 15-17 of SPLAG, that is *Sphere Packings, Lattices and Groups* by Conway and Sloane. Self-dual is typically called "unimodular," see the bottom of page 53. Easier introductions that will still lie in comfortable territory are *Lattices and Codes* by Wolfgang Ebeling, also *From Error Correcting Codes Through Sphere Packings to Simple Groups* by Thomas M. Thompson. Note that your Thompson is J. G. Thompson. 


See <http://en.wikipedia.org/wiki/Unimodular_lattice>


As to ***exact*** numbers of representations, I actually recommend a much earlier book, *The Arithmetic Theory of Quadratic Forms* by Burton W. Jones. I see that in *Rational Quadratic Forms* by Cassels, he does three squares on page 150, Lemma 6.4, then four squares on page 152, Lemma 6.5. 


In a few famous cases, notably four squares and eight squares, the exact number of representations has a fairly clean expression, due to Jacobi. <http://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem> 


See <http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/unimodular.html>


Finally, you might try <https://cstheory.stackexchange.com/> with a more obviously computer-centric version of your question. 


