A general criterion for linearly unrelated sequences (Q1881434)

In an earlier paper, the first author [Pac. J. Math. 190, 299--310 (1999; Zbl 1005.11033)] introduced the notion of linearly unrelated sequences as follows. The sequences \(\underline{a}_i:=(a_{i,n})\in \mathbb{R}_+^\mathbb{N} \enspace (i=1,\dots,K)\) are called {linearly unrelated} if, for every sequence \((c_n)\in \mathbb{N}^\mathbb{N}\), the \(K+1\) numbers 1 and \(\sum_{n=1}^\infty 1/(a_{i,n}c_n)\) \((i=1,\dots,K)\) are linearly independent over \(\mathbb{Q}\). In the same paper, he gave a sufficient condition for linear unrelatedness depending, inter alia, on divisibility properties. The main aim of the present paper is to prove another sufficient condition, which is free from divisibility hypotheses but depends only on size conditions. More precisely, the main result is as follows. For \(i=1,\dots,K\), let \(\underline{a}_i, \underline{b}_i \in \mathbb{N}^\mathbb{N}, \underline{a}_i\) non-decreasing and satisfying \(\limsup(\log_2 a_{1,n})/(K+1)^n = \infty\). Let \(\alpha,\varepsilon \in \mathbb{R}_+\) with \(\alpha<1\) be given. For \(n\) large enough, assume \(a_{1,n} \geq n^{1+ \varepsilon}\), \(\max(\log_2 b_{i,n},|\log_2(a_{1,n}/a_{i,n})|) \leq (\log_2 a_{1,n})^\alpha\) for \(i=1,\dots,K\), and finally \(a_{i,n}/b_{i,n}=o(a_{j,n}/b_{j,n})\) for \(1\leq j<i\leq K, \enspace \log_2\) denoting logarithm to base 2. Then the \(K\) sequences \((a_{i,n}/b_{i,n})\in \mathbb{Q}^\mathbb{N}\) are linearly unrelated. Several applications are given, and two open problems are proposed.