Almost sure asymptotics for the number variance of dilations of integer sequences

Let (xₙ)ₙ₌₁^∞ be a sequence of integers. We study the number variance of dilations (αxₙ)ₙ₌₁^∞ modulo 1 in intervals of length S, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of S, subject to certain regularity assumptions imposed upon (xₙ)ₙ₌₁^∞. For the important special case xₙ = p(n), where p is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range 0 ≤ S ≤ (log N)⁻ᶜ, for a suitable absolute constant $c>0$. For more general sequences (xₙ)ₙ₌₁^∞, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations (xₙ)ₙ₌₁N.