Factor Complexity of the Most Significant Digits of~a^nd

2024/02/25 by Golafshan, Mehdi, Mitrofanov, Ivan
#11A63 Secondary: 68R15 #11K06 #11K16 #37B10 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary:11K31

paper · doi:10.48550/arxiv.2402.16210

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 \((w_n)\), where \(w_n\) is the most significant digit of \(a^nd\) when expressed in base \(b\). We prove that the complexity function of the sequence \((w_n)\) is, up to finitely many exceptions, a polynomial function.

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