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.


Ana Zumalacarregui

Research Area



Thu, 06/04/2017 - 3:00pm


RC-4082, The Red Centre, UNSW