Optimizing Properties of Balanced Words

In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along these lines known to date.

💡 Research Summary

The paper surveys a collection of optimization problems across several mathematical and physical domains in which the optimal solutions are given by balanced binary words, often in the form of Sturmian sequences. After recalling the definition of a balanced word— a finite or infinite 0‑1 word w such that any two subwords of equal length differ in the number of 1’s by at most one— the author notes that non‑periodic infinite balanced words are called Sturmian. Sturmian sequences are characterized by the complexity function p_w(n)=n+1 and can be generated by the formula w_n = ⌊(n+1)γ+δ⌋ – ⌊nγ+δ⌋ for a prescribed 1‑ratio γ∈