On a simultaneous approximation of logarithms and algebraic powers of algebraic numbers (Q1905251)

Let \(\theta\in \mathbb{C}\) be an arbitrary transcendental number. Denote \(\mathbb{Q}_1= \mathbb{Q}(\theta)\) and \(\mathbb{J}_1= \mathbb{Z}[\theta]\). Let \(\mathbb{Q}^*_1\) be an algebraic extension of \(\mathbb{Q}_1\) of finite degree generated by the numbers \(\theta\) and \(\omega_1\), where \(\omega_1\) is a root of an irreducible polynomial in \(\mathbb{J}_1[x]\). For any \(\eta\in \mathbb{J}_1\), \(\eta= G_m \theta^m+\cdots+ G_1\theta+ G_0\), \(G_i\in \mathbb{Z}\) \((0\leq i\leq m)\), \(G_m\neq 0\), we define \(v(\eta)= m+ \log\max_{0\leq i\leq m} |G_i|+ 1\); and for any \(\kappa\in \mathbb{Q}^*_1\), \(\kappa= (D_{\nu- 1} \omega^{\nu- 1}_1+\cdots+ D_1\omega_1+ D_0)/D\), where \(D, D_0,\dots, D_{\nu- 1}\in \mathbb{J}_1\) have no common divisor in \(\mathbb{J}_1\), we define \(v(\kappa)= \max(v(D), v(D_0),\dots, v(D_{\nu- 1}))\). Strengthening the method used in his previous paper [Math. Notes 48, 1258-1266 (1990); translation from Mat. Zametki 48, No. 6, 126-136 (1990; Zbl 0739.11028)], the author proves the following theorem: Let \(a_1\), \(a_2\), \(a_3\) be algebraic numbers, and \(\beta\) be an algebraic irrational number. Denote \(\delta_i= \log a_i\) \((i= 1, 2,3)\) and \(\delta_{3+ j}= a^\beta_j\) \((j= 1, 2, 3)\), and for \(\eta_k\in \mathbb{Q}^*_1\) \((k= 1,\dots, 6)\) set \(W= \max(v(\eta_1),\dots, v(\eta_6))\). If \(\delta_1\), \(\delta_2\), \(\delta_3\) are linearly independent over \(\mathbb{Q}\), then the inequality \[ \sum^6_{i= 1} |\delta_i- \eta_i|< \exp(- W^{40}\log^{- 2}W) \] has only a finite number of solutions in \(\eta_1,\dots, \eta_6\in \mathbb{Q}^*_1\).