A quadratic Roth theorem for sets with large Hausdorff dimensions

TL;DR

This paper proves a quadratic Roth theorem for sets with large Hausdorff dimension, establishing the existence of three-point configurations without Fourier decay assumptions.

Contribution

It introduces a new result showing three-point patterns in sets with large Hausdorff dimension, removing previous Fourier decay conditions.

Findings

Sets with large Hausdorff dimension contain three-point configurations.

Extension of two-point pattern results to three-point configurations.

Advancement in geometric measure theory and harmonic analysis.

Abstract

Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations under the largeness of sets, which are sometimes specified via the ball condition and Fourier decay. Recently, Kuca-Orponen-Sahlsten and Bruce-Pramanik proved Sarkozy-like theorems, which remove the Fourier decay condition and show that sets with large Hausdorff dimensions contain two-point patterns. This paper explores the existence of a three-point configuration that relies solely on the Hausdorff dimension.