Relative density of sets generated by fixed-parameter extrapolation
For $\lambda \in \mathbb{C}$ let $Q_\lambda$ be the smallest subset of $\mathbb{C}$ that contains $0$ and $1$ and is closed under $$(a, b) \mapsto (1 - \lambda) a + \lambda b,$$ that is, the set of points obtained from $0$ and $1$ by finitely many applications of this operation; no topological closure is taken. Put $X_\lambda = \mathbb{R}$ if $\lambda \in \mathbb{R} \setminus [0, 1]$ and $X_\lambda = \mathbb{C}$ if $\lambda \notin \mathbb{R}$. Fenner, Green and Homer showed that for such $\lambda$ the set $Q_\lambda$ is either discrete or dense in $X_\lambda$, that it can be discrete only if $\lambda$ is an algebraic integer, and that $Q_\lambda$ is relatively dense for a number of algebraic parameters, among them $\lambda = 1 + \varphi$ with $\varphi$ the golden ratio.
Problem. Is $Q_\lambda$ relatively dense in $X_\lambda$ for every $\lambda \in \mathbb{C} \setminus [0, 1]$, that is, is there for each such $\lambda$ a finite $R_\lambda$, which may depend on $\lambda$, such that every $z \in X_\lambda$ satisfies $|z - q| \le R_\lambda$ for some $q \in Q_\lambda$? If not, for which $\lambda$ does this hold?
Source: Fenner, Green and Homer, Open Question 11.3 of their 2026 paper (Open Question 17.5 in arXiv:1212.2889v6, 2020). Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- NT
- Keywords
- AIM Open Applied Problems, aperiodic order, fixed-parameter extrapolation, relatively dense sets
References
@article{fenner2026,
author = {Fenner, Stephen and Green, Frederic and Homer, Steven},
title = {Fixed-Parameter Extrapolation and Aperiodic Order},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {1},
year = {2026},
pages = {1--74},
doi = {10.1007/s00454-025-00816-4},
eprint = {1212.2889},
archivePrefix = {arXiv},
}
@misc{aim-extrapolation-relative-density,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Relative density of sets generated by fixed-parameter extrapolation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/630-extrapolation-relative-density.md},
note = {Entry 630 at commit 24edaad, 2026-10-15},
}
Resolution claims
Starting from $0$ and $1$, repeatedly apply the operation $(a,b)\mapsto(1-\lambda)a+\lambda b$, with a fixed complex parameter $\lambda$. We prove that the generated set has bounded holes in the real line for every real $\lambda\notin[0,1]$, and in the complex plane for every nonreal $\lambda$. This answers the relative-density question of Fenner, Green and Homer. Their characterization by integer polynomials allows a …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/relative_density.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00788, "Relative density of sets generated by fixed-parameter extrapolation", clopen.solutions.