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 a strong honest definition for every formula $\phi (x;y)$ with $\lvert {x}\rvert =1$ , providing a rare example of concrete distal decompositions.