Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

TL;DR

This paper investigates the logical and arithmetical complexity of natural statements about Ulam sequences, demonstrating their low-level definability and independence from certain set-theoretic assumptions, with implications for model theory.

Contribution

It provides a uniform coding of Ulam sequences into first-order arithmetic and analyzes the arithmetical hierarchy level of their rigidity and regularity properties.

Findings

Rigidity and regularity statements are arithmetical and low in the hierarchy.

These properties are absolute between models of ZFC with the same natural numbers.

Model-theoretic tameness properties follow from combinatorial rigidity.

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).