Scott analysis, linear orders, and almost periodic functions

Given a countable structure, the Scott complexity measures the difficulty of characterizing the structure up to isomorphism. In this paper, we consider the Scott complexity of linear orders. For any limit ordinal $\lambda$, we construct a linear order $L_{\lambda }$ whose Scott complexity is ${\Sigma }_{\lambda +1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity ${\Sigma }_{\lambda +1}$, and our construction gives new examples, for example, rigid structures, of this complexity. Moreover, we can construct the linear orders $L_{\lambda }$ so that not only does $L_{\lambda }$ have Scott complexity ${\Sigma }_{\lambda +1}$, but there are continuum-many structures $M{\equiv }_{\lambda }{L}_{\lambda }$ and all such structures also have Scott complexity ${\Sigma }_{\lambda +1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity ${\Pi }_{\lambda +1}$ that is only $\lambda$-equivalent to structures with Scott complexity ${\Pi }_{\lambda +1}$. Our construction is based on functions $f:Z\to N\cup \left\{\infty \right\}$ that are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations.