Nonrepetitive sequences on arithmetic progressions

A sequence $S=s_{1}s_{2}…_{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every $k\geqslant 1$ and every $c\geqslant 1$ there exist arbitrarily long sequences over at most $(1+\frac{1}{c})k+18k^{c/c+1}$ symbols whose subsequences indexed by arithmetic progressions with common differences from the set ${1,2,…,k}$ are nonrepetitive. This improves a previous bound obtained in \cite{Grytczuk Rainbow}. Our approach is based on a technique introduced recently in \cite{GrytczukKozikMicek}, which was originally inspired by a constructive proof of the Lov'{a}sz Local Lemma due to Moser and Tardos \cite{MoserTardos}. We also discuss some related problems that can be successfully attacked by this method.

💡 Research Summary

The paper studies a natural generalization of Thue’s non‑repetitive sequences: a sequence S is called k‑mode non‑repetitive if every subsequence obtained by taking terms along an arithmetic progression with common difference d∈{1,…,k} contains no adjacent identical blocks. Let M(k) denote the smallest alphabet size that allows arbitrarily long k‑mode non‑repetitive sequences. Thue’s theorem gives M(1)=3, while a simple counting argument shows M(k)>k+2. It has been conjectured that M(k)=k+2 for all k, but the best known upper bound before this work was M(k)≤33k (Grytczuk, 2011).

The authors dramatically improve this bound by constructing, for any integer k≥1, arbitrarily long k‑mode non‑repetitive sequences over at most 2k+O(√k) symbols. Their method is based on a constructive version of the Lovász Local Lemma (LLL) due to Moser and Tardos, adapted to the non‑repetitive setting in a recent paper by Grytczuk, Kozik and Micek. The core of the approach is a randomized algorithm that builds the sequence one position at a time, choosing a symbol uniformly from a prescribed list L_i (each list of size at least 2k+10√k). Whenever a repetition appears, the algorithm erases the longest repeated block that contains the most recently placed symbol, and then resumes filling the smallest empty position. This “backtracking” guarantees that after each erasure the current partial sequence remains k‑mode non‑repetitive.

To prove termination, the authors encode every possible execution of the algorithm into a log consisting of five components:

1. R – a lattice path in the upper‑right quadrant of ℤ² from (0,0) to (2M,0) using steps (1,1) (when a symbol is placed) and (1,−1) (when a symbol is erased). Such paths are counted by the Catalan number C_M.
2. D – a sequence of differences d∈{1,…,k} attached to each peak of R, indicating the common difference of the arithmetic progression that caused the repetition.
3. O – a sequence of signs (±1) indicating whether the algorithm erased the left or right half of the repeated block.
4. P – a sequence of block lengths (the number of erased symbols) associated with each peak.
5. S – the final partial sequence after M random choices.

Each log uniquely determines the first M random choices made in line (3) of the algorithm, establishing an injective mapping from random choice sequences to logs. Consequently, the number of possible logs must be at least (10√k)^M (since each random choice has at least 10√k admissible symbols). On the other hand, the authors bound the total number of logs from above by

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