Threshold functions for systems of equations in random sets

Speaker: 

Ana Zumalacarregui

Affiliation: 

UNSW

Date: 

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

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

We will study the existence of certain additive structures in random sets of integers. More precisely, let $Mx=0$ 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 $Mx=0$".

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.

School Seminar Series: