Kolmogorov complexity, Lovasz local lemma and critical exponents

D. Krieger and J. Shallit have proved that every real number greater than 1 is a critical exponent of some sequence. We show how this result can be derived from some general statements about sequences whose subsequences have (almost) maximal Kolmogorov complexity. In this way one can also construct a sequence that has no “approximate” fractional powers with exponent that exceeds a given value.

💡 Research Summary

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The paper revisits the classical result of Krieger and Shallit that every real number α > 1 can be realized as the critical exponent of some infinite word. Instead of the original combinatorial constructions, the authors derive the same conclusion from two modern probabilistic tools: (1) the existence of infinite sequences whose every long prefix has Kolmogorov complexity close to its length, and (2) the Lovász Local Lemma (LLL), which guarantees the simultaneous avoidance of a family of low‑probability “bad” events with limited dependencies.

The first technical ingredient is a “high‑complexity” sequence X. For each n, let Aₙ be the event that the prefix X