Forbidden substrings, Kolmogorov complexity and almost periodic sequences

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.

💡 Research Summary

The paper addresses a classic combinatorial avoidance problem: given a family of “forbidden” binary strings Fₙ, each of length n and of size at most 2^{αn} with a fixed constant α < 1, does there exist an infinite binary sequence ω that never contains any of these forbidden substrings? The authors answer affirmatively and provide two fundamentally different proofs—one based on the Lovász Local Lemma (LLL) and another on Kolmogorov complexity—then combine these ideas to construct an almost‑periodic sequence with the same avoidance property, finally extending the whole framework to multidimensional arrays.

1. Classical LLL proof.
For each position i and length n, define the bad event A_{i,n} that the block ω