Variations on a theme of K. Mahler, I.

Abstract. For an integer \(n\) denote \((n)_g\) the sequence of digits of the \(g\)-ary representation of \(n\). Mahler [17] proved that the number \(0.(1)_{10}(g)_{10}(g^2)_{10}\ldots\) is irrational for any \(g\ge 2\). It has many generalizations and refinements. Here we prove further generalizations. In the first direction we replace the sequence of powers by weighted sums of elements of a finitely generated multiplicative semigroup of a number field. In the second direction, the base \(g\) is replaced by an algebraic integer. As a byproduct, we prove a Mahler-type result replacing the sequence of powers by a fixed coordinate of solutions of a norm form equation.