Forbidden substrings, Kolmogorov complexity and almost periodic sequences

TL;DR

The paper proves the existence of infinite binary sequences avoiding certain forbidden substrings using Kolmogorov complexity and combinatorial methods, and constructs almost periodic sequences with similar properties, generalizing to multidimensional cases.

Contribution

It introduces a Kolmogorov complexity approach to forbidden substring problems and constructs almost periodic sequences with these properties, extending previous results to multidimensional sequences.

Findings

Existence of infinite sequences avoiding forbidden substrings for given bounds.

Equivalence of combinatorial and Kolmogorov complexity proofs.

Generalization of methods to multidimensional sequences.

Abstract

Assume that for some $\alpha<1$ and for all nutural $n$ a set $F_n$ of at most $2^{\alpha n}$ "forbidden" binary strings of length $n$ is fixed. Then there exists an infinite binary sequence $\omega$ that does not have (long) forbidden substrings. We prove this combinatorial statement by translating it into a statement about Kolmogorov complexity and compare this proof with a combinatorial one based on Laslo Lovasz local lemma. Then we construct an almost periodic sequence with the same property (thus combines the results of Levin and Muchnik-Semenov-Ushakov). Both the combinatorial proof and Kolmogorov complexity argument can be generalized to the multidimensional case.