  We here construct (large) local and small global-in-time regular unique
solutions to the fractional Euler alignment system in the whole space ${\mathbb
R}^d$, in the case where the deviation of the initial density from a constant
is sufficiently small. Our analysis strongly relies on the use of Besov spaces
of the type $L^1(0,T;\dot B^s_{p,1})$, which allow to get time independent
estimates for the density even though it satisfies a transport equation with no
damping. Our choice of a functional setting is not optimal but aims at
providing a transparent and accessible argumentation.
