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Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

Frank Gilson

TL;DR

<3-5 sentence high-level summary> This work investigates the logical complexity and absoluteness of rigidity phenomena for Ulam sequences, focusing on the family U(a,b) and the rigidity patterns described in prior work. It develops a uniform arithmetisation of U(a,b) and shows that rigidity, regularity, and density statements live at low levels of the arithmetical hierarchy (Σ^0_2 or Π^0_3), making their truth values absolute under forcing and independent of CH or large cardinals. Consequently, expansions (N,+,U_{a,b}) are tame (NIP, dp-minimal) and, under rigidity, definable in Presburger arithmetic, precluding interpretation of full multiplication. The results provide a robust framework linking additive combinatorics with logical definability and model theory, with implications for other additively defined sets and potential decidability in tame regimes.

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.\ $Σ^0_2$ or $Π^0_3$). 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).

Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

TL;DR

<3-5 sentence high-level summary> This work investigates the logical complexity and absoluteness of rigidity phenomena for Ulam sequences, focusing on the family U(a,b) and the rigidity patterns described in prior work. It develops a uniform arithmetisation of U(a,b) and shows that rigidity, regularity, and density statements live at low levels of the arithmetical hierarchy (Σ^0_2 or Π^0_3), making their truth values absolute under forcing and independent of CH or large cardinals. Consequently, expansions (N,+,U_{a,b}) are tame (NIP, dp-minimal) and, under rigidity, definable in Presburger arithmetic, precluding interpretation of full multiplication. The results provide a robust framework linking additive combinatorics with logical definability and model theory, with implications for other additively defined sets and potential decidability in tame regimes.

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 . For each pair of coprime integers we view the associated Ulam sequence as a recursive subset of and consider expansions of the form . 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 are all arithmetical and lie at low levels of the arithmetical hierarchy (e.g.\ or ). As a consequence, these statements are absolute between transitive models of 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 model-theoretically, showing that combinatorial rigidity implies tameness properties (NIP, dp-minimality, non-interpretability of multiplcation).

Paper Structure

This paper contains 45 sections, 23 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ulam sequences and rigidity
  3. Main results and significance
  4. Logical background for non-logicians
  5. Technical overview
  6. Main results
  7. Uniform arithmetisation.
  8. Complexity of rigidity.
  9. Complexity of regularity and simple combinatorial properties.
  10. Complexity of density statements.
  11. Absoluteness.
  12. Model-theoretic consequences.
  13. Organisation of the paper
  14. Notation and conventions
  15. Motivations
  16. ...and 30 more sections

Key Result

Lemma 2.3

Let $M$ and $N$ be transitive models of $\mathrm{ZFC}$ with the same natural numbers, i.e. $\mathbb{N}^M = \mathbb{N}^N = \mathbb{N}$. Then for every arithmetical sentence $\varphi$ in the language of arithmetic, we have

Figures (1)

  • Figure 1: Logical complexity for rigidity, regularity and density statements.

Theorems & Definitions (87)

  • Remark 1.1: Status of rigidity for $U(a,b)$ with $a>1$
  • Definition 2.1
  • Remark 2.2: Higher-dimensional generalisations
  • Lemma 2.3: Absoluteness for arithmetical sentences
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7: The family $U(1,n)$: mask-free decompositions
  • ...and 77 more