On the $b$-ary expansion of a real number whose irrationality exponent is close to 2
Yann Bugeaud, Dong Han Kim
TL;DR
This work links the arithmetic complexity of a real number’s base-$b$ expansion to its irrationality exponent. By translating $\mu(\xi)$ into lower bounds on the return-time based repetition metrics ${\rm rep}$ and ${\rm Rep}$, it derives explicit, improved lower bounds for the block-complexity growth $p(n,\xi,b)/n$, valid for $2 \le \mu(\xi) < \mu_1\approx 2.246$ (liminf) and $2 \le \mu(\xi) < \mu_2\approx 2.324$ (limsup). Key technical contributions include a sharper bound linking ${\rm Rep}(\mathbf{x})$ to ${\rm rep}(\mathbf{x})$, and the deduction of the main inequalities via Sturmian-type results, strengthening prior results of BuKim17. The results imply, in particular, nontrivial complexity growth for all irrational numbers with $\mu(\xi)=2$, with concrete constants, and illustrate the intricate interplay between Diophantine approximation and the combinatorial structure of digit sequences. The Fibonacci word example highlights sharpness and structural phenomena behind these bounds and situates the work within the broader study of base-$b$ normality and Sturmian sequences.
Abstract
Let $b \ge 2$ be an integer and $ξ$ an irrational real number. We establishes that, if the irrationality exponent of $ξ$ is less than $2.324 \ldots$, then the $b$-ary expansion of $ξ$ cannot be `too simple', in a suitable sense. This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017].