Algebraic independence of power series generated by linearly independent positive numbers (Q1773445)

The main result of this paper is the Theorem: Let \(m\) be a positive integer and let \(\omega_1, \omega_2, \dots ,\omega_m\) be positive real numbers. For every \(i=1,2,\dots ,m\) and \(d=2,3,4,\dots\) denote \(f_{id}(z)=\sum_{k=0}^\infty a_{idk} z^{[\omega_id^k]}\) where \(a_{idk}\) (\(k=0,1,2,3,\dots\)) are in a finite set of nonzero algebraic numbers for every \(i\) and for every \(d\). If the numbers \(\omega_1, \omega_2, \dots ,\omega_m\) are linearly independent over the rational numbers then the numbers \(f_{id}(\alpha)\) (\(i=1,2,\dots ,m\); \(d=2,3,4,\dots \)) are algebraically independent for any algebraic number \(\alpha\) with \(0<\mid \alpha \mid <1\). The proof is based on the Mahler's method. Some consequences are included.