Transcendence of certain sequences of algebraic numbers (Q6590164)

Let \(D\) and \(d\) be two positive integers. Let \(K\) be a number field of dimension \(d\) and \(x_1,\dots ,x_D\in K\). Consider real numbers \(\alpha, \delta, \varepsilon >0\), \(\beta, \eta_1\geq 0\) and \(\eta_2, \gamma \geq 1\) such that \(\alpha <1\), \(\beta <\frac {\varepsilon}{1+\varepsilon}\) and \(\eta_1\leq (d-1)\gamma +\beta\). Let \(\{a_n\}_{n=1}^\infty\) be a sequence of non-zero numbers given by \(a_n=\sum_{i=1}^D a_{i,n}x_i\) with \(a_{i,n}\in\mathbb Z\) such that for all sufficiently large \(n\) we have \(n^{1+\varepsilon}\leq | a_n|\leq | a_{n+1}|\), \(|\mathcal N_K(a_n)| \geq | a_n|^{\eta_1}2^{-\log_2^\alpha| a_n|}\) and \(\gcd(a_{1,n},\dots ,a_{D,n})| \mathcal N_K(\frac {a_n}{\gcd(a_{1,n},\dots ,a_{D,n})})| \leq | a_n|^{\eta_2}2^{\log_2^\alpha| a_n|}\) where \(\mathcal N_K(x)\) denotes the field norm of \(x\). Let \(\{b_n\}_{n=1}^\infty\) be a sequence of positive integers such that for some fixed \(\zeta\in\mathbb C\) each \(i=1,\dots ,D\) and all sufficiently large \(n\) we have \(b_n\leq | a_n|^{\beta}2^{\log_2^\alpha| a_n|}\), \(| a_{i,n}|\leq | a_n|^\gamma 2^{\log_2^\alpha| a_n|}\) and \(\mathfrak R(\zeta a_n)>0\). Then the author proves that for every sequence \(\{c_n\}_{n=1}^\infty\) of positive integers the sum of the series \(\sum_{n=1}^\infty \frac {b_n}{c_na_n}\) is irrational if \(\limsup_{n\to\infty} | a_n|^{(\frac {d(\gamma +\beta)}{1-\beta}+1)^{-n}}=\infty\) and it is transcendental if \(\limsup_{n\to\infty} | a_n|^{(\frac {d^2(\gamma +\beta)}{1-\beta}+1)^{-n}}=\infty\).\N\NThe paper contains also two other similar results. The proofs are based on the Schmidt's subspace theorem.