Definability of Canonical Products with Sparse Zero Sets

Definitions and mathematical statement

The supplied record poses two definability questions, but does not specify a product formula or all definability conventions. The following makes a standard literal interpretation explicit; these conventions require source review.

Let $\mathcal R=({\mathbb{R}};0,1,+,\cdot,<)$ be the ordered real field. A function on a definable domain is definable in a structure if its graph is definable in that structure. In the conditional formulation below, real parameters are allowed; the record does not specify whether parameters are permitted.

For an integer $p\geq 0$, put \[ E_p(w)=(1-w)\exp\!\left(\sum_{j=1}^{p}\frac{w^j}{j}\right), \] where the sum is zero when $p=0$. Taking the listed sequence to consist of positive zeros, taking indices $n=1,2,\ldots$, and using the least admissible genus, define, for each fixed real $s$ with $0<s<1$, \[ p_s=\min\{p\in{\mathbb{Z}}_{\geq0}:s(p+1)>1\}, \qquad P_s(z)=\prod_{n=1}^{\infty}E_{p_s}\!\left(\frac{z}{n^s}\right), \quad z\in{\mathbb{C}}. \] Under these explicit conventions, the first question is: \[ \text{For every }s\in(0,1),\text{ is } \{(x,y)\in{\mathbb{R}}^2:x>0,\ y=P_s(x)\} \text{ definable in }\mathcal R\text{?} \] This asks about each fixed $s$; it does not impose uniform definability as $s$ varies. Neither endpoint $s=0$ nor $s=1$ is included.

For each fixed real $\alpha>1$, the analogous genus-zero, positive-zero convention gives \[ Q_\alpha(z)=\prod_{n=1}^{\infty}\left(1-\frac{z}{\alpha^n}\right), \qquad D_\alpha=\alpha^{{\mathbb{Z}}}=\{\alpha^k:k\in{\mathbb{Z}}\}. \] If the intended added function is its restriction to the positive real axis, write \[ \mathcal M_\alpha= \left(\mathcal R;\operatorname{graph}(Q_\alpha|_{(0,\infty)})\right). \] For $m\geq1$, let $\operatorname{Def}_m(\mathcal M_\alpha)$ be the collection of subsets of ${\mathbb{R}}^m$ definable in this structure, with the parameter convention above. The second question requests a description, for every $\alpha>1$, of the definable sets in the expansion by the associated canonical product, beyond the set $D_\alpha$, which the supplied statement describes as definable. It does not give a specific proposed classification or a precise criterion for "beyond."

The product formulas and the positive-axis domain for the second expansion are conditional specifications, not recovered source facts. The record does not settle the indexing, normalization, genus convention, or whether the second expansion uses a function on all of ${\mathbb{R}}$. The intended ambient structure in the first question also needs confirmation: the literal pure-real-field wording and prescribed infinite positive zero sequence require reconciliation with the supplied assessment that the question is open. No alternative ambient structure or corrected target is asserted here.

Short English statement

For each $0<s<1$, the record asks whether the canonical product associated with $(n^s)$, restricted to $(0,\infty)$, is definable in the real field. For each $\alpha>1$, it also asks which sets become definable after adjoining the canonical product associated with $(\alpha^n)$, beyond $\alpha^{{\mathbb{Z}}}$. The displayed products give an explicitly conditional interpretation; the intended first question, product conventions, and scope of the second classification need human source review. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003498$, source problem code catalogue entry, title Definability of Canonical Products with Sparse Zero Sets, in the collection Oberwolfach Reports Open Problems. The record gives proposed year 2017 and names no individual proposer.
• Supplied report citation: O-Minimality and its Applications to Number Theory and Analysis (2018), Oberwolfach Reports, pp. 1349--1420, DOI 10.4171/owr/2017/22. https://doi.org/10.4171/owr/2017/22. The supplied literature-source label is "Oberwolfach Reports 2018." The report is described as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Miller and Speissegger, "canonical products." https://arxiv.org/abs/1812.05547.
• Supplied literature source: Miller and Speissegger, "published canonical-products results." https://doi.org/10.4153/S0008439519000572.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that canonical-product expansions have been analyzed for important parameter ranges, while the stated question for every $0<s<1$ and the requested expansion remain open. This is supplied dated curation derived from the OWR material, not independently verified literature assessment; the status and cited claims require independent verification.