A curved three-point pattern problem for fractal sets on the real line

TL;DR

This paper investigates the presence of curved three-point configurations in fractal subsets of the real line, establishing conditions based on Hausdorff dimension and content for their existence.

Contribution

It introduces new results showing that large Hausdorff dimension or content guarantees curved three-point patterns for a broad class of nonlinear functions.

Findings

Large Hausdorff dimension ensures the existence of curved three-point patterns.

Hausdorff content bounded away from zero also guarantees such patterns.

Includes nonlinear functions like t^k log(1+t) in the pattern class.

Abstract

We study the occurrence of curved three-point configurations in fractal subsets of the real line. We prove that if \(E \subset [0,1]\) is a compact set with sufficiently large Hausdorff dimension, then \(E\) contains a curved three-point progression associated with a broad class of nonlinear functions. Our approach can also show the existence of the curved three-point pattern under the assumption that the Hausdorff content of \(E\) is bounded away from zero. The class of functions includes, in addition to polynomials with vanishing constant term, nonlinear functions such as \[ t^k \log(1+t), \quad \forall k \geq 1. \]