On the number of 3APs in fractal sets

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 $M$.