On the number of 3APs in fractal sets

TL;DR

This paper investigates conditions under which fractal sets contain many three-term arithmetic progressions, linking geometric measure theory with harmonic analysis to extend previous results on the structure of such sets.

Contribution

It provides a new result on the abundance of 3-term arithmetic progressions in fractal sets based on Fourier decay and measure bounds, generalizing prior work by Laba and Pramanik.

Findings

Establishes conditions for large sets of 3APs in fractals

Connects Fourier decay bounds to arithmetic progression structure

Extends previous results to broader measure assumptions

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 {\L}aba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $\delta$, on the mass of the measure $\mu$ 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 $\delta$ and $M$.