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Discrete sets definable in strong expansions of ordered Abelian groups

Alfred Dolich, John Goodrick

Abstract

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation taking D to D' n times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden 2, we show that any definable unary discrete set must be definable in some elementary extension of the structure (R; <, +, Z) (Theorem 1.3).

Discrete sets definable in strong expansions of ordered Abelian groups

Abstract

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation taking D to D' n times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden 2, we show that any definable unary discrete set must be definable in some elementary extension of the structure (R; <, +, Z) (Theorem 1.3).
Paper Structure (7 sections, 41 theorems, 81 equations)

This paper contains 7 sections, 41 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notations and convention
  3. Difference Sets for Discrete sets in strong OAGs
  4. Fine Structure for Discrete Sets in the Burden $2$ Case
  5. Establishing Proposition \ref{['weak-union-arith']}
  6. Deducing Theorem \ref{['union-arith']} from Proposition \ref{['weak-union-arith']}
  7. Proof Theorem \ref{['def_in_G']} and an example

Key Result

Theorem 1.1

Suppose that $\mathcal{R}$ is a definably complete ordered Abelian group, $D \subseteq R$ is definable and discrete, and $D^{(n)}$ is infinite. Then the burden of $\mathcal{R}$ is at least $n+1$. If $\mathcal{R}$ is densely ordered, then the burden is greater than $n+1$.

Theorems & Definitions (105)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.6
  • Remark 2.2
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 95 more