Piatetski-Shapiro primes in the intersection of multiple Beatty sequences (Q2080683)

The paper investigates whether there are primes in the intersection of the Piatetski-Shapiro sequence \((\lfloor n^c\rfloor)_n\) (\(c > 1\)) with multiple Beatty sequences. A Beatty sequence \(\mathcal{B}_{\alpha, \beta}\) is of the form \((\lfloor \alpha n + \beta\rfloor)_n\). The authors show that, if \(c \in (1,12/11)\), \(\alpha_1, \alpha_2 > 1\) are irrational and of finite type (a concept that captures how well an irrational number can be approximated by rationals), \(\beta_1, \beta_2 \in \mathbb{R}\) and \(1\), \(\alpha_1^{-1}\), \(\alpha_2^{-1}\) are \(\mathbb{Q}\)-linearly independent, then the intersection of the primes with the Piatetski-Shapiro sequence and \(\mathcal{B}_{\alpha_1, \beta_1} \cap \mathcal{B}_{\alpha_2, \beta_2}\) is non-empty and they provide an asymptotic formula for its size. The authors also improve the quality of an asymptotic formula by Harman who considered the intersection of the primes with the Piatetski-Shapiro sequence and a single Beatty sequence. They also provide an asymptotic formula when the intersection with an arbitrary finite number of Beatty sequences is considered, but they only sketch the proof of that result.