$p$-Adic Definability Rings with Finite Additive Descriptions

Definitions and mathematical statement

Let $p$ be a prime, and define the localization \[ {\mathbb{Z}}_{(p)}=\left\{\frac{m}{d}:m,d\in{\mathbb{Z}},\ d\ne0,\ p\nmid d\right\}\subseteq{\mathbb{Q}}. \] For $x\in{\mathbb{Q}}$, put \[ \gamma_p(x)=\frac{x^p-x}{p\left((x^p-x)^2-1\right)}, \qquad S_p=\gamma_p({\mathbb{Q}})=\{\gamma_p(x):x\in{\mathbb{Q}}\}. \] This rational function is defined at every rational number: a rational solution of $x^p-x=\pm1$ would be an integer by the rational root theorem, whereas $x^p-x\equiv0\pmod p$ for every integer $x$. Let $R_p={\mathbb{Z}}_{(p)}[S_p]$ denote the smallest subring of ${\mathbb{Q}}$ containing ${\mathbb{Z}}_{(p)}$ and $S_p$, and define \[ \Gamma_p({\mathbb{Q}})=\left\{\frac{a}{1+pb}:a,b\in R_p,\ 1+pb\ne0\right\}. \] For a positive integer $n$, define the $n$-fold sumset \[ \Sigma_n(S_p)=\sum_{i=1}^{n}S_p =\left\{\sum_{i=1}^{n}\gamma_p(x_i):x_1,\ldots,x_n\in{\mathbb{Q}}\right\}. \] Repetitions are allowed; since $\gamma_p(0)=0$, sums with fewer than $n$ terms can be padded with zeros.

The supplied existence question is whether \[ \forall\,p\text{ prime}\quad\exists\,n_p\in{\mathbb{Z}}_{\ge1}\quad \Gamma_p({\mathbb{Q}}) =\left\{\frac{a}{1+pb}:a,b\in\Sigma_{n_p}(S_p),\ 1+pb\ne0\right\}. \] The integer $n_p$ may depend on $p$, but must work simultaneously for every element of $\Gamma_p({\mathbb{Q}})$. The nonzero-denominator conditions make explicit the meaning of the inverses in the source.

The supplied dated curation labels the problem partially solved and reports that existence is proved, while a requested direct constructive proof and explicit bounds remain unavailable. These are supplied literature assessments, not independently verified conclusions. The displayed source statement asks only for existence; it does not specify what counts as a direct constructive proof or the intended form of explicit bounds.

Short English statement

For each prime $p$, ask whether a finite number $n_p$ suffices so that every element of the ring-defined set $\Gamma_p({\mathbb{Q}})$ has the displayed fractional representation with both parameters sums of $n_p$ values of $\gamma_p$. The supplied curation reports existence as proved and constructive proofs and bounds as remaining targets; that assessment requires independent verification.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30003250$, exact source problem code catalogue entry; title: $p$-Adic Definability Rings with Finite Additive Descriptions. Collection: Oberwolfach Reports Open Problems. The registry records proposed year 2016 and supplies no named proposer.
• Supplied report citation: Definability and Decidability Problems in Number Theory (2017), Oberwolfach Reports, pp. 2793--2866, DOI 10.4171/owr/2016/49. https://doi.org/10.4171/owr/2016/49. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Anscombe--Dittmann--Fehm, p-adic Siegel theorem. https://doi.org/10.1002/mana.201900173.
• Supplied literature source: Oberwolfach Reports 2017. https://ems.press/content/serial-article-files/46652.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the status partially solved and reports that existence is proved but a direct constructive proof and explicit bounds remain unavailable. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are source metadata and do not constitute independent mathematical verification of this draft.