Abstract:

A real number is normal to base m if its base-m expansion behaves statistically in the same manner as a sequence of digits chosen independently and with uniform distribution from {0,...,m-1}. Let q>1 be such that gcd(q,m)=1. In 1995 Host proved that if P is a \times q invariant measure on [0,1] satisfying some natural assumptions, then P almost every x is normal to base m. This result is related to classical Theorems of Cassels and Schmidt from around 1960, and to Furstenberg's \times 2, \times 3 Conjecture. In 2001 Lindenstrauss showed that it holds assuming only that q does not divide any power of m. In 2015 Hochman and Shmerkin showed that it holds in the "correct" generality, i.e. assuming only that q and m are independent. In this talk, we shall present a simultaneous version of this result. That is, for P typical x, we show that the orbit of (x,x) under the diagonal toral endomorphism (\times q, \times m) equidistributes for the product measure of P with the Lebesgue measure.

A real number is normal to base m if its base-m expansion behaves statistically in the same manner as a sequence of digits chosen independently and with uniform distribution from {0,...,m-1}. Let q>1 be such that gcd(q,m)=1. In 1995 Host proved that if P is a \times q invariant measure on [0,1] satisfying some natural assumptions, then P almost every x is normal to base m. This result is related to classical Theorems of Cassels and Schmidt from around 1960, and to Furstenberg's \times 2, \times 3 Conjecture. In 2001 Lindenstrauss showed that it holds assuming only that q does not divide any power of m. In 2015 Hochman and Shmerkin showed that it holds in the "correct" generality, i.e. assuming only that q and m are independent. In this talk, we shall present a simultaneous version of this result. That is, for P typical x, we show that the orbit of (x,x) under the diagonal toral endomorphism (\times q, \times m) equidistributes for the product measure of P with the Lebesgue measure.

## Date:

Wed, 15/05/2019 - 16:00 to 17:00

## Location:

Manchester Building (Hall 2), Hebrew University Jerusalem