Schmidt's conjecture on normality for dimension two. (Q1865803)

In a natural extension of the definition for the reals, a vector \({\mathbf x}\in\mathbb{R}^n\) is said to be \(S\)-normal for a real \(n\times n\) matrix \(S\) if the sequence \((\{S^n{\mathbf x}\})\) is uniformly distributed \(\text{mod\,}1\) in each coordinate. The analogue of Borel's theorem that almost all real numbers are normal no longer holds as there are many regular matrices for which no \({\mathbf x}\in\mathbb{R}^2\) is normal. Matrices \(S\) for which the set \(B(S)\) of \(S\)-normal points in \(\mathbb{R}^n\) has full Lebesgue measure are called ergodic. A rational non-singular matrix \(T\) is called almost integer if all of its eigenvalues are algebraic integers. Ergodic almost integer matrices have no eigenvalues that are roots of unity. It is straightforward to show that if for ergodic matrices \(S\), \(T\), there are natural numbers \(p\), \(q\) such that \(S^p= T^q\), then \(B(S)= B(T)\). Under the additional assumptions that the eigenvalues lie outside the closed unit disc and that \(ST= TS\), W. M. Schmidt extended his one-dimensional converse result to \(n\) dimensions by proving that otherwise there are uncountably many points in \(\mathbb{R}^n\) which are normal in base \(S\) (resp. \(T\)) but not normal in base \(T\) (resp. \(S\)) [\textit{W. M. Schmidt}, J. Reine Angew. Math. 214/215, 227--260 (1964; Zbl 0135.10803)]. He conjectured that the result held without the additional assumptions. In previous work [\textit{G. Brown} and \textit{W. Moran}, Invent. Math. 111, 449--463 (1993; Zbl 0773.11049)], the conjecture was proved for commuting \(2\times 2\) almost integer matrices. In the present paper the authors establish the conjecture for \(n= 2\). The proof uses measures constructed from Riesz products and involves results from harmonic analysis, Diophantine approximation and uniform distribution. The hope is expressed that a proof for \(n\geq 2\) can be obtained using suitable Riesz products.