Directive-sequence conditions for S-adic transcendence

Characterize the conditions on a directive sequence of substitutions that ensure every word in the associated S-adic shift yields a transcendental number $(w)_\beta$ for every algebraic base $\beta$ with $|\beta|>1$.

Background

The paper studies S-adic shifts generated by sequences of substitutions and focuses on systems with pure discrete spectrum, represented geometrically through Rauzy fractals and tilings. Its main theorem supplies sufficient conditions involving recurrence, unimodularity, tiling with compact sets, and dendricity. The broader problem of identifying which directive-sequence conditions guarantee transcendence for every word and every algebraic base remains open.

References

Which conditions on directive sequences~$\mathbf{\sigma}$ ensure that every word $w$ in the $S$-adic shift~$X_{\mathbf{\sigma}}$ provides a transcendental number $(w)_{\beta}$ for each algebraic base $\beta$ with $|\beta| > 1$?

— Transcendence of dendric beta-expansions with pure discrete spectrum  (2610.00856 - Seo et al., 1 Oct 2026) in Question B, Section 1, Introduction and main result