Distal expansions of Presburger arithmetic by a sparse predicate

Abstract

We prove that the structure $(\mathbb{Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq \mathbb{N}$. We do so by constructing strong honest definitions for representative formulas of the theory, providing a rare example of concrete distal decompositions.

24 pages; proof restructured to give explicit strong honest definitions