On the values of quadratic forms on difference sets or on non-periodic Bohr sets (or what did I do this summer)

Abstract: I will be talking about a recent progress on the following question: Given a quadratic form Q in N variables (of a mixed signature) over the integers, show that the set of values:

1) Q(E-E) for a large set E inside Z^N contains a non-trivial subgroup of the integers.

2) Q(B) = Q(Z^N) for any non-periodic Bohr set B in Z^N.

Some parts of the talk are based on the joint work (in progress) with Kamil Bulinski (Sydney).