An explicit lower bound for the block complexity of an algebraic number (Q993423)

In [J. Number Theory 67, No. 2, 146--161 (1997; Zbl 0895.11029)] \textit{S. Ferenczi} and \textit{C. Mauduit} showed that the complexity \(p(n,\xi,b)\) of a non-rational algebraic number \(\xi\) -- that is, the number of different blocks of length \(n\) in the \(b\)-ary expansion of \(\xi\) -- satisfies \(p(n,\xi,b) - n \to \infty\). The purpose of this paper is to quantify this property in an effective way. It is show that \(p(n,\xi,b) \geq (1+1/M)n\), where \(M\) is a \textit{huge} but explicit number in terms of the degree and the height of \(\xi\). The proof is based on the Quantitative Subspace Theorem of Wolfgang Schmidt.