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This is a Quantitative Study study.

October 3, 2025Open Access

The limit of repetition thresholds of rich sequences

Key Points

The research confirms that as the number of symbols increases, the repetition threshold for sequences tends to 2.
In 2020, Currie, Mol, and Rampersad established the repetition threshold for binary sequences, which serves as a foundation.
This investigation builds upon earlier findings by exploring repetition thresholds for richer classes of sequences.
The conjecture regarding the approach of repetition thresholds to 2 is now supported by novel proofs.

Abstract

The repetition threshold of a class of sequences is the smallest number r such that a sequence from the class contains no repetition with exponent > r. We focus on the class Cd of d-ary sequences rich in palindromes. In 2020, Currie, Mol, and Rampersad determined the repetition threshold for C₂. In 2024, Currie, Mol and Peltomäki found the repetition threshold for C₃ and conjectured that the repetition threshold for Cd tends to 2 with d growing to infinity. Here we verify their conjecture.

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Cite This Study

Dvořáková et al. (Sun,) studied this question.

synapsesocial.com/papers/68e02f3cf0e39f13e7fa262c https://doi.org/https://doi.org/10.48550/arxiv.2509.06104

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