A three-dimensional corner configuration involving the Omega function

Abstract

Let $\Omega (n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset {\mathbb{N}}^3$ has positive upper Banach density, then there are $(x,y,z)\in {\mathbb{N}}^3$ and $d\in \mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega (d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$\frac{1}{N}\sum _{n=1}^N{T}_1^n{f}_1{T}_2^n{f}_2{S}^{\Omega (n)}g$$ associated with three commuting transformations by isotropy factors and nilpotent structures in ${\mathbb{Z}}^2$-actions.

36 pages.Comments are welcome