On the distribution of \(\alpha p\) modulo one in the intersection of two Piatetski-Shapiro sets (Q6624945)

In this article, two classical topics of analytic number theory are combined:\N\NThe first topic is counting primes in the sequences \(\lfloor n^{1/\gamma}\rfloor, 0 < \gamma < 1\), following the seminal work of \textit{I. I. Piatetski-Shapiro} [Mat. Sb., Nov. Ser. 33(75), 559--566 (1953; Zbl 0053.02702)]. For various regions of \(\gamma\), not only the existence of infinitely many primes in the respective sequence, but also asymptotic formulae for the expected number of primes occurring have been proven since then. This was extended by \textit{D. Leitmann} [Monatsh. Math. 94, 33-44 (1982; Zbl 0478.10026)] who was the first to establish an asymptotic formula also for the intersection of two Piatetski-Shapiro sequences \((\lfloor n^{1/\gamma_1}\rfloor)_{n \in \mathbb{N}}, (\lfloor m^{1/\gamma_2}\rfloor)_{m \in \mathbb{N}}\).\N\NThe second topic is the irrational prime rotation and its Diophantine properties: This problem goes back to \textit{I. M. Vinogradov} [Tr. Mat. Inst. Steklova 23, 110 p. (1947; Zbl 0041.37002)] who proved that for any irrational \(\alpha\) and arbitrary real \(\beta\), there are infinitely many primes such that \(\lVert p \alpha + \beta \rVert \leq p^{-1/5 + \varepsilon}\). This result was subsequently improved, with the current record (at least in the homogeneous case \(\beta = 0\)) being \textit{K. Matomäki}'s result [Math. Proc. Camb. Philos. Soc. 147, No. 2, 267--283 (2009; Zbl 1196.11101)] who showed \(\lVert p \alpha \rVert \leq p^{-1/3 + \varepsilon}\).\N\NThe first article that combined the two topics is due to \textit{S. Dimitrov} [Indian J. Pure Appl. Math. 54, No. 3, 858--867 (2023; Zbl 1517.11085)] who combined the Piatetski-Shapiro result with Vinogradov's question, obtaining that for any \(11/12 < \gamma < 1\) and any real \(\beta\), there are infinitely many primes \(p\) in the respective Piatetski-Shapiro sequence that satisfy\N\[\N\lVert p \alpha + \beta \rVert \leq p^{-\tfrac{12\gamma-11}{26}}(\log p)^6.\N\]\NIn the article under review, the authors take a further step, taking Vinogradov's question and combine it with Leitmann's approach of intersecting two Piatetski-Shapiro sequences: Their main result (Theorem 1.1) states that for any \(0 < \gamma_1,\gamma_2 < 1\) with \(23/12 < \gamma_1 + \gamma_2 < 2\) and any real \(\beta\), there exist infinitely many primes \(p\) in the intersection of the two Piatetski-Shapiro sequences \((\lfloor n^{1/\gamma_1}\rfloor)_{n \in \mathbb{N}}, (\lfloor m^{1/\gamma_2}\rfloor)_{m \in \mathbb{N}}\) such that\N\[\N\lVert p \alpha + \beta \rVert \leq p^{-\tfrac{12(\gamma_1 + \gamma_2)-23}{38}+ \varepsilon}.\N\]\NThe proof methods are classical tools from analytic number theory such as estimates on exponential sums over primes.