Kolmogorov Complexity, Lovász 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 [1]. 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. 1 Kolmogorov complexity of subsequences Let ω = ω0ω1 . . . be an infinite binary sequence. For any finite set A ⊂ N let ω(A) be a binary string of length #A formed by ωi with i ∈ A (in the same order as in ω). We want to construct a sequence ω such that strings ω(A) have high Kolmogorov complexity for all simple A. (See [3] for the definition and properties of Kolmogorov complexity. We use prefix complexity and denote it by K, but plain complexity can also be used with minimal changes.) Theorem 1. Let γ be a positive real number less than 1. Then there exists a sequence ω and an integer N such that for any finite set A of cardinality at least N the inequality