Characterizing discrete 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 $$a \star_\lambda b = (1 - \lambda) a + \lambda b.$$ Each point of $Q_\lambda$ is obtained by finitely many operations; no topological closure is taken. Call $\lambda$ a strong PV number if it is an algebraic integer and every Galois conjugate other than $\lambda$ and $\overline{\lambda}$ lies in the real interval $(0, 1)$; integers and nonreal quadratic algebraic integers qualify. Fenner, Green and Homer introduced these numbers and showed that $Q_\lambda$ is uniformly discrete when $\lambda$ is strong PV, and that $\lambda$ is an algebraic integer whenever $Q_\lambda$ is discrete, extending results of Pinch for real $\lambda$.
Problem. Is $Q_\lambda$ discrete if and only if $\lambda$ is a strong PV number? For these sets discreteness is equivalent to uniform discreteness: there is a $\delta > 0$ with $|x - y| \ge \delta$ for all distinct $x, y \in Q_\lambda$.
Source: Fenner, Green and Homer (2026), Conjecture 11.1; for totally real λ, conjectured earlier by Pinch (1985).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- NT
- Keywords
- AIM Open Applied Problems, aperiodic order, fixed-parameter extrapolation, Pisot numbers
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},
}
@misc{fenner2020,
author = {Fenner, Stephen and Green, Frederic and Homer, Steven},
title = {Fixed-Parameter Extrapolation and Aperiodic Order},
year = {2020},
eprint = {1212.2889},
archivePrefix = {arXiv},
note = {Extended version},
}
@article{pinch1985,
author = {Pinch, R. G. E.},
title = {{$a$}-convexity},
journal = {Math. Proc. Cambridge Philos. Soc.},
volume = {97},
number = {1},
year = {1985},
pages = {63--68},
doi = {10.1017/S0305004100062587},
}
@misc{aim-extrapolation-discreteness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Characterizing discrete sets generated by fixed-parameter extrapolation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/629-extrapolation-discreteness.md},
note = {Entry 629 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 resulting set is discrete exactly when $\lambda$ is an algebraic integer whose other conjugates, apart from $\lambda$ and $\overline\lambda$, are real and lie in $(0,1)$. This gives the converse to the known sufficient condition for separated extrapolation sets. The …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/extrapolation_discreteness.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00787, "Characterizing discrete sets generated by fixed-parameter extrapolation", clopen.solutions.