On the number of 3APs in fractal sets

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 Łaba and Pramanik’s result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $δ$, on the mass of the measure $μ$ 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 $δ$ and $M$.

💡 Research Summary

The paper investigates the existence and quantitative abundance of three‑term arithmetic progressions (3APs) in two distinct settings: finite subsets of the integers and compact fractal subsets of ℝⁿ. The central objects of study are the set of step‑sizes (or “lengths”) of 3APs that lie entirely inside a given set, denoted
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