Additive structure of non-monogenic simplest cubic fields

We consider Shanks’ simplest cubic fields $K$ for which the index $[{\mathcal{O}}_K:\mathbb{Z}[\rho ]]$ of a root $\rho$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of ${\mathcal{O}}_K$ is $6$.