Quantitative bounds in the polynomial Szemer\'edi theorem: the homogeneous case

TL;DR

This paper establishes quantitative bounds for the polynomial Szemerédi theorem in the case of homogeneous polynomials of the same degree, including configurations like perfect kth power progressions.

Contribution

It provides new quantitative bounds specifically for homogeneous polynomial configurations in the polynomial Szemerédi theorem.

Findings

Quantitative bounds are derived for homogeneous polynomial configurations.

Includes progressions with common difference as perfect kth powers.

Advances understanding of polynomial configurations in additive combinatorics.

Abstract

We obtain quantitative bounds in the polynomial Szemer\'edi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations include arithmetic progressions with common difference equal to a perfect kth power.