Uniformity Aspects of SL(2,R) Cocycles and Applications to Schrödinger Operators Defined over Boshernitzan Subshifts

The paper proves that for subshifts satisfying Boshernitzan’s condition (B): (a) the set of sampling functions with no non-uniform hyperbolicity, equivalently Σ = Z, is a dense Gδ; and (b) in the aperiodic case, zero-measure spectrum is a dense Gδ. This is stated as Theorem 1.6 and proved by combining (i) Gδ-structure of uniform families (Theorem 1.4), (ii) density of locally constant cocycles and their uniformity under (B) (Lemma 6.1), (iii) the Johnson–Lenz relations linking Σ, Z, and NUH, and (iv) upper semicontinuity of Lebesgue measure of the spectrum (Proposition 4.1). The candidate solution reaches the same conclusions. For (a) it gives an alternative Gδ proof via an explicit covering argument using upper semicontinuity of the Lyapunov exponent and openness of uniform hyperbolicity; density is obtained via locally constant approximants and uniformity under (B). For (b) it uses Avila–Damanik’s generic absence of absolutely continuous spectrum plus Ishii–Pastur–Kotani to first get Leb(Z)=0 generically, and then intersects with the dense Gδ from (a) to conclude Leb(Σ)=0 generically. Thus both are correct; the paper’s proof and the model’s proof differ in technique for the Gδ and zero-measure steps.