Transcendental series of reciprocals of Fibonacci and Lucas numbers (Q2085767)

Let \(\alpha\) be a real quadratic unit with \(\vert\alpha\vert>1\) and let \(\beta\) be its \(\mathbb{Q}\)-conjugate. Let \((a_n)_{n\geq 1},(b_n)_{n\geq 1},(c_n)_{n\geq 1}\) be real sequences for which \(a_n\in\mathbb{Q},b_n,c_n\in\mathbb{Q}(\alpha)\), \(u_n:=a_n\alpha^n-b_n\beta^n\in\mathbb{Q}\) for every \(n\geq 1\), and with \(\lim_{n\rightarrow\infty}h(a_n)/n=\lim_{n\rightarrow \infty}h(b_n)/n=\lim_{n\rightarrow\infty}h(c_n)/n=0\), where \(h\) denotes the absolute logarithmic Weil height. Let \(c>2\) and let \((n_k)_{k\geq 1}\) be a sequence of positive integers such that \(n_{k+1}/n_k\geq c, u_{n_k}\neq 0\) and \(c_{n_k}\neq 0\) for every \(k\). The main result of the paper then is that \(\sum_{k=1}^\infty c_{n_k}/u_{n_k}\) is transcendental. Take for example \(\alpha=(1+\sqrt{5})/2,a_n=b_n=0\), if \(n\not\in\{n_k; k\geq 1\}\), \(a_n=b_n=1/\sqrt{5}\), otherwise, and \(c_n=1\) for every \(n\geq 1\). The main result applied with this data implies the transcendental counterpart of a question due to \textit{P. Erdős} and \textit{R. L. Graham} [Old and new problems and results in combinatorial number theory. Genève: L'Enseignement Mathématique, Université de Genève (1980; Zbl 0434.10001)] as a special instance, namely that \(\sum_{k=1}^\infty 1/F_{n_k}\) is transcendental. More generally, the main result implies that the sum \(\sum_{k=1}^\infty 1/f_k\) is transcendental if \(f_k\in\{F_{n_k},L_{n_k}\}\) for every \(k\), i.e., if we mix Fibonacci and Lucas numbers randomly along a sequence \((n_k)_{k\geq 1}\). Observe that \(c>2\) is best possible since \(\sum_{k=1}^\infty1/F_{2^k}=7-\sqrt{5}/2\). (Here, \(F_n\) is the \(n\)th Fibonacci and \(L_n\) the \(n\)th Lucas number respectively.) The proof follows by several applications of the subspace theorem in a tricky and innovative way, in which the author uses (and promotes!) the idea of choosing a ``minimal expression'' before applying the Subspace theorem in order to get the desired conclusion from the resulting linear relation. We mention that a first application of the Subspace theorem in the mentioned way proves that the sum is in \(\mathbb{Q}(\alpha)\), which is used in the further rounds of applying the new method in order to prove the main theorem. The paper is very well-written and carefully explains at each step what one would usually do, what the problem then is, and how this problem is circumvented.