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On a continuous Sárközy type problem

Borys Kuca, Tuomas Orponen, Tuomas Sahlsten

Abstract

We prove that there exists a constant $\varepsilon > 0$ with the following property: if $K \subset \mathbb{R}^{2}$ is a compact set which contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\mathrm{dim}_\mathrm{H} K \leq 2 - \varepsilon$.

On a continuous Sárközy type problem

Abstract

We prove that there exists a constant with the following property: if is a compact set which contains no pair of the form for , then .

Paper Structure

This paper contains 9 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Proof outline
  4. Notations
  5. Acknowledgements
  6. Measures with a spectral gap
  7. Finding measures with a spectral gap
  8. Concluding the proof of Theorem \ref{['mainThm']}
  9. The proof of Theorem \ref{['finite field bound']}

Key Result

Theorem 1

There exists an absolute constant $\epsilon > 0$ such that the following holds. Let $K \subset \mathbb{R}^{2}$ be a compact set with Hausdorff dimension $\dim_{\mathrm{H}} K \geq 2 - \epsilon$. Then, there exist $x \in K$ and $z \neq 0$ such that $x + (z,z^{2}) \in K$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2: Frostman's lemma
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['mainThm']}
  • ...and 4 more