Gδ-dense binary sequences producing specific spectral types and dynamics

Identify whether there exists a dense Gδ subset of binary sequences c=(c_k)∈{0,1}^N such that, for the one-dimensional crystal with ν=1 and weight function w(k)=c_k/k^2 (extended symmetrically by w(−k)=w(k)), the associated operator H_Γ achieves a specific spectral type; additionally, determine the corresponding dynamical behavior.

Background

This problem asks for a topological genericity statement over binary choices of long-range edges with 1/k^2 decay, seeking spectral-type universality on a Gδ-dense set and understanding of dynamics in that set.

It refines the previous random setting to a deterministic Baire category framework.