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Complete type amalgamation for non-standard finite groups

Amador Martin-Pizarro, Daniel Palacín

Abstract

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. We also obtain a model-theoretic proof of Roth's theorem on the existence of arithmetic progressions of length $3$ for subsets of positive density in suitable definably amenable groups, such as countable amenable abelian groups without involutions and ultraproducts of finite abelian groups of odd order.

Complete type amalgamation for non-standard finite groups

Abstract

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. We also obtain a model-theoretic proof of Roth's theorem on the existence of arithmetic progressions of length for subsets of positive density in suitable definably amenable groups, such as countable amenable abelian groups without involutions and ultraproducts of finite abelian groups of odd order.

Paper Structure

This paper contains 5 sections, 22 theorems, 81 equations.

Table of Contents

  1. Randomness and Fubini
  2. Forking and measures
  3. On $3$-amalgamation and solutions of $xy=z$
  4. Ultra-quasirandomness revisited
  5. Local ultra-quasirandomness

Key Result

Lemma \oldthetheorem

Let $b$ be weakly random over a set of parameters $C$ and $a$ be weakly random over $C, b$. The pair $(a,b)$ is weakly random over $C$.

Theorems & Definitions (79)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 69 more