A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such functions can be shown to be dense in tori with dimension equal to the number of independent frequencies. In this paper, we develop theoretical and computational techniques to study the persistent homology of such sets. Specifically, we provide parameter optimization schemes for sliding windows of quasiperiodic functions, and present theoretical lower bounds on their Rips persistent homology. The latter leverages a recent persistent K\"{u}nneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.

翻译：函数若其基频在有理数域上线性无关，则称为准周期函数。在适当参数下，此类函数的滑动窗口点云可证明在维数等于独立频率数量的环面上稠密。本文发展理论和计算技术以研究此类集合的持续同调。具体而言，我们为准周期函数的滑动窗口提供参数优化方案，并给出其Rips持续同调的理论下界。后者利用了近期提出的持续Künneth公式。通过计算示例及音乐音频样本中不协和音检测的应用，对该理论进行了阐释。