Threshold functions for systems of equations in random sets

Abstract: 

We will study the existence of certain additive structures in random sets of integers. More precisely, let Mx=0Mx=0 be a linear system defining our structure (kk-arithmetic progressions, kk-sums, Bh[g]Bh[g] sets or Hilbert cubes, for example) and AA be a random set on {1,...,n}{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 "AmAm contains a non-trivial solution of Mx=0Mx=0".

Furthermore, we will show that the number of solutions in the threshold scale converges to a Poisson distribution whose parameter depends on the volume of certain polytopes arising from the system under study.

Joint work with J. Rue and C. Spiegel.