Density regularity of $\{x,x+y,xy\}$ in the integers

Abstract

Fix $k,s\in \mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in \mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),x{y}^s\}$ for arbitrarily large $x>y\ge 2$. This provides a density strengthening of Moreira's result on partition regularity for $\{x,x+y,xy\}$.

arXiv comment24 pages, This version corrects a mistake appearing in Proposition 2.3 of the previous version