The dose-dependent risk of cancer is in textbooks as an example of linearity. While it is hard to shown for low values of exposure because any measurements/estimates would drown in noise and errors, there is little doubt that the relation shown for higher values is just as valid at the lower end. That also conforms with the stochastic model of how radiation causes cancers and the mutagenic effects of low-dose radiation seen in vitro.
We're not absolutely confident that there's not some threshold dose. LNT is a decent conservative assumption for calculating the harms from low doses and reasonable to use for regulatory purposes.
Even so, radiation hormesis has been noted in lab models, etc (slight benefits from low doses).
