On a linear form of powers of rationals (Q1897894)

The author proves the following theorem of simultaneous approximation to powers of rationals by using simultaneous Padé approximations of \((1- r_j z)^{-k}\), \(j=1, \dots, m\). Let \(B\), \(r_1< r_2< \dots< r_m= r\) be positive integers, \(0< \varepsilon< 1/100m\). Suppose that \(f_j: \mathbb N\to \mathbb Z\), \(j=1, \dots, m\) are any functions with \[ 0< \max_{1\leq j\leq m} |f_j (k) |< c(r, m, B)\exp (rk/3 \root{m+1}\of {B}) \] and \(\varepsilon \root{m+1}\of {B} > 3rm\log (Be)\). Then there exists an effective \(k_0\in \mathbb N\), such that for \(k> k_0\) the following inequality holds \[ \biggl|f_1 (k) \biggl( {B\over {B-r_1}} \biggr)^k+ \cdots+ f_m (k) \biggl( {B\over {B- r_m}} \biggr)^k \biggr|> e^{-\varepsilon k}. \] In the special case, if \(m=1\), \(r= B-A\), \(f_1 (k)\equiv 1\) the author gives an effective estimation of fractional parts of powers of rational numbers.