Covering lattice points by subspaces and
counting point-hyperplane incidences††thanks: The first and the third author acknowledge the support of the grants GAČR 14-14179S of Czech Science Foundation, ERC Advanced Research Grant no 267165 (DISCONV), and GAUK 690214 of the Grant Agency of the Charles University. The first author is also supported by the grant SVV–2016–260332.
Let and be integers with . Let be a -dimensional lattice and let be a -dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of -dimensional linear subspaces needed to cover all points in . In particular, our results imply that the minimum number of -dimensional linear subspaces needed to cover the -dimensional grid is at least and at most , where is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach . We also find tight bounds for the minimum number of -dimensional affine subspaces needed to cover .
We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer . For and , we show that there is an integer such that for all positive integers the following statement is true. There is a set of points in and an arrangement of hyperplanes in with no in their incidence graph and with at least incidences if is odd and incidences if is even.
M. Balko, J. Cibulka, and P. Valtr
In this paper, we study the minimum number of linear or affine subspaces needed to cover points that are contained in the intersection of a given lattice with a given 0-symmetric convex body. We also present an application of our results to the problem of estimating the maximum number of incidences between a set of points and an arrangement of hyperplanes. Consequently, this establishes a new lower bound for the time complexity of so-called partitioning algorithms for Hopcroft’s problem. Before describing our results in more detail, we first give some preliminaries and introduce necessary definitions.
For linearly independent vectors , the -dimensional lattice with basis is the set of all linear combinations of the vectors with integer coefficients. We define the determinant of as , where is the matrix with the vectors as columns. For a positive integer , we use to denote the set of -dimensional lattices , that is, lattices with .
A convex body is symmetric about the origin if . We let be the set of -dimensional compact convex bodies in that are symmetric about the origin.
For a positive integer , we use the abbreviation to denote the set . A point of a lattice is called primitive if whenever its multiple is a lattice point, then is an integer. For , let be the -dimensional Lebesgue measure of . We say that is the volume of . The closed -dimensional ball with the radius , , centered in the origin is denoted by . If , we simply write instead of . For , we use to denote the Euclidean norm of .
Let be a subset of . We use and to denote the affine hull of and the linear hull of , respectively. The dimension of the affine hull of is denoted by .
For functions , we write if there is a fixed constant such that for all . We write if there is a fixed constant such that for all . If the constants and depend on some parameters , then we emphasize this by writing and , respectively. If and , then we write .
1.2 Covering lattice points by subspaces
We say that a collection of subsets in covers a set of points from if every point from lies in some set from .
Let , , , and be positive integers that satisfy . We let be the maximum size of a set such that every -dimensional affine subspace of contains at most points of . Similarly, we let be the maximum size of a set such that every -dimensional linear subspace of contains at most points of . We also let be the minimum number of -dimensional linear subspaces of necessary to cover .
In this paper, we study the functions , , and and their generalizations to arbitrary lattices from and bodies from . We mostly deal with the last two functions, that is, with covering lattice points by linear subspaces. In particular, we obtain new upper bounds on (Theorem 2.1), lower bounds on (Theorem 2.2), and we use the estimates for and to obtain improved lower bounds for the maximum number of point-hyperplane incidences (Theorem 2.4). Before doing so, we first give a summary of known results, since many of them are used later in the paper.
The problem of determining is essentially solved. In general, the set can be covered by affine -dimensional subspaces and thus we have an upper bound . This trivial upper bound is asymptotically almost tight for all fixed , , and some , as Brass and Knauer  showed with a probabilistic argument that for every there is an such that for each positive integer we have
For fixed and , the upper bound is known to be asymptotically tight in the cases and . This is shown by considering points on the modular moment surface for and the modular moment curve for ; see .
Covering lattice points by linear subspaces seems to be more difficult than covering by affine subspaces. From the definitions we immediately get . In the case and fixed, Bárány, Harcos, Pach, and Tardos  obtained the following asymptotically tight estimates for the functions and :
In fact, Bárány et al.  proved stronger results that estimate the minimum number of -dimensional linear subspaces necessary to cover the set in terms of so-called successive minima of a given lattice and a body .
For a lattice , a body , and , we let be the th successive minimum of and . That is, . Since is compact, it is easy to see that the successive minima are achieved. That is, there are linearly independent vectors from such that for every . Also note that we have and .
Theorem 1.1 ()
For an integer , a lattice , and a body , we let for every . If , then the set can be covered with at most
-dimensional linear subspaces of , where is some absolute constant.
On the other hand, if , then there is a subset of of size
such that no -dimensional linear subspace of contains points from .
We note that the assumption is necessary; see the discussion in . Not much is known for linear subspaces of lower dimension. We trivially have for all with . Thus for some by (1). Brass and Knauer  conjectured that for fixed. This conjecture was refuted by Lefmann  who showed that, for all and with , there is an absolute constant such that we have for every positive integer . This bound is asymptotically smaller in than the growth rate conjectured by Brass and Knauer for sufficiently large and almost all values of with .
Covering lattice points by linear subspaces is also mentioned in the book by Brass, Moser, and Pach , where the authors pose the following problem.
Problem 1 ([6, Problem 6 in Chapter 10.2])
What is the minimum number of -dimensional linear subspaces necessary to cover the -dimensional lattice cube?
1.3 Point-hyperplane incidences
As we will see later, the problem of determining and is related to a problem of bounding the maximum number of point-hyperplane incidences. For an integer , let be a set of points in and let be an arrangement of hyperplanes in . An incidence between and is a pair such that , , and . The number of incidences between and is denoted by .
We are interested in the maximum number of incidences between and . In the plane, the famous Szemerédi–Trotter theorem  says that the maximum number of incidences between a set of points in and an arrangement of lines in is at most . This is known to be asymptotically tight, as a matching lower bound was found earlier by Erdős . The current best known bounds are 111 The lower bound claimed by Pach and Tóth [18, Remark 4.2] contains the multiplicative constant . This is due to a miscalculation in the last equation in the calculation of the number of incidences. The correct calculation is . This leads to . and .
For , it is easy to see that there is a set of points in and an arrangement of hyperplanes in for which the number of incidences is maximum possible, that is . It suffices to consider the case where all points from lie in an affine subspace that is contained in every hyperplane from . In order to avoid this degenerate case, we forbid large complete bipartite graphs in the incidence graph of and , which is denoted by . This is the bipartite graph on the vertex set and with edges where is an incidence between and .
With this restriction, bounding becomes more difficult and no tight bounds are known for . It follows from the works of Chazelle , Brass and Knauer , and Apfelbaum and Sharir  that the number of incidences between any set of points in and any arrangement of hyperplanes in with satisfies
We note that an upper bound similar to (2) holds in a much more general setting; see the remark in the proof of Theorem 2.4. The best general lower bound for is due to a construction of Brass and Knauer , which gives the following estimate.
Theorem 1.2 ()
Let be an integer. Then for every there is a positive integer such that for all positive integers and there is a set of points in and an arrangement of hyperplanes in such that and
For , this lower bound has been recently improved by Sheffer  in a certain non-diagonal case. Sheffer constructed a set of points in , , and an arrangement of hyperplanes in such that and .
2 Our results
In this paper, we nearly settle Problem 1 by proving almost tight bounds for the function for a fixed and an arbitrary from . For a fixed , an arbitrary , and some fixed , we also provide bounds on the function that are very close to the bound conjectured by Brass and Knauer . Thus it seems that the conjectured growth rate of is true if we allow to be (significantly) larger than .
We study these problems in a more general setting where we are given an arbitrary lattice from and a body from . Similarly to Theorem 1.1 by Bárány et al. , our bounds are expressed in terms of the successive minima , .
2.1 Covering lattice points by linear subspaces
First, we prove a new upper bound on the minimum number of -dimensional linear subspaces that are necessary to cover points in the intersection of a given lattice with a body from .
For integers and with , a lattice , and a body , we let for . If , then we can cover with -dimensional linear subspaces of , where
We also prove the following lower bound.
For integers and with , a lattice , and a body , we let for . If , then, for every , there is a positive integer and a set of size at least , where
such that every -dimensional linear subspace of contains at most points from .
We remark that we can get rid of the in the exponent if or ; for details, see Theorem 1.1 for the case and the proof in Section 4 for the case . Also note that in the definition of in Theorem 2.1 the minimum is taken over the set , while in the definition of in Theorem 2.2 the minimum is taken over . There are examples that show that cannot be replaced by in Theorem 2.1. It suffices to consider , , and let be the lattice for some large positive integer . Then , , , and thus . However, it is not difficult to see that we need at least 1-dimensional linear subspaces to cover , which is asymptotically larger than . On the other hand, and 1-dimensional linear subspaces suffice to cover . We thus suspect that the lower bound can be improved.
Since for every , we can apply Theorem 2.2 with and and obtain the following lower bound on .
Let and be integers with . Then, for every , there is an such that for every we have
The existence of the set from Theorem 2.2 is shown by a probabilistic argument. It would be interesting to find, at least for some value , some fixed , and arbitrarily large , a construction of a subset of of size such that every -dimensional linear subspace contains at most points from . Such constructions are known for and ; see [5, 19].
Let , , and be integers with . Then, for every , we have
2.2 Covering lattice points by affine subspaces
For affine subspaces, Brass and Knauer  considered only the case of covering the -dimensional lattice cube by -dimensional affine subspaces. To our knowledge, the case for general and was not considered in the literature. We extend the results of Brass and Knauer to covering .
For integers and with , a lattice , and a body , we let for . If , then the set can be covered with -dimensional affine subspaces of .
On the other hand, at least -dimensional affine subspaces of are necessary to cover .
2.3 Point-hyperplane incidences
As an application of Corollary 1, we improve the best known lower bounds on the maximum number of point-hyperplane incidences in for . That is, we improve the bounds from Theorem 1.2. To our knowledge, this is the first improvement on the estimates for in the general case during the last 13 years.
For every integer and , there is an such that for all positive integers and the following statement is true. There is a set of points in and an arrangement of hyperplanes in such that and
We can get rid of the in the exponent for . That is, we have the bounds for and for . For , our bound is the same as the bound from Theorem 1.2. For larger , our bounds become stronger. In particular, the exponents in the lower bounds from Theorem 2.4 exceed the exponents from Theorem 1.2 by for odd and by for even. However, the bounds are not tight. The exponents in the known bounds for for small values of are summarized in Table 1.
|Upper bounds [2, 5, 7, 22]|
|Lower bounds from Theorem 1.2|
|Lower bounds from Theorem 2.4|
In the non-diagonal case, when one of and is significantly larger that the other, the proof of Theorem 2.4 yields the following stronger bound.
For all integers and with and for , there is an such that for all positive integers and the following statement is true. There is a set of points in and an arrangement of hyperplanes in such that and
For example, in the case considered by Sheffer , Theorem 2.5 gives a slightly better bound than if we set, for example, . However, the forbidden complete bipartite subgraph in the incidence graph is larger than .
The following problem is known as the counting version of Hopcroft’s problem [5, 9]: given points in and hyperplanes in , how fast can we count the incidences between them? We note that the lower bounds from Theorem 2.4 also establish the best known lower bounds for the time complexity of so-called partitioning algorithms  for the counting version of Hopcroft’s problem; see  for more details.
In the proofs of our results, we make no serious effort to optimize the constants. We also omit floor and ceiling signs whenever they are not crucial.
3 Proof of Theorem 2.1
Here we show the upper bound on the minimum number of -dimensional linear subspaces needed to cover points from a given -dimensional lattice that are contained in a body from . We first prove Theorem 2.1 in the special case (Theorem 3.4) and then we extend the result to arbitrary .
3.1 Proof for balls
Before proceeding with the proof of Theorem 2.1, we first introduce some auxiliary results that are used later. The following classical result is due to Minkowski  and shows a relation between , , and the successive minima of and .
Theorem 3.1 (Minkowski’s second theorem )
Let be a positive integer. For every and every , we have
Theorem 3.2 ([12, Theorem 1.5])
Let be a positive integer. For every and every , we have
For and , let be linearly independent vectors such that for every . For , the vectors do not necessarily form a basis of [21, see Section X.5]. However, the following theorem shows that there exists a basis with vectors of lengths not much larger than the lengths of .
Theorem 3.3 (First finiteness theorem [21, see Lemma 2 in Section X.6])
Let be a positive integer. For every and every , there is a basis of with for every .
Now, let be a -dimensional lattice with . Throughout this section, we use to denote the th successive minimum for . Let be an integer with . We show the following result.
There is a constant such that the set can be covered with -dimensional linear subspaces of , where
This is the same expression as in the statement of Theorem 2.1. We have just chosen a different index notation, since we will work mostly in a dual setting in the proof, where this new expression becomes more natural. Let be an integer from such that , where is the parameter from the statement of Theorem 3.4.
In the rest of the section, we prove Theorem 3.4. However, since its proof is rather long and complicated, we first give a high-level overview.
We start by proving a weaker upper bound on the number of -dimensional subspaces of needed to cover (Corollary 3). This bound is obtained from Theorem 3.2 and Lemma 1, which states that, for each with , there is a suitable projection of on a -dimensional linear subspace such that the th successive minimum of the image of is in . The existence of such projections is proved using Minkowski’s second theorem and the First finiteness theorem. Theorem 1.1 and the bound from Corollary 3 then allows us to to assume and . The latter assumption can be used to obtain two estimates on products of successive minima of and (Lemma 2).
The proof of Theorem 3.4 is then carried out by induction on , starting with the case , in which we cover by hyperplanes. This initial step is treated essentially in the same way as in  and it is derived using the pigeonhole principle and results of Mahler  and Banaszczyk . In the resulting covering of by hyperplanes, the intersection of with a hyperplane from induces a lattice of lower dimension. We can thus apply the induction hypothesis on for each hyperplane . Using Minkowski’s second theorem and Lemma 2, we can show that the larger the norm of the normal vector of is, the sparser is (Corollary 4). Then we partition the hyperplanes from according to the lengths of their normal vectors and we sum the sizes of the coverings of by -dimensional subspaces for each . Combining Corollary 4, Theorem 3.2, and the bounds from Lemma 2, we finally show that the total sum is bounded from above by .
Let and be integers with . There is a positive integer and a projection of along vectors of onto a -dimensional linear subspace of such that is mapped to and such that for every .
If , then we set to be the identity on and . Thus we assume .
For , we set . For and a lattice , we show that there is a projection of along a vector onto a -dimensional linear subspace of such that is mapped to by and such that
for every . We let and, for every , we use the above-defined projection for and define . The statement of the lemma is then obtained by setting .
Let be a basis of such that for every . Such basis exists by the First finiteness theorem (Theorem 3.3). In particular,
Let and let be the linear subspace generated by . Let be the set . Note that is a -dimensional lattice with the basis .
We consider the projection onto along . That is, every is mapped to , where , , is the expression of with respect to the basis .
We show that for every . We have , since is a basis of and is a basis of . Let , , be the expression of with respect to and let be the Euclidean distance between and .