  This is a companion paper to arXiv:2312.10772. We deduce an equidistribution
theorem for periodic nilsequences and use this theorem to give two applications
in arithmetic combinatorics. The first application is quasi-polynomial bounds
for a certain complexity one polynomial progression, improving the iterated
logarithm bound previusly obtained. The second application is a proof of the
quasi-polynomial $U^4[N]$ inverse theorem. In work with Sah and Sawhney, we
obtain improved bounds for sets lacking nontrivial $5$-term arithmetic
progressions.
