Quantitative Entropy of Rational Multiplicative Orbits

Definitions and mathematical statement

The supplied record concerns a quantitative covering bound for rational multiplicative orbits. Several conventions needed for a fully determined statement are absent. The following is a precise provisional interpretation; the domain restrictions and covering convention stated here require confirmation against the source.

Assume that $a,b\in{\mathbb{Z}}$, $a,b\geq2$, are multiplicatively independent, meaning that \[ a^u b^v=1,\qquad (u,v)\in{\mathbb{Z}}^2, \quad\Longrightarrow\quad u=v=0. \] Write $\mathbb T={\mathbb{R}}/{\mathbb{Z}}$, with metric \[ d_{\mathbb T}(x+{\mathbb{Z}},y+{\mathbb{Z}})=\min_{j\in{\mathbb{Z}}}|x-y-j|. \] For a nonempty finite set $B\subseteq\mathbb T$ and $\delta>0$, provisionally define its covering number by \[ N(B,\delta)=\min\left\{m\in{\mathbb{Z}}_{\geq1}:\ \exists x_1,\ldots,x_m\in\mathbb T,\quad B\subseteq\bigcup_{i=1}^{m}\{y\in\mathbb T:d_{\mathbb T}(y,x_i)\leq\delta\}\right\}. \] Here $\log$ denotes the natural logarithm. The symbol $N$ used for an exponent below is distinct from the covering-number notation $N(B,\delta)$.

Under these provisional conventions, the question is whether, for every such pair $(a,b)$, there exist positive real constants $N,M,K$, depending only on $a,b$, such that for every $s\in{\mathbb{Z}}$ with $s\geq2$ and every $r\in{\mathbb{Z}}$ satisfying \[ \gcd(s,ab)=\gcd(s,r)=1, \] the finite set \[ A_{r,s}=\left\{a^n b^k\frac{r}{s}+{\mathbb{Z}}:\ n,k\in{\mathbb{Z}}_{\geq0},\quad n,k\leq M\log s\right\}\subseteq\mathbb T \] satisfies \[ \log N\left(A_{r,s},(\log s)^{-N}\right)\geq K\log\log s. \]

The supplied statement does not specify whether $a,b$ must be positive and greater than one, whether exponents start at zero or one, the allowed types and signs of $N,M,K$, the covering-number convention, or whether the denominator quantifier is over all admissible $s\geq2$ or only sufficiently large denominators. Thus the displayed interpretation is not an assertion that these conventions have been verified. In particular, the source's unqualified phrase "integers $a,b$" has been restricted provisionally above. The case $s=1$ cannot be included in the displayed formula because $\log\log s$ is undefined there.

The supplied background reports restricted results under a size condition on the denominator, but supplies neither that condition nor a precise restricted theorem. No such condition is incorporated into the question here.

Short English statement

The question asks whether rational orbits generated by two multiplicatively independent integer multipliers must require at least $(\log s)^K$ covering balls at radius $(\log s)^{-N}$, using exponents bounded by $M\log s$, with constants depending only on the multipliers. The precise formulation above uses explicitly provisional conventions because the supplied record leaves several domains and quantifiers unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003116$, title "Quantitative Entropy of Rational Multiplicative Orbits," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Diophantische Approximationen (2017), Oberwolfach Reports, pp. 1099--1187, DOI 10.4171/owr/2016/21. Supplied source link: https://doi.org/10.4171/owr/2016/21.
• Supplied literature source, labeled "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that the general quantitative Furstenberg covering-number lower bound is not proved, with only restricted results under a denominator size condition recorded in the report. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired. These are supplied editorial claims, not mathematical verification performed for this draft.