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A non-linear Roth theorem for thick Cantor sets

Alex McDonald, Micah Nguyen

TL;DR

The paper proves a non-linear Roth-type theorem for thick Cantor sets: if $K\subset\mathbb{R}$ is Cantor with Newhouse thickness $\tau(K)>1$ and $f:\mathbb{R}\to\mathbb{R}$ is $C^1$ with $f(0)=0$ and $1/\tau<f'(0)<\tau$, then there exist $x\in K$ and $t>0$ such that $\{x-t,x,x+f(t)\}\subset K$. The approach generalizes the linear case ($f(t)=t$) by combining the Newhouse Gap Lemma with a local-to-global thickness control under near-constant derivatives, and uses a careful decomposition around the largest gap to create interleaving between $K$-pieces and their images. The work discusses necessity of the $f(0)=0$ condition, contemplates relaxing derivative bounds, and presents a degenerate example showing that certain non-degenerate strategies fail if the derivative behavior is not suitably controlled. Overall, the results connect fractal geometry, gap/bridge analysis, and non-linear configurations, broadening the scope of pattern existence in thick Cantor sets and suggesting avenues for future non-linear extensions.

Abstract

We prove that for any function $f$ satisfying certain mild conditions and any Cantor set $K$ with Newhouse thickness greater than $1$, there exists $x\in K$ and $t>0$ such that \[ \{x-t,x,x+f(t)\}\subset K. \] This is an extension of previous work on the existence of three-term arithmetic progressions in Cantor sets to the non-linear setting.

A non-linear Roth theorem for thick Cantor sets

TL;DR

The paper proves a non-linear Roth-type theorem for thick Cantor sets: if is Cantor with Newhouse thickness and is with and , then there exist and such that . The approach generalizes the linear case () by combining the Newhouse Gap Lemma with a local-to-global thickness control under near-constant derivatives, and uses a careful decomposition around the largest gap to create interleaving between -pieces and their images. The work discusses necessity of the condition, contemplates relaxing derivative bounds, and presents a degenerate example showing that certain non-degenerate strategies fail if the derivative behavior is not suitably controlled. Overall, the results connect fractal geometry, gap/bridge analysis, and non-linear configurations, broadening the scope of pattern existence in thick Cantor sets and suggesting avenues for future non-linear extensions.

Abstract

We prove that for any function satisfying certain mild conditions and any Cantor set with Newhouse thickness greater than , there exists and such that This is an extension of previous work on the existence of three-term arithmetic progressions in Cantor sets to the non-linear setting.

Paper Structure

This paper contains 8 sections, 8 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic properties of Newhouse thickness
  3. Proof of Theorem \ref{['main']}
  4. The linear case
  5. The non-linear case
  6. Necessity of derivative assumptions
  7. Goals for future work: The non-degenerate case
  8. A critical example in the degenerate case

Key Result

Theorem 1.3

Let $K\subset \mathbb{R}$ be a Cantor set with thickness $\tau>1$, and let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function satisfying $f(0)=0$ and $\frac{1}{\tau}<f'(0)<\tau$. Then, there exist $x\in \mathbb{R},t> 0$ such that

Figures (4)

  • Figure 1: An "interior" bridge of $K\cap B$
  • Figure 2: An "endpoint" bridge of $K\cap B$
  • Figure 3: The convex hulls of $K_1$ (red) and $K_2$ (blue).
  • Figure 4: The set $K$ in Theorem \ref{['squareexample']} with bridges labeled.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1: Newhouse Gap Lemma
  • Lemma 2.2: MT22, Lemma 3.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • ...and 8 more