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A low-energy decomposition theorem

Antal Balog, Trevor D. Wooley

Abstract

We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.

A low-energy decomposition theorem

Abstract

We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.

Paper Structure

This paper contains 5 sections, 15 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Permissible low-energy decomposition exponents
  3. Low-energy decompositions
  4. Low-energy decompositions in finite fields
  5. Higher order energies

Key Result

Theorem 1.1

Let $A$ be a finite subset of the real numbers. Then, with ${\delta}=\frac{2}{33}$, there exist disjoint subsets $B$ and $C$ of $A$, with $A=B\cup C$, and

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 14 more