Distinct Distances on Non-Ruled Surfaces and Between Circles
Abstract We improve the current best bound for distinct distances on non-ruled algebraic surfaces in $${\mathbb {R}}^3$$ R...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2023
|
| Online Access: | https://hdl.handle.net/1721.1/147921 |
| _version_ | 1826201867449794560 |
|---|---|
| author | Mathialagan, Surya Sheffer, Adam |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Mathialagan, Surya Sheffer, Adam |
| author_sort | Mathialagan, Surya |
| collection | MIT |
| description | Abstract
We improve the current best bound for distinct distances on non-ruled algebraic surfaces in
$${\mathbb {R}}^3$$
R
3
. In particular, we show that n points on such a surface span
$$\Omega (n^{32/39-{\varepsilon }})$$
Ω
(
n
32
/
39
-
ε
)
distinct distances, for any
$${\varepsilon }>0$$
ε
>
0
. Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in
$${\mathbb {R}}^3$$
R
3
. Consider two point sets of respective sizes m and n, such that each set lies on a distinct circle in
$${\mathbb {R}}^3$$
R
3
. We characterize the cases when the number of distinct distances between the two sets can be
$$O(m+n)$$
O
(
m
+
n
)
. This includes a new configuration with a small number of distances. In any other case, we prove that the number of distinct distances is
$$\Omega (\min {\{m^{2/3}n^{2/3},m^2,n^2\}})$$
Ω
(
min
{
m
2
/
3
n
2
/
3
,
m
2
,
n
2
}
)
. |
| first_indexed | 2024-09-23T11:58:07Z |
| format | Article |
| id | mit-1721.1/147921 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:58:07Z |
| publishDate | 2023 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1479212024-01-22T21:28:36Z Distinct Distances on Non-Ruled Surfaces and Between Circles Mathialagan, Surya Sheffer, Adam Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Abstract We improve the current best bound for distinct distances on non-ruled algebraic surfaces in $${\mathbb {R}}^3$$ R 3 . In particular, we show that n points on such a surface span $$\Omega (n^{32/39-{\varepsilon }})$$ Ω ( n 32 / 39 - ε ) distinct distances, for any $${\varepsilon }>0$$ ε > 0 . Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in $${\mathbb {R}}^3$$ R 3 . Consider two point sets of respective sizes m and n, such that each set lies on a distinct circle in $${\mathbb {R}}^3$$ R 3 . We characterize the cases when the number of distinct distances between the two sets can be $$O(m+n)$$ O ( m + n ) . This includes a new configuration with a small number of distances. In any other case, we prove that the number of distinct distances is $$\Omega (\min {\{m^{2/3}n^{2/3},m^2,n^2\}})$$ Ω ( min { m 2 / 3 n 2 / 3 , m 2 , n 2 } ) . 2023-02-07T12:55:00Z 2023-02-07T12:55:00Z 2022-11-14 2023-02-07T04:28:54Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/147921 Mathialagan, Surya and Sheffer, Adam. 2022. "Distinct Distances on Non-Ruled Surfaces and Between Circles." en https://doi.org/10.1007/s00454-022-00449-x Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature application/pdf Springer US Springer US |
| spellingShingle | Mathialagan, Surya Sheffer, Adam Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title | Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title_full | Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title_fullStr | Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title_full_unstemmed | Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title_short | Distinct Distances on Non-Ruled Surfaces and Between Circles |
| title_sort | distinct distances on non ruled surfaces and between circles |
| url | https://hdl.handle.net/1721.1/147921 |
| work_keys_str_mv | AT mathialagansurya distinctdistancesonnonruledsurfacesandbetweencircles AT shefferadam distinctdistancesonnonruledsurfacesandbetweencircles |