Consecutive pure cubic fields with large class number (Q6624943)

It has been shown by \textit{G. Cherubini} et al. [Int. Math. Res. Not. 2023, No. 14, 12052--12063 (2023; Zbl 1532.11152)] that for every given \(k\ge2\) and \(\varepsilon>0\) there exist at least \(x^{1/2-\varepsilon}\) integers \(0<d\le x\) such the class-number of the fields \(\mathbb Q\left(\sqrt{d+j}\right)\) for \(j=1,2,\dots k\) exceeds\N\[\Nc(k)\frac{\sqrt d\log\log d}{\log d}\N\]\Nwith positive \(c(k)\).\N\NThe authors establish an analogue of this result for pure cubic fields. They prove that for every given \(k\ge2\) and \(\varepsilon>0\) there exist at least \(x^(1/3-\varepsilon)\) integers \(0<d\le x\) such the class-number of the fields \(\mathbb Q\left(\sqrt[ 3]{d+j}\right)\) for \(j=1,2,\dots k\) exceeds\N\[\Na(k)\frac{\sqrt d\log\log d}{\log d}\N\]\Nwith positive \(a(k)\).