Applications of measure rigidity to recurrence

Speaker
Alexander Fish, School of Mathematics and Statistics, University of Sydney, Australia
Date
22/01/2017 - 15:00 - 14:00Add to Calendar 2017-01-22 14:00:00 2017-01-22 15:00:00 Applications of measure rigidity to recurrence   We present a new approach (joint with M. Bjorklund (Chalmers))  for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.   By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k. By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers. seminar room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
seminar room
Abstract

 

We present a new approach (joint with M. Bjorklund (Chalmers))  for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.  

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

Last Updated Date : 16/01/2017