In this talk I will present a dynamical approach to understanding the combinatorial structure of leading‐digit sequences by connecting flows on homogeneous spaces with factor complexity. Our work shows that many naturally occurring digit sequences—notably those arising from exponential, factorial, and Gamma function contexts—exhibit surprisingly structured behavior, as measured by the growth of their complexity functions.
First, we consider sequences formed by taking the most significant digit of numbers of the form a nda\,n^d (with a>0a>0 and a fixed base b>5b>5 under a multiplicative independence condition). By leveraging unipotent dynamics on a torus—viewed as a homogeneous space—we demonstrate that the number of distinct factors of length kk in such sequences eventually agrees with a polynomial P(k)P(k) of degree d(d+1)/2d(d+1)/2 (up to finitely many exceptions). This result, obtained via a careful analysis of the partitioning of the torus by families of parallel hyperplanes, illuminates the link between arithmetic properties and the geometry of flows.
In a complementary direction, I will discuss leading-digit words generated by factorial and Gamma sequences. Here, a detailed geometric investigation yields sharp upper and lower bounds on the factor complexity. In particular, we distinguish between “significant” factors—those appearing in the natural geometric partition—and “residual” factors, for which we prove that their number grows asymptotically like Θ(k2lnk)\Theta(k^2\ln k). These findings reveal a rich interplay between recurrence properties, uniform distribution, and symbolic dynamics.
By combining tools from ergodic theory, Diophantine approximation, and combinatorics on words, this talk will illustrate how dynamical systems methods can provide deep insight into arithmetic phenomena, bridging ideas from continued fraction algorithms and homogeneous dynamics to the modern study of automatic sequences and complexity measures.