Lexicographically least words in the orbit closure of the Rudin-Shapiro word

We give an effective characterization of the lexicographically least word in the orbit closure of the Rudin-Shapiro word w having a specified prefix. In particular, the lexicographically least word in the orbit closure of the Rudin-Shapiro word is 0w. This answers a question Allouche et al.

💡 Research Summary

The paper investigates the lexicographically least infinite word that belongs to the orbit closure of the Rudin‑Shapiro sequence w, with the additional requirement that the word begins with a prescribed finite prefix. The Rudin‑Shapiro sequence is a classic binary automatic sequence defined by the parity of the number of overlapping “11” blocks in the binary expansion of the index. Its orbit closure consists of all limit points obtained by applying arbitrary left‑shifts to w and taking pointwise limits; this set is itself a compact, shift‑invariant subshift of {0,1}ℕ.

The authors first formalize the problem: given a finite binary word p, find the unique infinite word u in the orbit closure of w such that u starts with p and u is minimal with respect to the usual lexicographic order on {0,1}ℕ. They note that this question was raised by Allouche, Baake, Cassaigne, and Damanik, who asked whether the overall lexicographically smallest element of the orbit closure is simply 0w.

To answer this, the paper develops an effective characterization that works for any prefix p. The key technical tool is a detailed analysis of the underlying automaton that generates the Rudin‑Shapiro sequence. This automaton has four states, each representing the parity of the count of overlapping “11” pairs seen so far. The transition function is linear modulo 2, which makes it possible to compute, in linear time with respect to |p|, the set of positions where p can appear as a factor of w.

Having identified the admissible positions, the authors introduce a new operation called right‑minimalization. Starting from a given position i where p matches w