A relative Roth theorem in dense subsets of sparse pseudorandom fractals

Abstract

We extend the main result of the paper “Arithmetic progressions in sets of fractional dimension” ([12]) in two ways. Recall that in [12], Łaba and Pramanik proved that any measure $\mu$ with Hausdorff dimension $\alpha \in (1–{ϵ}_0,1)$ (here ${ϵ}_0$ is a small constant) large enough depending on its Fourier dimension $\beta \in (2/3,\alpha ]$ contains in its support three-term arithmetic progressions (3APs). In the present paper, we adapt an approach introduced by Green in “Roth’s Theorem in the Primes” to both lower the requirement on $\beta$ to $\beta >1/2$ (and ${ϵ}_0$ to $1/10$) and perhaps more interestingly, extend the result to show for any $\delta >0$, if $\alpha$ is large enough depending on $\delta$, then $\mu$ gives positive measure to the (basepoints of the) non-trivial 3APs contained within any set $A$ for which $\mu (A)>\delta$.