Threshold functions for systems of equations on random sets

I will present a unified framework to deal with threshold
functions for the existence of certain combinatorial structures in random
sets. More precisely, let M·x=0 be a linear system defining our structure
(k-arithmetic progressions, k-sums, B_h[g] sets or Hilbert cubes, for
example), and A be a random set on {1,...,n} where each element is chosen
independently with the same probability.

I will show that, under certain natural conditions, there exists a
threshold function for the property "A^m contains a non-trivial solution of
Mx=0" which only depends on the dimensions of M. I will focus on the
behavior of the limiting distribution for the number of non-trivial
solutions in the threshold scale, and show that it follows a Poisson
distribution in terms of volumes of certain convex polytopes arising from
the linear system under study.

(Joint work with J. Rué)