2101.02056
On Long Arithmetic Progressions in Binary Morse-Like Words
Ibai Aedo, Uwe Grimm, Yasushi Nagai, Petra Staynova
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves, for the Thue–Morse word, A(2^n+1)=2^n+2 (Proposition 16) and A(2^n−1)=2^n+4 if n is even and 2^n otherwise (Proposition 17) . For the generalised Thue–Morse substitutions θ_{p,q}, it gives exact values for A_{p,q}(Q^n+1): Q^n+Q−2 if p,q>1; Q^n+Q−1 if min(p,q)=1; and Q^n+Q in the symmetric Thue–Morse case p=q=1 (Proposition 27) . The candidate’s solution derives exactly these values using a block/XOR-diagonal argument, while the paper uses a recognizability/block-substitution method; the results agree. (The paper also includes further results for d=Q^n−1 when p=q and bounds when p≠q, which the candidate did not aim to cover) .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper gives precise, well-motivated results on longest monochromatic arithmetic progressions in Thue–Morse and generalised Thue–Morse words, rederiving known bounds and adding an exact series at 2\^n+1, with a clean extension to θ\_{p,q}. The methods are standard in the area (recognisability, block substitutions) but are deployed effectively. Small clarifications would enhance readability, especially around boundary recognisability and explicit small examples.