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On the number of 3APs in fractal sets

Marc Carnovale, Steven Senger

TL;DR

This work addresses the presence and largeness of 3-term arithmetic progressions (3APs) inside fractal and sparse sets by importing Falconer-distance methods into the 3AP context. It replaces strong pointwise dimension/decay hypotheses with an ℓ^q bound on the Fourier transform and a mass lower bound, enabling an interpolation between discrete pseudorandom frameworks and fractal energy conditions. The authors prove two main results: an L^2-Falconer-type bound ensuring the 3AP-length set Δ^3(supp μ) has positive measure under suitable ℓ^q controls, and a mass-density statement via δ(μ) with L^2 density in higher dimensions. These results generalize Łaba–Pramanik-type conclusions to weaker hypotheses and establish a concrete bridge between additive combinatorics, fractal geometry, and harmonic analysis using tools like Bohr-set decompositions, arithmetic regularity, and Mattila’s distance-set framework. The findings advance understanding of how arithmetic patterns persist in fractal and sparse environments and open avenues for quantitative 3AP-density statements under flexible Fourier-analytic conditions.

Abstract

We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of Łaba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $δ$, on the mass of the measure $μ$ together with an upper bound, $M$ on the $L^q$ norm of its Fourier transform for some $q\in(2,3]$ depending on the parameters $δ$ and $M$.

On the number of 3APs in fractal sets

TL;DR

This work addresses the presence and largeness of 3-term arithmetic progressions (3APs) inside fractal and sparse sets by importing Falconer-distance methods into the 3AP context. It replaces strong pointwise dimension/decay hypotheses with an ℓ^q bound on the Fourier transform and a mass lower bound, enabling an interpolation between discrete pseudorandom frameworks and fractal energy conditions. The authors prove two main results: an L^2-Falconer-type bound ensuring the 3AP-length set Δ^3(supp μ) has positive measure under suitable ℓ^q controls, and a mass-density statement via δ(μ) with L^2 density in higher dimensions. These results generalize Łaba–Pramanik-type conclusions to weaker hypotheses and establish a concrete bridge between additive combinatorics, fractal geometry, and harmonic analysis using tools like Bohr-set decompositions, arithmetic regularity, and Mattila’s distance-set framework. The findings advance understanding of how arithmetic patterns persist in fractal and sparse environments and open avenues for quantitative 3AP-density statements under flexible Fourier-analytic conditions.

Abstract

We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of Łaba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, , on the mass of the measure together with an upper bound, on the norm of its Fourier transform for some depending on the parameters and .
Paper Structure (17 sections, 36 theorems, 188 equations)

This paper contains 17 sections, 36 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Integer sets
  3. Fractal sets
  4. Integer results
  5. Background and tools
  6. Proofs
  7. 3AP counts of $L^2$ functions
  8. Regularity Lemmas
  9. Fractal results
  10. Proof of Theorem \ref{['thm:MAIN']}
  11. Support Lemma
  12. Mass Lemma
  13. Non-singularity Lemma
  14. Proof of Theorem \ref{['thm:L2FalconerResult']}
  15. Pointwise decay
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that $E\subset[0,1]$ is a closed set which supports a probabilistic measure $\mu$ with the following properties: (A) $\mu([x,x+\epsilon])\leq C_1\epsilon^\alpha$ for all $0<\epsilon\leq 1$, (B) $|\widehat{\mu}(k)| \leq C_2 (1 - \alpha)^{-B} |k|^{-\frac{\beta}{2}}$ for all $k \ne 0$, where $0<

Theorems & Definitions (69)

  • Theorem 1.1: Laba and Pramanik
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Theorem 2.1: Gowers gowers
  • Theorem 2.2
  • Definition 2.3
  • ...and 59 more