On simultaneous Diophantine approximation (Q2145923)

Let \(\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_m)\in {\mathbb R}^m\) be such that \(1,\alpha_1,\ldots,\alpha_m\) are linearly independent over \({\mathbb Q}\) and for a positive integer \(x\) put \[ b(x)=\max\{ \| \alpha_1 x\|,\ldots, \|\alpha_m x\|\}. \] By Dirichlet's theorem for any positive real \(q\) there is \(x\in {\mathbb Z}_{\ge 1}\), \(x\le e^{mq}\) such that \(b(x)\le e^{-q}\). In particular, \(x b(x)^m\le 1\) holds for infinitely many positive integers \(x\). Let \(1=x_1<x_2<\cdots\) be the positive integers which are best possible for the above inequality, namely that for all \(x<x_j\) we have \(b(x)\ge b(x_{j-1})\). Call \(\boldsymbol{\alpha}\) badly approximable if \(x b(x)^m\gg 1\). Set \(b_j:=b(x_j)\) and consider the following conditions on \(\boldsymbol{\alpha}\): \begin{itemize} \item[(i)] \({\alpha}\) is badly approximable. \item[(ii)] \(x_{j+1}/x_j\ll 1.\) \item[(iii)] \(b_j/b_{j+1}\ll 1\). \end{itemize} When \(m=1\) it is known that the above conditions are equivalent and recent work of \textit{R. Akhunzhanov} and \textit{N. Moshchevitin} [Math. Z. 301, No. 2, 1573--1602 (2022; Zbl 1492.11110)] shows that for every \(m\ge 1\), (i) implies (ii) and (iii). The main result of the paper is to show that for every \(m\) there are \(\boldsymbol{\alpha}\) with (iii) but not (i) or (ii). The proof uses a theorem of \textit{D. Roy} [Ann. Math. (2) 182, No. 2, 739--786 (2015; Zbl 1328.11076)] on parametric geometry of numbers.