Approximate polynomial structure in additively large sets

TL;DR

This paper demonstrates that subsets of natural numbers with positive logarithmic Banach density contain sets close to geometric progressions, advancing understanding of approximate polynomial structures in large sets.

Contribution

The authors improve previous bounds and develop new density conditions that guarantee the existence of approximate powers of arithmetic progressions in large sets.

Findings

Subsets with positive logarithmic Banach density contain sets near geometric progressions.

Improved bounds on the approximation factor for geometric progressions.

New density conditions linking large sets to approximate polynomial structures.

Abstract

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explored.