Mahler's $Z$-number problem in its generalized form

Theorem E gives $Z_{3/2}(\tfrac27,\tfrac57)=\emptyset$: a window of length $\tfrac37=0.428571\ldots$ at base $3/2$, beating the $31/81$ of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of $\xi$, where that result needs algebraicity. Mahler's problem itself is untouched: for which parameters $Z$ is empty remains open, and this is a record window rather than a classification.

Theorem E gives $Z_{3/2}(\tfrac27,\tfrac57)=\emptyset$: a window of length $\tfrac37=0.428571\ldots$ at base $3/2$, beating the $31/81$ of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of $\xi$, where that result needs algebraicity. Mahler's problem itself is untouched: for which parameters $Z$ is empty remains open, and this is a record window rather than a classification.