We investigate the cosmological implications of an effective gravitational action, inspired by Sakharov's idea of induced gravity, containing non-local contributions from the operator $\left(\Box +β\right)^{-1} R$. The $β$ term is a novel feature in the panorama of non-local models of gravity, and arises naturally within Sakharov theory from the potential of a non-minimally coupled scalar field after a spontaneous symmetry breaking takes place. In this class of models the non-local contribution can acquire an oscillatory behaviour, thereby avoiding the divergence of the Hubble parameter at late times, which is another characteristic feature of the non-local models treated in the literature. Furthermore, the effective gravitational coupling $G_{\rm eff}$ inherits the oscillatory behaviour, resulting in alternating epochs of stronger and weaker gravity. This framework is argued to have potentially interesting implications for the $H_0$ and the $σ_8$ tensions.

v2, minor changes to match the version accepted for publication in Physics of the Dark Universe. 10 pages + references, 5 figures, comments are welcome