On the number variance of sequences with small additive energy

For a real-valued sequence (x_n)_n=1^∞, denote by S_N(ℓ) the number of its first N fractional parts lying in a random interval of size ℓ:=L/N, where L=o(N) as N→∞. We study the variance of S_N(ℓ) (the number variance) for sequences of the form x_n=αa_n, where (a_n)_n=1^∞ is a sequence of distinct integers. We show that if the additive energy of the sequence (a_n)_n=1^∞ is bounded from above by N^3−ε/L² for some ε>0, then for almost all α, the number variance is asymptotic to L (Poissonian number variance). This holds in particular for the sequence x_n=αn^d,d≥2 whenever L=N^β with 0≤β<1/2.