A relative Roth theorem in dense subsets of sparse pseudorandom fractals

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 \( \alpha \in (1 – \epsilon_0, 1) \) (here \( \epsilon_0 \) 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 \( \epsilon_0 \) 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 \( \mu(A) > \delta \).