Scott analysis, linear orders and almost periodic functions

For any limit ordinal λ, we construct a linear order L_λ whose Scott complexity is Σλ+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 Σλ+1, and our construction gives new examples, e.g., rigid structures, of this complexity. Moreover, we can construct the linear orders L_λ so that not only does L_λ have Scott complexity Σλ+1, but there are continuum-many structures M ≡_λ L_λ and all such structures also have Scott complexity Σλ+1. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity Πλ+1 that is only λ-equivalent to structures with Scott complexity Πλ+1. Our construction is based on functions f Z→ N∪ \∞\ which are almost periodic but not periodic, such as those arising from shifts of the p-adic valuations.