Scott analysis, linear orders, and almost periodic functions

Abstract 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 , we construct a linear order whose Scott complexity is . 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 , and our construction gives new examples, for example, rigid structures, of this complexity. Moreover, we can construct the linear orders so that not only does have Scott complexity , but there are continuum‐many structures and all such structures also have Scott complexity . In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity that is only ‐equivalent to structures with Scott complexity . Our construction is based on functions that are almost periodic but not periodic, such as those arising from shifts of the ‐adic valuations.