The time delay (or Sliding Window) embedding is a technique from dynamical systems to reconstructattractors from time series data. Recently, descriptors from Topological Data Analysis (TDA) — specifically, persistence diagrams — have been used to measure the shape of said attractors in applications including periodicity and quasiperiodicity quantification. Despite their utility, the fast computation of persistence diagrams of sliding windows embeddings is still poorly understood. In this work, we present theoretical and computational schemes to approximate the persistence diagrams of sliding window embeddings from quasiperiodic functions. We do so by combining the Three Gap Theorem from number theory with the Persistent Künneth formula from TDA, and derive fast and provably correct persistent homology approximations. The input to our procedure is the spectrum of the signal, and we provide numerical as well as theoretical evidence of its utility to capture the shape of toroidal attractors. We begin our efforts by documenting the stability of using the Three Gap Theorem to compute persistence diagrams. The theorem relies on the continued fraction expansion of an irrational parameter. In turn, this expansion yields the k-th convergent which lie at the core of the result. This relation is leveraged to show that the number of matching continued fraction expansion terms on two parameter values can be used to bound the bottleneck distance of their corresponding persistence diagrams. This stability is then extended to the number of matching terms in their decimal expression. This is valuable since in practice we extract our parameters using the Fast Fourier Transform (FFT). Our results show exponential decay in bottleneck distance with respect to matching decimal terms. Ultimately, they validate the reliability of algorithms relying on continued fraction information, like our method 3G. In the experiments presented here, our method took less than a second to run, on average. In linear computational time, it approximates persistence diagrams, which usually take exponential computational time. We demonstrated its performance by applying it to dynamical systems from a wide range of scientific disciplines. We are able to successfully approximate persistence diagrams within known error bounds and our work contributes to the implementation of TDA on larger data sets.
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