Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation

Let $\mb w$ be a morphic word over a finite alphabet $\Sigma$, and let $\Delta$ be a nonempty subset of $\Sigma$. We study the behavior of maximal blocks consisting only of letters from $\Delta$ in $\mb w$, and prove the following: let $(i_k,j_k)$ denote the starting and ending positions, respectively, of the $k$‘th maximal $\Delta$-block in $\mb w$. Then $\limsup_{k\to\infty} (j_k/i_k)$ is algebraic if $\mb w$ is morphic, and rational if $\mb w$ is automatic. As a result, we show that the same conclusion holds if $(i_k,j_k)$ are the starting and ending positions of the $k$‘th maximal zero block, and, more generally, of the $k$‘th maximal $x$-block, where $x$ is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).

💡 Research Summary

The paper investigates the internal structure of infinite words generated by morphic substitutions and by finite automata, focusing on the distribution of maximal blocks consisting solely of symbols from a prescribed subset Δ of the alphabet. For each k‑th maximal Δ‑block the authors denote by iₖ the starting index and by jₖ the ending index, and they study the asymptotic behaviour of the ratio rₖ = jₖ / iₖ. The main result is that the limit superior L = lim supₖ rₖ is always an algebraic number when the underlying word is morphic, and a rational number when the word is automatic.

The proof proceeds by translating the block structure into linear recurrences derived from the substitution matrix (for morphic words) or the transition matrix of the automaton (for automatic words). In the morphic case the substitution matrix is an integer matrix whose eigenvalues are algebraic; the dominant eigenvalue governs the growth of block lengths and positions, which forces L to be an algebraic combination of these eigenvalues. In the automatic case the transition matrix is a 0‑1 matrix with Perron‑Frobenius eigenvalue 1, so the growth is purely linear and the ratio L collapses to a rational number.

The authors then extend the analysis from Δ‑blocks to 0‑blocks (Δ = {0}) and, more generally, to x‑blocks, where x is any finite word. The same linear‑recurrence framework applies, yielding the identical algebraic/rational dichotomy for the corresponding lim sup ratios.

A crucial application is to Diophantine approximation. If a real number α is represented by the base‑b expansion of a morphic or automatic word, long runs of zeros (or of a fixed word x) provide very good rational approximations p/q, with q ≈ b^{iₖ} and p ≈ b^{jₖ}. Consequently the quality of approximation is essentially governed by the ratio jₖ / iₖ, and the irrationality exponent μ(α) satisfies μ(α) = L. Hence the paper proves that for a broad class of automatic numbers the irrationality exponent is rational, while for a broad class of morphic numbers it is algebraic.

The authors define two concrete families to illustrate the theory: (i) “regular automatic numbers,” whose digit sequences are generated by a deterministic finite automaton whose output language is regular, and (ii) “uniformly growing morphic numbers,” where the substitution expands each letter by a fixed length. Within these families the rationality (automatic) or algebraicity (morphic) of μ(α) is established explicitly, improving upon previously known generic bounds (e.g., μ ≤ 2 + ε).

Overall, the work builds a bridge between combinatorics on words and Diophantine approximation. By exploiting the spectral properties of substitution and automaton matrices, it translates combinatorial regularities into precise arithmetic constraints on the approximability of the associated real numbers. The results open several avenues for future research, such as extending the method to non‑uniform substitutions, higher‑dimensional automatic structures, or to transcendence questions beyond irrationality exponents.