Factor Complexity of the Most Significant Digits of~$a^{n^d}$

Abstract

We investigate unipotent dynamics on a torus and apply these techniques to the following problem. Let \(d\) be a positive integer, and let \(a > 0\) be a real number. For an integer \(b \geqslant 5\), such that \(a\) and \(b\) are multiplicatively independent, consider the sequence \((\mathbf{w}_n)\), where \(\mathbf{w}_n\) is the most significant digit of \(a^{n^d}\) when expressed in base \(b\). We prove that the complexity function of the sequence \((\mathbf{w}_n)\) is, up to finitely many exceptions, a polynomial function.