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We introduce a wide class of deterministic subsets of primes of zero relativedensity and we prove Roth's Theorem in these sets, namely, we show that anysubset of them with positive relative upper density contains infinitely manynon-trivial three-term arithmetic progressions. We also prove that theHardy--Littlewood majorant property holds for these subsets of primes. Notably,our considerations recover the results for the Piatetski--Shapiro primes forexponents close to $1$, which are primes of the form $\lfloor n^c\rfloor$ for afixed $c>1$.

Roth's Theorem and the Hardy--Littlewood majorant problem for thin subsets of primes