A separation lemma on sub-lattices

TL;DR

This paper proves a deterministic version of Bourgain's separation lemma for sub-lattices in a hyperbolic Lorentzian setting, extending classical results and aiding the construction of quasi-periodic solutions to nonlinear PDEs.

Contribution

It establishes a fixed-frequency separation lemma on sub-lattices, generalizing classical lattice results to a Lorentzian setting, which was previously only valid for positive measure sets.

Findings

Proves a deterministic separation lemma on sub-lattices in Lorentzian space.

Extends classical lattice partition results to hyperbolic Lorentzian geometry.

Facilitates the existence proof of quasi-periodic solutions to nonlinear Klein-Gordon equations.

Abstract

We prove that Bourgain's separation lemma, Lemma~20.14 [B2] 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 holds on the affine spaces. Previously Bourgain's lemma was not deterministic, and 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 $(1, -1, -1)$, and could be of independent interest. Combined with the method in [W2], this should lead to the existence of quasi-periodic solutions to the nonlinear Klein-Gordon equation with the usual polynomial nonlinear term $u^{p+1}$.