Kolmogorov complexity, Lovasz local lemma and critical exponents

TL;DR

This paper links Kolmogorov complexity and the Lovasz local lemma to construct sequences with prescribed critical exponents, extending understanding of sequence complexity and fractional powers.

Contribution

It introduces a novel method to derive critical exponents of sequences using Kolmogorov complexity and probabilistic combinatorics, providing new sequence constructions.

Findings

Sequences can be constructed with prescribed critical exponents.

Sequences can have no approximate fractional powers exceeding a certain exponent.

The approach connects complexity theory with combinatorial properties of sequences.

Abstract

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.