For 27/14 < γ1 + γ2 < 2 with 1/2 < γ2 < γ1 < 1, infinitely many primes p in the intersection of two Piatetski-Shapiro sets satisfy ||α p² + β|| < p to the power -(14(γ1+γ2)-27)/43 + ε.
Rivat, Autour d’un th´ eor´ eme de Pjatecki ˘ ı–˘Sapiro: Nombres premiers dans la suite [nc], Th´ ese de Doc- torat, Universit´ e Paris–Sud
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On the distribution of $\alpha p^2$ modulo one in the intersection of two Piatetski--Shapiro sets
For 27/14 < γ1 + γ2 < 2 with 1/2 < γ2 < γ1 < 1, infinitely many primes p in the intersection of two Piatetski-Shapiro sets satisfy ||α p² + β|| < p to the power -(14(γ1+γ2)-27)/43 + ε.