Efficient equidistribution of periodic nilsequences and applications

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.