  We completely resolve the global Cauchy problem for the multi-dimensional
Euler-Riesz equations, where the interaction forcing is given by $\nabla
(-\Delta)^{-\sigma/2}\rho$ for some $\sigma \in (0,2)$. We construct the
global-in-time unique solution to the Euler-Riesz system in a $H^s$ Sobolev
space under a smallness assumption on the initial density and a dispersive
spectral condition on the initial velocity. Moreover, we investigate the
algebraic time decay of convergences for the constructed solutions. Our results
cover the both attractive and repulsive cases as well as the whole regime
$\sigma \in (0,2)$.
