Roth–type theorem with unequal powers of Piatetski–Shapiro primes

Let [Formula: see text] be nonzero integers satisfying [Formula: see text]. We consider solutions of the unequal powers equation [Formula: see text], where [Formula: see text] and [Formula: see text] belong to two certain sets [Formula: see text] and [Formula: see text] of Piatetski-Shapiro primes corresponding to [Formula: see text], respectively. We prove a Roth-type result that for [Formula: see text] and [Formula: see text], whenever the densities of [Formula: see text] and [Formula: see text] satisfy a certain lower bound condition, the equation has nontrivial solutions.