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Fourier Dimension and Translation Invariant Linear Equations

Angel D. Cruz

Abstract

We consider a translation invariant linear equation in four variables with integer coefficients of the form: $ax_1 +bx_2= cy_1+dy_2$. The main result of the paper states that any set on the real line with Fourier dimension greater than 1/2 must contain a nontrivial solution of such an equation.

Fourier Dimension and Translation Invariant Linear Equations

Abstract

We consider a translation invariant linear equation in four variables with integer coefficients of the form: . The main result of the paper states that any set on the real line with Fourier dimension greater than 1/2 must contain a nontrivial solution of such an equation.

Paper Structure

This paper contains 17 sections, 12 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Nontrivial Solutions and Genus
  3. Proof Techniques and Layout of the Paper
  4. Construction of Measures
  5. Proof of Lemma \ref{['lemma: Fourier side of the limit']}
  6. Proof of Lemma \ref{['lemma: constant for linear functional']}
  7. Proof of Proposition \ref{['prop: summary of measure properties']} Part $(a)$
  8. Proof of Proposition \ref{['prop: summary of measure properties']} Part $(b)$
  9. Denseness of $\mathcal{V}$ in $C([0,1]^3)$
  10. Proof of Theorem \ref{['theorem: containment result']}
  11. Proof of Theorem \ref{['theorem: containment result']}
  12. Proof of Continuity of $F^{[f]}$
  13. Verification of Requirements for the dominated convergence theorem in Theorem \ref{['theorem: containment result']}
  14. Avoiding Set
  15. Notation & Construction
  16. ...and 2 more sections

Key Result

Theorem 1

A more general version of Theorem theorem: containment result holds for multivariate translation-invariant equations. This is currently work in progress phdthesis. Let $E\subset [0,1]$ be a Borel set with $\text{dim}_\mathbb{F}(E)>\frac{1}{2}$. Then there exists an equation of the form 4variable equ

Theorems & Definitions (28)

  • Definition 1: Fourier Dimension
  • Theorem 1
  • Theorem 2
  • Definition 2: Genus
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • proof
  • proof
  • ...and 18 more