Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

Abstract

We analyse the logical complexity and absoluteness of natural statements about Ulam sequences, with particular emphasis on the rigidity phenomena introduced by Hinman, Kuca, Schlesinger and Sheydvasser for the family $U(1,n)$. For each pair of coprime integers $a<b$ we view the associated Ulam sequence $U(a,b)$ as a recursive subset of $\mathbb{N}$ and consider expansions of the form $(\mathbb{N},+,\mathrm{U}_{a,b})$. Our first main result is a uniform coding of Ulam sequences and of the ``interval with periodic mask'' patterns appearing in rigidity conjectures into first-order arithmetic. Using this, we show that the strong rigidity, regularity (eventual periodicity of gaps), and density statements for $U(a,b)$ are all arithmetical and lie at low levels of the arithmetical hierarchy (e.g.\ $\Sigma^0_2$ or $\Pi^0_3$). As a consequence, these statements are absolute between transitive models of $\mathrm{ZFC}$ with the same natural numbers: their truth value cannot be changed by forcing, and is independent of the Continuum Hypothesis and large cardinal axioms. We also study the expansions $(\mathbb{N},+,\mathrm{U}_{a,b})$ model-theoretically, showing that combinatorial rigidity implies tameness properties (NIP, dp-minimality, non-interpretability of multiplcation).