Efficient equidistribution of periodic nilsequences and applications

TL;DR

This paper proves an equidistribution theorem for periodic nilsequences and applies it to improve bounds in polynomial progressions and the inverse theorem in additive combinatorics.

Contribution

It introduces a new equidistribution result for periodic nilsequences and applies it to obtain quasi-polynomial bounds in key problems in arithmetic combinatorics.

Findings

Quasi-polynomial bounds for a complexity one polynomial progression

Proof of the quasi-polynomial $U^4[N]$ inverse theorem

Improved bounds for sets lacking 5-term arithmetic progressions

Abstract

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.