A separation lemma on sub-lattices

Abstract

Abstract We prove that Bourgain’s separation lemma [J. Bourgain, Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Ann. of Math. Stud. 158, Princeton University, Princeton, 2005] holds at fixed frequencies and their neighborhoods, on sub-lattices, sub-modules of the dual lattice associated with a quasi-periodic Fourier series in two dimensions. And, by extension, it holds on the affine spaces. Previously Bourgain’s lemma was not deterministic, and it is valid only for a set of frequencies of positive measure. The new separation lemma generalizes classical lattice partition-type results to the hyperbolic Lorentzian setting, with signature ( 1 , - 1 , - 1 ) {(1,-1,-1)} , and could be of independent interest. Combined with the method in [W.-M. Wang, Quasi-periodic solutions to a nonlinear Klein–Gordon equation with a decaying nonlinear term, preprint 2021, https://arxiv.org/abs/1609.00309], this should lead to the existence of quasi-periodic solutions to the nonlinear Klein–Gordon equation with the usual polynomial nonlinear term u p + 1 {u^{p+1}} .