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On the structure of approximate rings

Krzysztof Krupiński, Simon Machado

Abstract

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.

On the structure of approximate rings

Abstract

By a [-]approximate subring of a ring we mean an additively symmetric subset such that is covered by finitely many [resp.\ ] additive translates of . We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.

Paper Structure

This paper contains 18 sections, 55 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Locally compact compact model and overview of the methods
  3. Auxiliary structural applications of locally compact models
  4. Preliminaries
  5. Model theory
  6. Approximate subrings
  7. Improving the target space
  8. Escape norm
  9. Structure of finite approximate subrings
  10. Applications of the main structural results
  11. Sum-product phenomenon
  12. A ring-theoretic counterpart of Gromov's theorem
  13. Sum-product for real algebras
  14. The cut-and-project construction
  15. Notions of arithmeticity
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\epsilon > 0$. There exists a non-decreasing unbounded function $f\colon \mathbb{N} \rightarrow \mathbb{N}$ such that the following holds. Let $R$ be a ring and $X \subseteq R$ be a finite subset. Define the set $X' := 4(X-X) + (X-X) \cdot 4(X-X)$. Then: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (137)

  • Theorem 1.1: Sum-product phenomenon in all rings
  • Theorem 1.2
  • Theorem 1.3: Structure of finite approximate subrings
  • Theorem 1.4
  • Theorem 1.6
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Proposition 1.12
  • ...and 127 more