A three-dimensional corner configuration involving the Omega function
Zhuowen Guo, Rongzhong Xiao, Shuhao Zhang
Abstract
Let Ω(n) denote the number of prime factors of n, counted with multiplicity. We prove that if A⊂N3 has positive upper Banach density, then there are (x,y,z)∈N3 and d∈N such that (x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+Ω(d))∈ A. To establish the above result, we give an L2-decoupling theorem for the triple ergodic averages 1NΣn=1N T1n f1\,T2n f2\,SΩ(n)g associated with three commuting transformations by isotropy factors and nilpotent structures in Z2-actions.
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