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Distal Expansions of the Integers and the $p$-adic Fields

Koki Okura

Abstract

This paper investigates expansions of distal structures by a unary subset that arises as the image of a projection map. We first provide a sufficient condition for such an expansion to remain distal. Based on this criterion, we establish the distality of three kinds of expansions involving the integers or the $p$-adic fields. Let $R$ be an almost sparse sequence. We prove that $(\mathbb{Z};<,+,R)$ is distal, thereby answering a question posed by Tong. Furthermore, we show the distality of $(\mathbb{Q}_p;+,\cdot,p^{\mathbb{Z}})$ and $(\mathbb{Q}_p;+,\cdot,p^{\mathbb{Z}},p^R)$.The latter provides an example of a NIP expansion of the $p$-adic field without the rationality of the Poincaré series.

Distal Expansions of the Integers and the $p$-adic Fields

Abstract

This paper investigates expansions of distal structures by a unary subset that arises as the image of a projection map. We first provide a sufficient condition for such an expansion to remain distal. Based on this criterion, we establish the distality of three kinds of expansions involving the integers or the -adic fields. Let be an almost sparse sequence. We prove that is distal, thereby answering a question posed by Tong. Furthermore, we show the distality of and .The latter provides an example of a NIP expansion of the -adic field without the rationality of the Poincaré series.
Paper Structure (16 sections, 38 theorems, 26 equations)

This paper contains 16 sections, 38 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Distality
  4. Generalities on distality
  5. Criterion for distality
  6. (Z;<,+) with an almost sparse sequence
  7. Basic properties of almost sparse sequences
  8. Back-and-forth
  9. Variable separation
  10. Distality
  11. (Qp;+,×,pZ) and (Qp;+,×,pZ,pR)
  12. Facts and brief history
  13. Elementary behavior of (Qp;+,×,pZ)
  14. Back-and-forth for (Qp;+,×,pZ,pR)
  15. Variable separation
  16. ...and 1 more sections

Key Result

Theorem 1

Let $R$ be an almost sparse sequence of integers. Let $p^\mathbb{Z} = \left\{ p^z \mid z \in \mathbb{Z} \right\}$ and $p^R = \left\{ p^r \mid r \in R \right\}$. Then, the following structures are distal:

Theorems & Definitions (98)

  • Theorem
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • ...and 88 more