Non-Constructive Upper Bounds for Repetition Thresholds

Abstract

We study the power of entropy compression in proving avoidance results in combinatorics on words. Namely, we analyze variants of a simple algorithm that transforms an input word into a word avoiding repetitions of prescribed type. This transformation can be made reversible by adding the log of the run of the algorithm to the output. Counting distinct logs, it is possible to conclude that a given repetition is avoidable over all sufficiently large alphabets. We introduce two methods of counting logs. Applying them to ordinary, undirected, and conjugate repetitions, we prove, in all cases, the results of type “\((1+\frac{1}{d})\)-powers are avoidable over \(d+O(1)\) letters”. These results are closer to the optimum than is usually expected from purely information-theoretic considerations. In the final part, we present experimental results obtained by the mentioned transformation algorithm in the extreme case of \((d+1)\)-ary words avoiding \((1+\frac{1}{d})^+\!\)-powers.