The Three Gap Theorem is a result in number theory: place n points on a circle at positions determined by multiples of an irrational number, and the gaps between consecutive points take at most three distinct values. This was conjectured by Steinhaus in the 1950s and proved independently by multiple mathematicians. It is a statement about how irrationals distribute themselves — a fact about arithmetic, seemingly distant from any application.
Suarez Salas and Perea connect it to something entirely different: computing the topology of attractors reconstructed from time series. In sliding window embeddings — the standard method for reconstructing dynamics from a single observed variable — quasiperiodic signals produce toroidal attractors. Characterizing these attractors requires computing persistent homology, which is computationally expensive for long time series.
The Three Gap Theorem provides a shortcut. Because the gaps between embedded points on the reconstructed torus take only three values, the combinatorial structure of the persistence diagram simplifies dramatically. Combined with the Persistent Künneth formula — which decomposes the homology of a product space into the homologies of its factors — this yields fast, provably correct approximations to the full persistence diagram.
The structural point: a theorem about how points space themselves on a circle turns out to be exactly the right tool for computing the topology of high-dimensional reconstructed attractors. Number theory and topological data analysis are solving the same structural problem from opposite ends — one counting gaps, the other counting holes. The connection is not metaphorical; the gaps are the topology.