This paper proves the following theorem.
Every set of points in spans unit distances.
This is a small improvement over the previous bound of , which was proved independently by Kaplan, Matous̆ek, Patáková, and Sharir in  and by the author in . However, it is still far from the conjectured optimal bound of .
To put Theorem 1.1 in context, we will give a brief history of incidence geometry in Euclidean space. In , Kővári, Sós, and Turán showed that if is a bipartite graph with edge sets of size and that does not contain an induced copy of , then has at most edges. This theorem can be used to prove many results in incidence geometry. For example, since every pair of distinct points uniquely determines a line, there are incidences between points and lines in the plane. Similarly, since at most two unit spheres can pass through any three points in , there are unit distances spanned by points in .
However, the incidence theorems given by the Kővári-Sós-Turán theorem are frequently not sharp. For example, Szemerédi and Trotter proved in  that points and lines in the plane can have at most incidences, and this is sharp. To do this, they employed a technique now known as “partitioning + Kővári-Sós-Turán.” In short, they decomposed the plane into a union of open connected sets (called “cells”), plus a “boundary.” Each point in the plane lies in at most one of these cells. Each line can intersect several of these cells, but the number of cells that each line can intersect is controlled. Szemerédi and Trotter then examined the collection of points and lines inside each cell, applied the Kővári-Sós-Turán theorem, and summed the resulting contribution over all cells in the partition.
In , Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl systematically extended this technique to prove incidence theorems in the plane and in higher dimensions. Amongst many other results, they proved that points in span unit distances, where is a very slowly growing function. In , Guth and Katz developed a new partitioning theorem that has led to a revolution in combinatorial geometry. Amongst many other results, this new partitioning theorem allows one to slightly sharpen the methods from  to show that points in span unit distances. This was done independently by Kaplan, Matous̆ek, Patáková, and Sharir in  and by the author in .
Although many technical difficulties still abound, the “partitioning + Kővári-Sós-Turán” technique is now well understood and has been used to make progress on a wide variety of incidence problems. With the notable exception of the Szemerédi-Trotter theorem for points and lines, however, this technique rarely yields bounds that are conjectured to be sharp.
In , Aronov and Sharir developed a new method for proving incidence bounds for points and circles in the plane that gives stronger results than the “partitioning + Kővári-Sós-Turán” method. Aronov and Sharir “cut” a set of circles into “pseudo-segments” (a set of Jordan arcs are called pseudo-segments if they have the same combinatorial properties as line segments. In particular, each pair of points in the plane is incident to at most one arc). They then applied a variant of the Szemerédi-Trotter theorem to this set of pseudo-segments. In , Sharir and the author extended this cutting method from circles to general algebraic curves. This yielded an incidence theorem for points and curves in the plane that is stronger than the one given by the “partitioning + Kővári-Sós-Turán” method.
The unit distance problem in can be re-cast as an incidence problem involving points and circles in . In , Sheffer, Sharir, and the author used the “partitioning + Kővári-Sós-Turán” method to obtain a new bound for incidences between points and circles in three dimensions (this bound is stronger than the one from , because the three-dimensionality of the point-circle arrangement is exploited). Perhaps unsurprisingly, this point-circle bound is exactly what is needed111Recovering the bound for the unit distance problem using the bound from  still requires some careful arguments, since the bound from from  contains several terms that depend on the number of circles contained in a common plane or sphere, so this degeneracy must be carefully controlled. to recover the existing bound for the unit distance problem in .
In the present paper, we extend the cuttings method developed in  from plane curves to circles in . This gives us a new incidence bound for points and circles in that is stronger than the one from , and this in turn yields a new, improved bound on the number of unit distances in .
As is often the case with incidence bounds in higher dimensions, there are delicate issues regarding degeneracy. Many of the incidence bounds for points and curves in are stronger than the corresponding incidence bounds for points and curves in the plane. On the face of it, this appears suspicious, since any arrangement of points and curves in the plane can be embedded in , and the number of incidences remains unchanged. To obtain stronger incidence theorems, we must prohibit these types of degenerate configurations from occurring. Much of the technical complexity of this paper comes from navigating between the possible “degenerate” and “non-degenerate” configurations of points and circles.
Finally, we remark that it is not always possible to improve upon the bound for the unit distance problem if the Euclidean metric is replaced by a different metric. In Example 1.1 below, we give a semi-algebraic metric in which points can span unit distances. The metric from Example 1.1 can also be modified so that it is smooth (though after doing so, it is no longer semi-algebraic).
Let be the metric whose unit ball is given by , i.e.
Let be a positive integer. Let and let . Let Then for each point with and , there are at least points with ; indeed, every point with , , and will be contained in . Thus spans at least unit distances.
The key property of the Euclidean metric that we use (which is lacking in Example 1.1) is that there is a one-to-one correspondence between circles in of radius and pairs of distinct unit spheres whose centers have distance less than one. Translates of the unit paraboloid from Example 1.1 do not have this property: it is possible for many such translates to intersect in a common curve.
Let be a set of points in and let be a set of sets in (usually the sets in will be algebraic varieties). We define
We define .
If , we define
We define the set of two-rich points
If , we define
Let and be functions. We say or if there exists an absolute constant so that . We say or if the constant can depend on the parameter . We say if there is an absolute constant so that . The meaning of the variable will always be apparent from context.
2.2 Real algebraic varieties
In this section we will recall some standard results about real algebraic varieties that will be needed in later sections. A real algebraic variety is a subset of that is the common zero-locus of a finite set of polynomials. If is an algebraic variety, the dimension of is the largest integer so that contains a subset homeomorphic to the -dimensional unit cube . Further details can be found in .
If is an algebraic variety of dimension , then there exists an orthogonal projection so that is an algebraic variety of dimension . Indeed, “most” orthogonal projections have this property—if we give the set of orthogonal projections the structure of a real algebraic variety, then the set of projections for which this statement fails is contained in a proper Zariski closed subset. If can be defined using polynomials of degree , then the projections discussed above can be defined using polynomials whose degree is bounded by a function that depends only on and .
The set is the ideal of polynomials that vanish on . If is an algebraic variety of dimension , then the singular locus is the set of points of that are singular in dimension . If is generated by the polynomials , then is the set of points for which the matrix has rank less than ; this set is independent of the choice of generators .
The set of regular points of is given by . If , then there is a (Euclidean) open set containing so that is a -dimensional smooth manifold. Again, further details can be found in . We have that , and if can be defined by polynomials of degree at most , then can be defined by polynomials of degree bounded by a function depending only on and . If is square-free, then each point is contained in the zero-locus of exactly one irreducible component of
. Furthermore, there is a non-zero vectorso that is contained in , and the later variety has dimension at most .
A semi-algebraic set is a subset of that satisfies a finite list of polynomial equalities and inequalities. In particular, a real algebraic variety is a semi-algebraic set. We define the complexity222This definition is not standard, but since we will only consider semi-algebraic sets of bounded complexity, any reasonable definition of complexity will suffice. of a semi-algebraic set to be the minimum value of so that can be defined with polynomial equalities and inequalities, each of which have degree . If is a semi-algebraic set, then the image of under a projection is also semi-algebraic. If has complexity , then the image of the projection has complexity that is bounded by a function of and .
The dimension of a semi-algebraic set can be defined analogously to that of a real algebraic set; see  for details. If is a semi-algebraic set of dimension and complexity then we can write , where is a smooth -dimensional manifold and is a real algebraic variety of dimension strictly smaller than . Furthermore, is defined by polynomials whose degree is bounded by a function of and .
2.3 Polynomial partitioning
The revolutionary discrete polynomial partitioning theorem developed by Guth and Katz in  has led to many new incidence theorems. Since then, the partitioning theorem has been extended by Guth in  from points to general algebraic varieties. We will use a variant of this theorem from , which is a corollary of [8, Theorem 0.3]. The following theorem allows us to partition sets of algebraic varieties using a partitioning polynomial in the variables that is independent of some of the variables.
Theorem 2.1 (, Corollary 2.4).
Let be a set of algebraic varieties in , each of which has dimension at most and is defined by polynomials of degree at most . Let . Then for each , there is a polynomial of the form of degree at most so that is a union of cells, and varieties from intersect each cell.
We will also need the “partitioning on an algebraic hypersurface” theorem from . The variant we will use here is [23, Theorem 2.3]. This result makes reference to a “real ideal.” We will not define this term here, but we will recall the following result:
Lemma 2.1 (Lemma 2.1, ).
Let be an irreducible polynomial. Then there is a polynomial with so that generates a real ideal and .
For our purposes, Lemma 2.1 says that we can assume without loss of generality that every irreducible polynomial generates an irreducible ideal.
Let be an irreducible polynomial of degree that generates a real ideal and let . Let be a set of points. Then for each , there exists a polynomial of degree at most so that does not vanish identically on , and is a union of cells, each of which contains points from .
2.4 Connected components of sign conditions
We will apply the partitioning theorems from Section 2.3 to partition into cells. The following theorem controls how many of these cells an algebraic variety can intersect.
Theorem 2.3 (Barone-Basu ).
Let be an algebraic variety of dimension that is defined by polynomials of degree at most . Let be a polynomial of degree at most . Then contains connected components.
2.5 Doubly-ruled surfaces
A (complex) algebraic curve is a complex algebraic variety that has dimension one. A complex surface is said to be doubly-ruled by curves of degree if at a generic point , there are at least two curves of degree containing and contained in . The next lemma says that algebraic surfaces that contain many intersecting curves must be doubly-ruled by curves.
For each , there is a constant so that the following holds. Let be an irreducible variety of degree , and let be a set of algebraic curves in , each of which has degree and is contained in . Suppose that no two curves share a common components, and that for at least of the curves . Then is doubly-ruled by curves of degree . In particular, .
We will begin by recalling several results from . These results make reference to the property of being “–flecnodal.” This is a technical concept that we will not define here, since the definition can be taken as a black-box when using results from . However, a definition can be found in [11, Section 9] (here is the Chow variety of curves of degree ).
Lemma 8.3 and Proposition 10.2 from  says that for each and , there is a number so that the following holds. Let be an irreducible surface of degree . Suppose that there exists a set of irreducible algebraic curves of degree that are contained in , and that for at least of the curves . Then is –flecnodal at a generic point.
Theorem 8.1 from  says that for each , there exists a number with the following property: If is an irreducible surface that is –flecnodal at a generic point, then is doubly-ruled by curves of degree .
Theorem 3.5 from  says that if is an irreducible surface that is doubly-ruled by curves of degree , then has degree at most .
Lemma 2.2 now follows by combining the above results. ∎
Lemma 2.2 can be used to understand the structure of surfaces that contain many real curves.
A set is called a real algebraic curve if is a real algebraic variety of dimension one. To each real algebraic curve we can associate a complex algebraic curve so that is Zariski dense in . In particular, if and are two real algebraic curves that do not share a common component, then the complex algebraic curves and also will not share a common component. If is a real algebraic curve, we define the degree of to be the degree of the complex curve .
If is a real algebraic curve, then the image of under “most” projections will be an algebraic curve of the same degree. To be more precise, if we give the set of rotations the structure of an algebraic variety, and if is the projection to the first two coordinates, then the set of projections for which the image fails to be an algebraic curve of the same degree is contained in a proper closed sub-variety.
Note that if is a set of real algebraic curves, then , where
For each , there is a constant so that the following holds. Let be an irreducible polynomial of of degree . Let be a set of real algebraic curves, each of degree at most , each of which is contained in , and no two of which share a common component. Then either , or for all but at most curves , we have .
We will be particularly interested in circles in . The following result from  says that for our purposes, the only interesting surfaces containing many circles are planes and spheres.
Lemma 2.3 (, Lemma 3.2).
Let be an irreducible polynomial of degree . Let be a set of circles contained in and let be a set of points. Then either is a plane or sphere, or
2.6 Existing incidence bounds for points, circles, and spheres
When bounding the number number of incidences between points and unit spheres in , we will have to deal with incidences between points and circles, which arise as the intersection of pairs of unit spheres. When understanding point-circle incidences in , we will be forced to understand degenerate configurations in which many circles lie on a common sphere. Thus it will be necessary for us to deal with incidences between points and (arbitrary) spheres.
The following two results are the current best bounds in this direction. Neither of these results are sharp, and improvements to either result would yield an improved bound on the unit distance problem. However, even if the (conjectured) best-possible point-circle and point-sphere bounds were known, this would not be enough to obtain the conjectured sharp bound on the unit distance problem using the methods from this paper.
Theorem 2.4 (Aronov, Koltun, and Sharir ).
Let be a set of points and let be a set of circles in . Then the number of point-circle incidences is
Let be a sphere and let . We say that is -non-degenerate with respect to a set of points if for every circle we have
Theorem 2.5 (Apfelbaum and Sharir ).
Let be a set of points in . Then for each , the number of –rich, -non-degenerate spheres is
3 Cutting circles in into pseudosegments
In , Sharir and the author showed that algebraic plane curves can be cut into pseudo-segments. In this section we will extend this result from plane curves to circles in . It is possible that a similar result holds for general algebraic curves in , but the proof from  uses topological arguments that do not generalize from plane curves to (general) space curves.
A set is called an open (resp. closed) Jordan arc if it is homeomorphic to the open interval (resp. closed interval . Unless stated otherwise, all Jordan arcs that we refer to will be open.
Let be a set of Jordan arcs in . We say is a set of pseudo-segments if for every pair of points , there is at most one curve from that contains both and .
Let be a set of algebraic space curves in , no two of which share a common component. Let be a set of Jordan arcs, each pair of which have finite intersection. We say that is a cutting of if each curve in can be written as a union of finitely many arcs from , plus finitely many points. If , we say that is a cutting of into pieces.
Rather than keeping track of the Jordan arcs obtained by cutting algebraic curves, it will sometimes be easier to keep track of the points that are removed from each curve to obtain the cutting. Thus if is a set of algebraic curves in , it will sometimes be helpful to think of a cutting as a set with the following properties:
For each , .
For each , let . Then each connected component of is a Jordan arc.
then is a cutting of in the sense of Definition 3.3 into pieces.
Conversely, if is a cutting of into Jordan arcs, then we can represent the cutting as a set with ; simply define
3.2 Vertical hypersurfaces
A vertical hypersurface is an algebraic variety of the form , where is independent of the variable. If is a vertical hypersurface then if and only if .
The following is an immediate consequence of Corollary 2.1.
Let be the projection to the first three coordinates. Let be an irreducible vertical hypersurface in of degree . Let be a set of space curves in of degree , and suppose that no two curves have projections to the hyperplane that share a common component. Then either is doubly-ruled by curves of degree , or the curves in can be cut into Jordan arcs whose projections to the hyperplane are disjoint.
Let . For each point , define to be the number of points of intersection of with the closed ray . Note that for each , is either a non-negative integer or is .
Let , let be the projection to the first three coordinates, and let
where denotes Zariski closure. Note that is a vertical hypersurface defined by a polynomial of degree at most . By [18, Section 6], the function is constant on each (Euclidean) connected component of (the result in  is the analogous statement in rather than , but the proof is identical; the same proof works in any dimension ).
3.3 Lifting space curves to
In this section we will describe a transformation that sends space curves in to space curves in . The extra dimension will encode information about the slope of the curve.
Let be an irreducible curve in and let be the projection of to the plane. Applying a generic rotation if necessary, we can assume that the image of this projection is an irreducible algebraic curve, and that the fiber above a generic point in the image of the projection has cardinality one. We can also assume that the degree of is the same as that of .
As discussed in Section 3.3 of , this set is the union of a set of vertical lines (i.e. lines whose projection to the hyperplane consists of a point), plus an irreducible space curve in that is not a vertical line. Call this irreducible curve . Intuitively, if , then if is the slope of at the point .
If is a set of irreducible algebraic curves in , define
Applying a generic rotation if necessary, we will assume that each set in is an irreducible algebraic curve.
3.4 Depth cycles
Let and be closed Jordan arcs in . We say that and form a depth cycle if there exist points and so that either and , or and . We say that and form a proper depth cycle if one of these inequalities is strict.
We say that the closed Jordan arcs and form a minimal depth cycle if and are the endpoints of ; and are the endpoints of ; and the projections of and to the hyperplane intersect only at the points and .
If and form a depth cycle, then there always exist closed Jordan arcs and that form a minimal depth cycle. The choice of and might not be unique, however.
In , Sharir and the author proved the following result:
Theorem 3.1 (, Theorem 1.2).
Let be a set of algebraic curves in , each of degree at most . Suppose that no two curves have projections to the plane that share a common component. Then the curves in can be cut into Jordan arcs so that the arcs contain no proper depth cycles.
Since vertical depth cycles are preserved under projections of the form where is a projection, Theorem 3.1 also allows us to cut algebraic curves in into Jordan arcs so that all depth cycles are eliminated.
Let be a set of algebraic curves in , each of degree at most . Suppose that no two curves have projections to the hyperplane that share a common component. Then the curves in can be cut into Jordan arcs so that the arcs contain no proper depth cycles.
If the algebraic curves satisfy certain non-degeneracy conditions, however, then a stronger bound is possible.
For each integer and each , there is a constant so that the following holds. Let be a set of algebraic curves in , each of degree at most . Suppose that no two curves have projections to the hyperplane that share a common component. Then there are sets , a set of vertical hypersurfaces in and a cutting with the following properties.
Each hypersurface has degree at most .
Each curve in is contained in (at least one) vertical hypersurface from .
Each contains at least curves from .
is a cutting of into Jordan arcs; each arc is disjoint from each hypersurface in .
The arcs from the cutting contain no proper depth cycles.
We will prove the result by induction on . If is small then the result is immediate provided we choose the constant sufficiently large; simply choose and cut the curves from at each singular point of each curve and at each point where two or more curves have the same projection to the hyperplane.
We will now discuss the induction step. By Theorem 2.1 (with and ) there is a partitioning polynomial of degree so that is a union of at most cells and at most curves from intersect each cell, where is a constant that depends only on . The number will be chosen later; it will depend on and , but not on .
Let and be closed Jordan arcs in , each of which is contained in a curve from (though and need not be contained in the same curve), and suppose that and form a minimal proper depth cycle. Then at least one of the following four things must happen:
and are entirely contained in the same cell.
and are each entirely contained in a cell, but these cells are different.
At least one of or intersects but is not contained in .
Both of and are contained in .
We will cut the curves in to eliminate each of these types of depth cycles.
3.4.1 Eliminating depth cycles of Type 1
We will first describe a procedure to eliminate all depth cycles of Type 1. Let be the set of cells. For each cell , apply the induction hypothesis to (recall from (1) that this is the set of curves that intersect ). We obtain sets , , and so that:
Each vertical hypersurface has degree at most .
Each curve in is contained in (at least one) vertical hypersurface from .
Each contains at least curves from .
is a cutting of into Jordan arcs; each arc is disjoint from each hypersurface in .
The arcs from the cutting contain no proper depth cycles.
We have that
If is chosen sufficiently large (depending only on and , which in turn depends only on ), then
Define . It would be good if , but all we know is that this bound is too weak to close the induction. We will “fix” this issue below.
Let be the vertical hypersurface defined in (2). For each component that has degree , use Lemma 3.1 to cut the curves of into Jordan arcs, no two of which form a proper depth cycle. Denote this cutting by and let where the union is taken over all irreducible components of that have degree . We have .
Let be the union of and the irreducible components of that have degree . We have that
Observe that if are distinct, then the projection of to the hyperplane is an algebraic curve of degree at most . In particular, since no two curves have projections that share a common component, at most curves can satisfy . Combining this observation with the inclusion-exclusion principle, we conclude that
where on the final line we used the fact that if is sufficiently large (compared to and ), then . Thus . Define
and define . Note that and satisfy the first four requirements from Lemma 3.2.
For each , use Corollary 3.1 to cut the curves in into at most
Jordan arcs, so that the resulting collection of arcs contains no depth cycles. The total number of cuts required to perform this step is
Call the resulting cutting .
Observe that at this point, if , and if and are Jordan arcs contained inside the same cell that form a proper depth cycle, then there must either be a point with , or there must be a point with . In other words, for each cell , all depth cycles of Type 1 have been eliminated. It remains to eliminate the other types of depth cycles.
We have .
The cutting eliminates all depth cycles of Type 3. Next we will argue that the cutting eliminates all depth cycles of Types 2 and 4. Let and be closed Jordan arcs that form a minimal depth cycle, and suppose that either each of and are entirely contained in distinct cells, or each of and are contained in . If and are the endpoints of and if are the endpoints of , then after interchanging the roles of and if necessary, we have and , and at least one of these inequalities must be strict. In particular, there must exist a point on either or where the function changes value. Suppose the point is on . Since every point of this type is contained in , we have that , where is the curve containing . If instead the point is on , then an identical argument applies with in place of .
where is the projection to the plane. We have that . Let . If is chosen sufficiently large (depending only on and ), then . By construction, is a cutting of the curves in into Jordan arcs so that all depth cycles are eliminated. Furthermore, each arc in the cutting is disjoint from each surface in . This completes the induction step and finishes the proof. ∎ The main motivation for [18, Theorem 1.2] is that there is a transformation from plane curves in to space curves in so that lenses (pairs of curves that intersect at two common points) become depth cycles. Thus [18, Theorem 1.2] allows one to cut a set of algebraic plane curves into pseudo-segments. The following lemma is a three-dimensional analogue of this result in the special case where the curves are circles.
Let and be circles in that intersect at the two (distinct) points and . Let and be the projection of and to the plane, and suppose that both and are ellipses (i.e. neither is a line segment).
Let (resp. ) be a closed Jordan arc with endpoints and , and suppose that (resp ) does not contain an -extremal point of (resp. ).
Let and be the Jordan arcs in that are contained in and , respectively, whose projections to the plane are and . Then and form a proper depth cycle.
First, since and intersect at two points, the two curves cannot be tangent at the points of intersection. In particular, if and are the endpoints of and if and