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Furstenberg-type estimates under mild non-concentration assumptions

Tuomas Orponen, Pablo Shmerkin

Abstract

We prove sharp $δ$-discretised versions of some variants of the Furstenberg set problem under weaker or different non-concentration assumptions compared to previous works.

Furstenberg-type estimates under mild non-concentration assumptions

Abstract

We prove sharp -discretised versions of some variants of the Furstenberg set problem under weaker or different non-concentration assumptions compared to previous works.
Paper Structure (13 sections, 13 theorems, 126 equations)

This paper contains 13 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and notation
  3. Dyadic scales, tubes, and nice configurations
  4. Uniform sets and branching functions
  5. Lipschitz decomposition lemma
  6. Multiscale decomposition of nice configurations
  7. Proof of Theorem \ref{['thm:Furstenberg-minimal-non-concentration']}
  8. Proof of Theorems \ref{['thm1']} and \ref{['thm2']}
  9. Sharpness of main results
  10. Sharpness of Theorem \ref{['thm:Furstenberg-minimal-non-concentration']} and Corollary \ref{['cor:projDeltaDiscretisedSingleScale']}.
  11. Sharpness of Theorem \ref{['thm1']}
  12. Sharpness of Theorem \ref{['thm2']}
  13. Ratios of arithmetic progressions

Key Result

Theorem 1.1

Let $s \in (0,1]$, $t \in [0,2]$, and let Then, there exist $\epsilon = \epsilon(s,t,\gamma) > 0$ and $\delta_{0} = \delta_{0}(s,t,\gamma) > 0$ such that the following holds for all $\delta \in (0,\delta_{0}]$. Let $(\mathcal{P},\mathcal{T})$ be a $(\delta,s,\delta^{-\epsilon},M)$-nice configuration, where $\mathcal{P}$ is a Frostman $(\del Moreover, $\delta_0,\epsilon$ can be chosen uniform over

Theorems & Definitions (25)

  • Theorem 1.1: Furstenberg set estimate
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1: Frostman and Katz-Tao $(\delta,s,C)$-sets
  • Lemma 2.2
  • Lemma 2.3
  • ...and 15 more