Can late-time extensions solve the $H_0$ and $σ_8$ tensions?

We analyze the properties that any late-time modification of the $Λ$CDM expansion history must have in order to consistently solve both the $H_0$ and the $σ_8$ tensions. Taking a model-independent approach, we obtain a set of necessary conditions that can be applied to generic late-time extensions. Our results are fully analytical and merely based on the assumptions that the deviations from the $Λ$CDM background remain small. For the concrete case of a dark energy fluid with equation of state $w(z)$, we derive the following general requirements: (i) Solving the $H_0$ tension demands $w(z)
15 pages, 6 figures. Companion paper to arXiv:2201.11623