  We consider interacting one-dimensional bosons in the universal low-energy
regime. The interactions consist of a combination of attractive and repulsive
parts that can stabilize quantum gases, droplets and liquids. In particular, we
study the role of effective three-body repulsion, in systems with weak
attractive pairwise interactions. Its low-energy description is often argued to
be equivalent to a model including only two-body interactions with non-zero
range. Here, we show that, at zero temperature, the equations of state in both
theories agree quantitatively at low densities for overall repulsion, in the
gas phase, as can be inferred from the $S$-matrix formulation of statistical
mechanics. However, this agreement is absent in the attractive regime, where
universality only occurs in the long-distance properties of quantum droplets.
We develop analytical tools to investigate the properties of the theory, and
obtain astounding agreement with exact numerical calculations using the
density-matrix renormalization group.
