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