Abstract

We show that rounding to a delta-net in SO(3) is not close to a group operation, thus confirming a conjecture of Gowers and Long.

Highlights

  • In a very interesting recent preprint [7], Gowers and Long considered somewhat associative binary operations, that is to say maps ◦ : X × X → X on a finite set X which satisfy the associativity relation x ◦ (y ◦ z) = (x ◦ y) ◦ z for a positive fraction of (x, y, z) ∈ X3

  • It is natural to ask whether more might be true, namely whether such a binary operation ◦ must agree with a group operation a positive fraction of the time

  • Recall that a K-approximate group is a subset B of some ambient group which is symmetric and such that B2 is covered by K left- translates of B

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Summary

Introduction

In a very interesting recent preprint [7], Gowers and Long considered somewhat associative binary operations, that is to say maps ◦ : X × X → X on a finite set X which satisfy the associativity relation x ◦ (y ◦ z) = (x ◦ y) ◦ z for a positive fraction of (x, y, z) ∈ X3. Their main result is that such operations are all near (in a sense they make precise) to the group operation on a metric group G.

An initial reduction
Outline of the rest of the argument
An application of Tao’s metric entropy BSG theorem
Case 1: a large element in the error set
Case 2: almost homomorphisms
Further comments and open questions
Full Text
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