---
_id: '267'
abstract:
- lang: eng
text: Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and
Li, we produce infinitely many smooth cubic surfaces defined over the field of
rational numbers that contain rational points.
acknowledgement: While working on this paper the author was supported by ERC grant
306457.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
full_name: Browning, Timothy D
id: 35827D50-F248-11E8-B48F-1D18A9856A87
last_name: Browning
orcid: 0000-0002-8314-0177
citation:
ama: Browning TD. Many cubic surfaces contain rational points. Mathematika.
2017;63(3):818-839. doi:10.1112/S0025579317000195
apa: Browning, T. D. (2017). Many cubic surfaces contain rational points. Mathematika.
Cambridge University Press. https://doi.org/10.1112/S0025579317000195
chicago: Browning, Timothy D. “Many Cubic Surfaces Contain Rational Points.” Mathematika.
Cambridge University Press, 2017. https://doi.org/10.1112/S0025579317000195.
ieee: T. D. Browning, “Many cubic surfaces contain rational points,” Mathematika,
vol. 63, no. 3. Cambridge University Press, pp. 818–839, 2017.
ista: Browning TD. 2017. Many cubic surfaces contain rational points. Mathematika.
63(3), 818–839.
mla: Browning, Timothy D. “Many Cubic Surfaces Contain Rational Points.” Mathematika,
vol. 63, no. 3, Cambridge University Press, 2017, pp. 818–39, doi:10.1112/S0025579317000195.
short: T.D. Browning, Mathematika 63 (2017) 818–839.
date_created: 2018-12-11T11:45:31Z
date_published: 2017-11-29T00:00:00Z
date_updated: 2024-03-05T11:49:27Z
day: '29'
doi: 10.1112/S0025579317000195
extern: '1'
external_id:
arxiv:
- '1701.00525'
intvolume: ' 63'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1701.00525
month: '11'
oa: 1
oa_version: Preprint
page: 818 - 839
publication: Mathematika
publication_identifier:
issn:
- 0025-5793
publication_status: published
publisher: Cambridge University Press
publist_id: '7635'
quality_controlled: '1'
status: public
title: Many cubic surfaces contain rational points
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 63
year: '2017'
...
---
_id: '1455'
abstract:
- lang: eng
text: First, a special case of Knaster's problem is proved implying that each symmetric
convex body in ℝ3 admits an inscribed cube. It is deduced from a theorem in equivariant
topology, which says that there is no S4 - equivariant map from SO(3) to S2, where
S4 acts on SO(3) on the right as the rotation group of the cube, and on S2 on
the right as the symmetry group of the regular tetrahedron. Some generalizations
are also given. Second, it is shown how the above non-existence theorem yields
Makeev's conjecture in ℝ3 that each set in ℝ3 of diameter 1 can be covered by
a rhombic dodecahedron, which has distance 1 between its opposite faces. This
reveals an unexpected connection between inscribing cubes into symmetric bodies
and covering sets by rhombic dodecahedra. Finally, a possible application of our
second theorem to the Borsuk problem in ℝ3 is pointed out.
acknowledgement: The research of the first author was partially supported by Trinity
College, Cambridge, and that of all the authors by grants 23444, T-030012 and A
046/96, respectively, from the Hungarian National Foundation for Scientific Research.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tamas
full_name: Hausel, Tamas
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Endre
full_name: Makai, Endre
last_name: Makai
- first_name: András
full_name: Szücs, András
last_name: Szücs
citation:
ama: Hausel T, Makai E, Szücs A. Inscribing cubes and covering by rhombic dodecahedra
via equivariant topology. Mathematika. 2000;47(1-2):371-397. doi:10.1112/S0025579300015965
apa: Hausel, T., Makai, E., & Szücs, A. (2000). Inscribing cubes and covering
by rhombic dodecahedra via equivariant topology. Mathematika. University
College London. https://doi.org/10.1112/S0025579300015965
chicago: Hausel, Tamás, Endre Makai, and András Szücs. “Inscribing Cubes and Covering
by Rhombic Dodecahedra via Equivariant Topology.” Mathematika. University
College London, 2000. https://doi.org/10.1112/S0025579300015965.
ieee: T. Hausel, E. Makai, and A. Szücs, “Inscribing cubes and covering by rhombic
dodecahedra via equivariant topology,” Mathematika, vol. 47, no. 1–2. University
College London, pp. 371–397, 2000.
ista: Hausel T, Makai E, Szücs A. 2000. Inscribing cubes and covering by rhombic
dodecahedra via equivariant topology. Mathematika. 47(1–2), 371–397.
mla: Hausel, Tamás, et al. “Inscribing Cubes and Covering by Rhombic Dodecahedra
via Equivariant Topology.” Mathematika, vol. 47, no. 1–2, University College
London, 2000, pp. 371–97, doi:10.1112/S0025579300015965.
short: T. Hausel, E. Makai, A. Szücs, Mathematika 47 (2000) 371–397.
date_created: 2018-12-11T11:52:07Z
date_published: 2000-06-01T00:00:00Z
date_updated: 2023-05-08T08:56:46Z
day: '01'
doi: 10.1112/S0025579300015965
extern: '1'
external_id:
arxiv:
- math/9906066
intvolume: ' 47'
issue: 1-2
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/9906066
month: '06'
oa: 1
oa_version: Preprint
page: 371 - 397
publication: Mathematika
publication_identifier:
issn:
- 0025-5793
publication_status: published
publisher: University College London
publist_id: '5745'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Inscribing cubes and covering by rhombic dodecahedra via equivariant topology
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 47
year: '2000'
...