Schmidt's conjecture on normality for commuting matrices (Q1802540)

The authors prove the commuting case of a longstanding conjecture due to W. M. Schmidt on a generalization of Borel normality. A vector \(x\in\mathbb{R}^ n\) is normal with respect to the \(n\times n\) matrix \(S\) or \(S\)-normal if the sequence \((S^ nx:n=1,2,\dots)\) is uniformly distributed mod 1, so that the space and time averages tend to the same limit over the \(n\)-dimensional torus \(\mathbb{T}^ n\). A matrix \(S\) is called ergodic if almost every \(x\in\mathbb{R}^ n\) is \(S\)-normal and two such ergodic systems generated by \(S\) and \(T\) are normally equivalent if \(S\)- normality and \(T\)-normality coincide. An \(n\times n\) invertible matrix \(T\) with rational entries is almost integral if its eigenvalues are algebraic integers; such matrices are ergodic iff their eigenvalue are not 0 or a root of unity. Let \(S,T\) be commuting almost integer ergodic matrices. Schmidt showed that if there exist positive integers \(p,q\) such that \(S^ p=T^ q\) then every \(S\)-normal \(x\) is \(T\)-normal and vice- versa; and that otherwise there exist uncountably many points in \(\mathbb{R}^ n\) normal in base \(S\) but not in base \(T\) and vice-versa, providing the eigenvalues of \(S\) and \(T\) had modulus \(>1\). He also gave a complete solution in the one-dimensional case. The authors prove the second part of the result without the restriction on the modulus of the eigenvalues by using Riesz product measures instead of measures drawn from the class of Bernoulli convolutions. A special case of their main result is that if two normally equivalent ergodic systems are generated by commuting matrices \(S\) and \(T\) then some power of \(S\) is a power of \(T\). Their arguments are based on those of Schmidt and provide a short proof of the one dimensional case. The authors state that they will be reporting on substantial progress in the noncommuting case.