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 \cite{GK}, as further developed by Elekes et al. \cite{EKS}, to obtain sharp bounds on the number of incidences between these helices or parabolas and points in $\reals^3$. 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 $k\ge 3$ 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 $k\ge 2$. If true, it would imply the lower bound $\Omega(s/\log s)$ on the number of distinct distances in the plane.
Deep Dive into Incidences in Three Dimensions and Distinct Distances in the Plane.
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 \cite{GK}, as further developed by Elekes et al. \cite{EKS}, to obtain sharp bounds on the number of incidences between these helices or parabolas and points in $\reals^3$. 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 $k\ge 3$ 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
second step, using more traditional space decomposition techniques. The final bound is still not as good as we would like it to be, but it shows that the case studied in this paper "behaves better" than its counterpart involving lines.
Distinct distances and incidences with helices. We offer the following novel approach to the problem of distinct distances.
(H1) Notation. Let S be a set of s points in the plane with x distinct distances. Let K denote the set of all quadruples (a, b, a ′ , b ′ ) ∈ S 4 , such that the pairs (a, b) and (a ′ , b ′ ) are distinct (although the points themselves need not be) and |ab| = |a ′ b ′ | > 0.
Let δ 1 , . . . , δ x denote the x distinct distances in S, and let
We have
x .
(H2) Rotations. We associate each (a, b, a ′ , b ′ ) ∈ K with a (unique) rotation (or, rather, a rigid, orientation-preserving transformation of the plane) τ , which maps a to a ′ and b to b ′ . A rotation τ , in complex notation, can be written as the transformation z → pz + q, where p, q ∈ C and |p| = 1.
Putting p = e iθ , q = ξ + iη, we can represent τ by the point (ξ, η, θ) ∈ R 3 . In the planar context, θ is the counterclockwise angle of the rotation, and the center of rotation is c = q/(1 -e iθ ), which is defined for θ = 0; for θ = 0, τ is a pure translation.
The multiplicity µ(τ ) of a rotation τ (with respect to S) is defined as |τ (S) ∩ S| = the number of pairs (a, b) ∈ S 2 such that τ (a) = b. Clearly, one always has µ(τ ) ≤ s, and we will mostly consider only rotations satisfying µ(τ ) ≥ 2. As a matter of fact, the bulk of the paper will only consider rotations with multiplicity at least 3. Rotations with multiplicity 2 are harder to analyze.
If µ(τ ) = k then S contains two congruent and equally oriented copies A, B of some k-element set, such that τ (A) = B. Thus, studying multiplicities of rotations is closely related to analyzing repeated (congruent and equally oriented) patterns in a planar point set; see [3] for a review of many problems of this kind.
Anti-rotations. In this paper we will also consider anti-rotations, which are rigid, orientationreversing transformations of the plane. Any anti-rotation can be represented as a rotation, followed by a reflection about some fixed line, e.g., the x-axis (so, in complex notation, this can be written as z → pz + q). Anti-rotations will be useful in certain steps of the analysis.
(H3) Bounding |K|. If µ(τ ) = k then τ contributes k 2 quadruples to K. Let N k (resp., N ≥k ) denote the number of rotations with multiplicity exactly k (resp., at least k), for k ≥ 2. Then
(H4) The main conjecture.
Conjecture 1. For any 2 ≤ k ≤ s, we have
Suppose that the conjecture were true. Then we would have
which would have implied that x = Ω(s/ log s). This would have almost settled the problem of obtaining a tight bound for the minimum number of distinct distances guaranteed to exist in any set of s points in the plane, since, as mentioned above, the upper bound for this quantity is O(s/ √ log s) [7].
We note that Conjecture 1 is rather deep; even the simple instance k = 2, asserting that there are only O(s 3 ) rotations which map (at least) two points of S to two other points of S (at the same distance apart), seems quite difficult. In this paper we establish a variety of upper bounds on the number of rotations and on the sum of their multiplicities. In particular, these results provide a partial positive answer, showing that N ≥3 = O(s 3 ); that is, the number of rotations which map a (degenerate or non-degenerate) triangle determined by S to another congruent (and equally oriented) such triangle, is O(s 3 ). Bounding N 2 by O(s 3 ) is still an open problem. See Section 5 for a simple proof of the weaker bound N ≥2 = O(s 10/3 ).
Lower bound. We next give a construction (suggested by Haim Kaplan) which shows: Lemma 2. There exist sets S in the plane of arbitrarily large cardinality, which determine Θ(|S| 3 ) distinct rotations, each mapping a triple of points of S to another triple of points of S. For each triple a, b, c ∈ {1, . . . , s} such that a + b -c also belongs to {1, . . . , s}, construct the rotation τ a,b,c which maps (a, 0) to (b, 0) and (c, 1) to (a + b -c, 1). Since the distance between the two source points is equal to the distance between their images, τ a,b,c is well (and uniquely) defined. Moreover, τ a,b,c maps the midpoint ((a + c)/2, 1/2) to the midpoint ((a + 2b -c)/2, 1/2).
We claim that the rotations τ a,b,c are all distinct. Indeed, suppose that two such rotations, τ a,b,c and τ a ′ ,b ′ ,c ′ , for distinct triples (a, b, c), (a ′ , b ′ , c ′ ), coincide; call the common rotation τ . We can represent τ as the rigid transformation which first translates the plane horizontally by distance b -a, so that (a, 0) is mapped to (b, 0), and then rotates it around (b, 0) by an appropriate angle 0 < θ < π, so that (c + b -a, 1) is mapped to (a + b -c, 1). Suppose first that a = a ′ . Since τ = τ a,b,c = τ a ′ ,b ′ ,c ′ , it m
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