Positive integers: counterexample to W. M. Schmidt's conjecture (Q1946752)

For a real number \(x\) let \(\| x\|\) denote the distance to the nearest integer. In 1976, \textit{W. M. Schmidt} [Monatsh. Math. 82, 237--245 (1976; Zbl 0337.10022)] proved that if \(\alpha_1\), \(\alpha_2\) are real numbers such that \(1,\alpha_1,\alpha_2\) are linearly independent over \({\mathbb Q}\), then there exists a sequence of vectors \(\{(x_1^{(n)},x_2^{(n)})\}_{n\geq 1}\) whose components are positive integers and such that the quantity \[ \| x_1^{(n)} \alpha_1+x_2^{(n)} \alpha_2\|\left( \max\{x_1^{(n)},x_2^{(n)}\}\right)^{\phi} \] tends to \(0\) with \(n\), where \(\phi=(1+{\sqrt{5}})/2\) is the golden section. In the same work, Schmidt conjectured that the same statement holds with \(\phi\) replaced by \(2-\varepsilon\) for any \(\varepsilon>0\). In the paper under review, the author refutes Schmidt's conjecture by constructing a counterexample \(\alpha_1,\alpha_2\) for the exponent \(\sigma\) equal the largest root of \(x^4-2x^2-4x+1=0\), a number about \(1.94696\ldots\). The proof is based on a new result of the author in the geometry of numbers which he calls fundamental lemma whose statement is one page long and whose proof occupies most of the paper. No new conjecture is offered as to what should be the optimal exponent (between \(\phi\) and \(\sigma\)) in Schmidt's theorem.