## Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract) (2010)

Citations: | 11 - 5 self |

### BibTeX

@MISC{Elekes10incidencesin,

author = {György Elekes and Micha Sharir},

title = {Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)},

year = {2010}

}

### OpenURL

### Abstract

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.

### Citations

1707 | The Probabilistic Method
- Alon, Spencer
- 1992
(Show Context)
Citation Context ...parabolas; since all these parabolas are distinct (recall that a pair of parabolas can meet in at most one rotation point), we have n1 ≥ 2cn 1/2 . Hence, using Chernoff’s bound, as in [6] (see, e.g., =-=[1]-=-), we obtain that, with positive probability, (a) |C s | ≤ 2tn. (b) Each parabola h ∗ ∈ C contains at least 1 2ctn1/2 rotations that lie on parabolas of C s . (To see (b), take a parabola h ∗ ∈ C and ... |

270 |
Basic Algebraic Geometry
- Shafarevich
- 1977
(Show Context)
Citation Context ...s flat points of C. We recall (see (H9)) that any pair of distinct h-parabolas which meet at a point have there distinct tangents. First, we note that, using a trivial application of Bézout’s theorem =-=[15]-=-, a trivariate polynomial p of degree d which vanishes at 2d+1 points that lie on a common h-parabola h ∗ ∈ C must vanish identically on h ∗ . Critical points and parabolas. A point a is critical (or ... |

129 | On sets of distances of n points
- Erdős
- 1946
(Show Context)
Citation Context ...ir Tel Aviv University and New York University 1. THE INFRASTRUCTURE The motivation for the study reported in this paper comes from the celebrated and long-standing problem, originally posed by Erdős =-=[8]-=- in 1946, of obtaining a sharp lower bound for the number of distinct distances guaranteed to exist in any set S of s points in the plane. Erdős has shown that a section of the integer lattice determi... |

109 | Crossing numbers and hard Erd�os problems in discrete geometry
- Székely
- 1997
(Show Context)
Citation Context .... Moser [12], Chung [4], and Chung et al. [5] proved that the number of distinct distances determined by s points in the plane is Ω(s 2/3 ), Ω(s 5/7 ), and Ω(s 4/5 /polylog(s)), respectively. Székely =-=[19]-=- managed to get rid of the polylogarithmic factor, while Solymosi and Tóth [17] improved this bound to Ω(s 6/7 ). This was a real breakthrough. Their analysis was subsequently refined by Tardos [21] a... |

70 |
Extremal problems in discrete geometry. Combinatorica 3
- Szemerédi, Trotter
- 1983
(Show Context)
Citation Context ...d in Ξi, which were not yet removed when processing previous surfaces. The number of incidences between these rotations and parabolas can be bounded by the classical Szemerédi-Trotter incidence bound =-=[20]-=- (see also (2)), which is O(m 2/3 i n2/3 i + mi + ni). Summing these bounds over all the special surfaces Ξi, and using Hölder’s inequality and the fact, established in Lemma 5, that ni ≤ s, we get an... |

41 |
Elementary Differential Geometry
- Pressley
- 2001
(Show Context)
Citation Context ...n, if τ is a regular geometrically flat point of p (with respect to three parabolas of C) then Π(p)(τ) = 0. Remarks. (1) Π(p) is one of the polynomials that form the second fundamental form of p; see =-=[6, 7, 9, 13]-=- for details. (2) Although most details are suppressed, it is important to note that for Proposition 9 to hold it is crucial that τ be incident to (at least) three parabolas of C. This is why we can o... |

30 |
A new entropy inequality for the Erdös distance problem, Contemp. Math. 342, Towards a theory of geometric graphs
- Katz, Tardos
- 2004
(Show Context)
Citation Context ...polylogarithmic factor, while Solymosi and Tóth [17] improved this bound to Ω(s 6/7 ). This was a real breakthrough. Their analysis was subsequently refined by Tardos [21] and then by Katz and Tardos =-=[11]-=-, who obtained the current record of Ω(s (48−14e)/(55−16e)−ε ), for any ε > 0, which is Ω(s 0.8641 ). In this paper we transform the problem of distinct distances in the plane to an incidence problem ... |

28 |
On the different distances determined by n points
- Moser
- 1952
(Show Context)
Citation Context ... √ log s) distinct distances, and conjectured this to be a lower bound for any planar point set. In spite of steady progress on this problem, reviewed next, Erdős’s conjecture is still open. L. Moser =-=[12]-=-, Chung [4], and Chung et al. [5] proved that the number of distinct distances determined by s points in the plane is Ω(s 2/3 ), Ω(s 5/7 ), and Ω(s 4/5 /polylog(s)), respectively. Székely [19] managed... |

27 |
Unit distances in the Euclidean plane
- Spencer, Szemerédi, et al.
- 1984
(Show Context)
Citation Context .../7 k>s7/9 s k2 3 5 =It is fairly easy to show that N≥2 is O(s 10/3 ), by noting that N≥2 can be upper bounded by O `P i |Ei|2´, where Ei is as defined in (H1). Using the upper bound |Ei| = O(s 4/3 ) =-=[18]-=-, we get N≥2 = O X |Ei| 2 ! = O(s 4/3 ) · O X ! |Ei| = O(s 10/3 ). i Thus, at the moment, N≥2 is the bottleneck in the above bound, and we only get the (weak) lower bound Ω(s 2/3 ) on the number of di... |

21 | The number of different distances determined by n points in the plane
- Chung
- 1984
(Show Context)
Citation Context ...stinct distances, and conjectured this to be a lower bound for any planar point set. In spite of steady progress on this problem, reviewed next, Erdős’s conjecture is still open. L. Moser [12], Chung =-=[4]-=-, and Chung et al. [5] proved that the number of distinct distances determined by s points in the plane is Ω(s 2/3 ), Ω(s 5/7 ), and Ω(s 4/5 /polylog(s)), respectively. Székely [19] managed to get rid... |

19 |
The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput
- Chung, Szemerédi, et al.
- 1992
(Show Context)
Citation Context ...conjectured this to be a lower bound for any planar point set. In spite of steady progress on this problem, reviewed next, Erdős’s conjecture is still open. L. Moser [12], Chung [4], and Chung et al. =-=[5]-=- proved that the number of distinct distances determined by s points in the plane is Ω(s 2/3 ), Ω(s 5/7 ), and Ω(s 4/5 /polylog(s)), respectively. Székely [19] managed to get rid of the polylogarithmi... |

15 | Algebraic methods in discrete analogs of the Kakeya problem
- Guth, Katz
(Show Context)
Citation Context ...arabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz =-=[9]-=-, as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3 . Applying these bounds, we obtain, among several o... |

13 | On distinct sums and distinct distances
- Tardos
(Show Context)
Citation Context ...y [19] managed to get rid of the polylogarithmic factor, while Solymosi and Tóth [17] improved this bound to Ω(s 6/7 ). This was a real breakthrough. Their analysis was subsequently refined by Tardos =-=[21]-=- and then by Katz and Tardos [11], who obtained the current record of Ω(s (48−14e)/(55−16e)−ε ), for any ε > 0, which is Ω(s 0.8641 ). In this paper we transform the problem of distinct distances in t... |

11 | Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom., submitted. (An earlier version appeared
- Aronov, Koltun, et al.
(Show Context)
Citation Context ...of such incidences translate back to sharp lower bounds on the number of distinct distances. Incidence problems in three dimensions between points and curves have been studied in several recent works =-=[2, 6, 16]-=-, and a major push in this direction has been made last year, with the breakthrough result of Guth and Katz [9], who have introduced methods from algebraic geometry for studying problems of this kind.... |

5 | Point-line incidences in space
- Sharir, Welzl
(Show Context)
Citation Context ...of such incidences translate back to sharp lower bounds on the number of distinct distances. Incidence problems in three dimensions between points and curves have been studied in several recent works =-=[2, 6, 16]-=-, and a major push in this direction has been made last year, with the breakthrough result of Guth and Katz [9], who have introduced methods from algebraic geometry for studying problems of this kind.... |

4 |
The joints problem in R n
- Quilodrán
(Show Context)
Citation Context ... conjectures. Nevertheless, we have made considerable progress on the incidence problem itself, which is the second purpose of the study in this paper. We show how to adapt the algebraic machinery of =-=[6, 9, 10, 14]-=- to derive sharp bounds for the incidence problem. These bounds are very similar to, and in fact even better than the bounds obtained in [6] for pointline incidences, where they have been shown to be ... |

4 | On a question of Bourgain about geometric incidences
- Solymosi, Tóth
(Show Context)
Citation Context ...t distances determined by s points in the plane is Ω(s 2/3 ), Ω(s 5/7 ), and Ω(s 4/5 /polylog(s)), respectively. Székely [19] managed to get rid of the polylogarithmic factor, while Solymosi and Tóth =-=[17]-=- improved this bound to Ω(s 6/7 ). This was a real breakthrough. Their analysis was subsequently refined by Tardos [21] and then by Katz and Tardos [11], who obtained the current record of Ω(s (48−14e... |

3 |
On lines, joints, and incidences in three dimensions, manuscript
- Elekes, Sharir
- 2009
(Show Context)
Citation Context ...jectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. =-=[6]-=-, to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3 . Applying these bounds, we obtain, among several other results, the upper bound O(s 3 ) on th... |

3 |
On lines and joints, Discrete Comput
- Kaplan, Sharir, et al.
(Show Context)
Citation Context ...ns, each of which is incident to at least three parabolas of C. Suppose further that no special surface contains more than q parabolas of C. Then m = O(n 3/2 + nq). Remarks. (1) The recent results of =-=[10, 14]-=- imply that the number of joints in a set of n h-parabolas is O(n 3/2 ). The proofs in [10, 14] are much simpler than the proof given below, but they do not apply to flat points (rotations) as does Th... |