For a real-valued sequence (xn)n=1∞, denote by SN(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 SN(ℓ) (the number variance) for sequences of the form xn=αan, where (an)n=1∞ is a sequence of distinct integers. We show that if the additive energy of the sequence (an)n=1∞ is bounded from above by N3−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 xn=αnd,d≥2 whenever L=Nβ with 0≤β<1/2.