Square-free words with square-free arithmetic subsequences: a correction at (3,9), and the first data on shifted triples

Abstract

For an infinite word w=w₀w₁w₂… over {0,1,2} and integers p ≥ 1, s ≥ 0, write sh(w, p)s for the subsequence of letters at positions congruent to s modulo p, and su(w, p)=sh(w, p)0. Call a triple (p,q,s) positive when some infinite square-free ternary word w has both su(w, p) and sh(w, q)s square-free, and a pair (p,q) positive when (p,q,0) is. Delépine, Ochem and Rosenfeld (arXiv:2605.29853v1) determine every pair with p,q ≤ 20 except (5,8), and ask which triples are positive. We prove that (3,9) is a positive pair, correcting the list of negative pairs printed in Proposition 28 of that preprint — its Table 1, its companion code and the argument here all agree that (3,9) is positive. We then give the first values for the shifted problem: the shifts reachable from s=0 by translation are exactly the multiples of gcd(p,q); 129 triples coming from the six cells (3,9), (3,12), (3,18), (3,24), (3,30), (4,36) are positive, settling each of those cells at every shift; (4,12,s) is positive if and only if 4 | s, which is the first witness for the assertion that for non-coprime pairs the starting positions matter; and 36 further triples are negative, each with its exact maximum admissible length. We also record 34 positive pairs with q>20 whose verdict follows from morphisms published in the companion code of the same preprint but which its own counting script classifies as unresolved. Every result proved below is machine-checked in Lean 4.

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