Covering Lattice Points by Subspaces and Counting Point–Hyperplane Incidences

Let d and k be integers with 1≤k≤d-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le k \le d-1$$\end{document}. Let Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in Λ∩K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \cap K$$\end{document}. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n×⋯×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times \cdots \times n$$\end{document} grid is at least Ω(nd(d-k)/(d-1)-ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \bigl (n^{d(d-k)/(d-1)-\varepsilon }\bigr )$$\end{document} and at most O(nd(d-k)/(d-1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\bigl (n^{d(d-k)/(d-1)}\bigr )$$\end{document}, where ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover Λ∩K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \cap K$$\end{document}. We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 3$$\end{document} and ε∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,1)$$\end{document}, we show that there is an integer r=r(d,ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=r(d,\varepsilon )$$\end{document} such that for all positive integers n, m the following statement is true. There is a set of n points in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} and an arrangement of m hyperplanes in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} with no Kr,r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{r,r}$$\end{document} in their incidence graph and with at least Ω((mn)1-(2d+3)/((d+2)(d+3))-ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \bigl ((mn)^{1-(2d+3)/((d+2)(d+3)) - \varepsilon }\bigr )$$\end{document} incidences if d is odd and Ω((mn)1-(2d2+d-2)/((d+2)(d2+2d-2))-ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \bigl ((mn)^{1-(2d^2+d-2)/((d+2)(d^2+2d-2)) -\varepsilon }\bigr )$$\end{document} incidences if d is even.


Introduction
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 0symmetric 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. A convex body K is symmetric about the origin 0 if K = −K . We let K d be the set of d-dimensional compact convex bodies in R d that are symmetric about the origin.

Preliminaries
For a positive integer n, we use the abbreviation [n] to denote the set {1, 2, . . . , n}. A point x of a lattice is called primitive if whenever its multiple λ · x is a lattice point, then λ is an integer. For K ∈ K d , let vol(K ) be the d-dimensional Lebesgue measure of K . We say that vol(K ) is the volume of K. The closed d-dimensional ball with the radius r ∈ R, r ≥ 0, centered at the origin is denoted by B d (r ). If r = 1, we simply write B d instead of B d (1). For x ∈ R d , we use x to denote the Euclidean norm of x.
Let X be a subset of R d . We use aff(X ) and lin(X ) to denote the affine hull of X and the linear hull of X, respectively. The dimension of the affine hull of X is denoted by dim(X ).
For functions f, g : N → N, we write f (n) ≤ O(g(n)) if there is a fixed constant c 1 such that f (n) ≤ c 1 · g(n) for all n ∈ N. We write f (n) ≥ (g(n)) if there is a fixed constant c 2 > 0 such that f (n) ≥ c 2 · g(n) for all n ∈ N. If the constants c 1 and c 2 depend on some parameters a 1 , . . . , a t , then we emphasize this by writing f (n) ≤ O a 1 ,...,a t (g(n)) and f (n) ≥ a 1 ,...,a t (g(n)), respectively. If f (n) ≤ O a 1 ,...,a t (n) and f (n) ≥ a 1 ,...,a t (n), then we write f (n) = a 1 ,...,a t (n).

Covering Lattice Points by Subspaces
We say that a collection S of subsets in R d covers a set of points P from R d if every point from P lies in some set from S.
Let d, k, n, and r be positive integers that satisfy 1 ≤ k ≤ d −1. We let a(d, k, n, r ) be the maximum size of a set S ⊆ Z d ∩ B d (n) such that every k-dimensional affine subspace of R d contains at most r − 1 points of S. Similarly, we let l(d, k, n, r ) be the maximum size of a set S ⊆ Z d ∩ B d (n) such that every k-dimensional linear subspace of R d contains at most r −1 points of S. We also let g(d, k, n) be the minimum number of k-dimensional linear subspaces of R d necessary to cover Z d ∩ B d (n).
In this paper, we study the functions a(d, k, n, r ), l(d, k, n, r ), and g(d, k, n) and their generalizations to arbitrary lattices from L d and bodies from K d . 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 g(d, k, n) (Theorem 2.1), lower bounds on l(d, k, n, r ) (Theorem 2.2), and we use the estimates for a(d, k, n, r ) and l(d, k, n, r ) to obtain improved lower bounds for the maximum number of point-hyperplane incidences (Theorem 2.6). 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 a(d, k, n, r ) is essentially solved. In general, the set Z d ∩ B d (n) can be covered by (2n + 1) d−k affine k-dimensional subspaces and thus we have an upper bound a(d, k, n, r ) ≤ (r − 1)(2n + 1) d−k . This trivial upper bound is asymptotically almost tight for all fixed d, k, and some r , as Brass and Knauer [5] showed with a probabilistic argument that for every ε > 0 there is an r = r (d, ε, k) ∈ N such that for each positive integer n we have a(d, k, n, r ) ≥ d,ε,k n d−k−ε . (1) For fixed d and r , the upper bound is known to be asymptotically tight in the cases k = 1 and k = d − 1. This is shown by considering points on the modular moment surface for k = 1 and the modular moment curve for k = d − 1; see [5]. Covering lattice points by linear subspaces seems to be more difficult than covering by affine subspaces. From the definitions we immediately get l(d, k, n, r ) ≤ (r − 1) g(d, k, n). In the case k = d − 1 and d fixed, Bárány et al. [4] obtained the following asymptotically tight estimates for the functions l(d, d − 1, n, d) and g(d, d − 1, n): and In fact, Bárány et al. [4] proved stronger results that estimate the minimum number of (d − 1)-dimensional linear subspaces necessary to cover the set ∩ K in terms of so-called successive minima of a given lattice ∈ L d and a body K ∈ K d . For a lattice ∈ L d , a body K ∈ K d , and i ∈ [d], we let λ i ( , K ) be the ith successive minimum of and K . That is, λ i ( , K ) := inf{λ ∈ R : dim( ∩(λ·K )) ≥ i}. Since K is compact, it is easy to see that the successive minima are achieved. That is, there are linearly independent vectors v 1 , . . . , v d from such that v i ∈ λ i ( , K ) · K for every i ∈ [d]. Also note that we have λ 1 ( , K ) ≤ · · · ≤ λ d ( , K ) and λ 1 (Z d , B d (n)) = · · · = λ d (Z d , B d (n)) = 1/n. Theorem 1.1 [4] For an integer d ≥ 2, a lattice ∈ L d , and a body K ∈ K d , we let We note that the assumption λ d ≤ 1 is necessary; see the discussion in [4]. Not much is known for linear subspaces of lower dimension. We trivially have l(d, k, n, r ) ≥ a(d, k, n, r ) for all d, k, n, r with 1 ≤ k ≤ d −1. Thus l(d, k, n, r ) ≥ d,ε,k n d−k−ε for some r = r (d, ε, k) by (1). Brass and Knauer [5] conjectured that l(d, k, n, k+1) = d,k n d(d−k)/(d−1) for d fixed. This conjecture was refuted by Lefmann [14] who showed that, for all d and k with 1 ≤ k ≤ d − 1, there is an absolute constant c such that we have l(d, k, n, k + 1) ≤ c · n d/ k/2 for every positive integer n. This bound is asymptotically smaller in n than the growth rate conjectured by Brass and Knauer for sufficiently large d and almost all values of k with 1 ≤ k ≤ d − 1.
Covering lattice points by linear subspaces is also mentioned in the book by Brass, Moser, and Pach [6], where the authors pose the following problem. Problem 1.2 [6, Problem 6 in Chapter 10.2] What is the minimum number of kdimensional linear subspaces necessary to cover the d-dimensional n × · · · × n lattice cube?

Point-Hyperplane Incidences
As we will see later, the problem of determining a(d, k, n, r ) and l(d, n, k, r ) is related to a problem of bounding the maximum number of point-hyperplane incidences. For an integer d ≥ 2, let P be a set of n points in R d and let H be an arrangement of m hyperplanes in R d . An incidence between P and H is a pair ( p, H ) such that p ∈ P, H ∈ H, and p ∈ H . The number of incidences between P and H is denoted by I(P, H).
We are interested in the maximum number of incidences between P and H. In the plane, the famous Szemerédi-Trotter theorem [22] says that the maximum number of incidences between a set of n points in R 2 and an arrangement of m lines in R 2 is at most O (mn) 2/3 + m + n . This is known to be asymptotically tight, as a matching lower bound was found earlier by Erdős [8]. The current best known bounds are ≈ 1.27(mn) 2/3 + m + n [18] 1 and ≈ 2.44(mn) 2/3 + m + n [1].
For d ≥ 3, it is easy to see that there is a set P of n points in R d and an arrangement H of m hyperplanes in R d for which the number of incidences is maximum possible, that is I(P, H) = mn. It suffices to consider the case where all points from P lie in an affine subspace that is contained in every hyperplane from H. In order to avoid this degenerate case, we forbid large complete bipartite graphs in the incidence graph of P and H, which is denoted by G (P, H). This is the bipartite graph on the vertex set P ∪ H and with edges { p, H } where ( p, H ) is an incidence between P and H.
With this restriction, bounding I(P, H) becomes more difficult and no tight bounds are known for d ≥ 3. It follows from the works of Chazelle [7], Brass and Knauer [5], and Apfelbaum and Sharir [2] that the number of incidences between any set P of n points in R d and any arrangement H of m hyperplanes in R d with K r,r G(P, H) 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 2.6. The best general lower bound for I(P, H) is due to a construction of Brass and Knauer [5], which gives the following estimate. Theorem 1.3 [5] Let d ≥ 3 be an integer. Then for every ε > 0 there is a positive integer r = r (d, ε) such that for all positive integers n and m there is a set P of n points in R d and an arrangement H of m hyperplanes in R d such that K r,r G(P, H) and For d ≥ 4, this lower bound has been recently improved by Sheffer [20] in a certain non-diagonal case. Sheffer constructed a set P of n points in R d , d ≥ 4, and an arrangement H of m = n (3−3ε)/(d+1) hyperplanes in R d such that K (d−1)/ε,2 G(P, H) and I(P, H) ≥ (mn) 1−2/(d+4)−ε .
Thus it seems that the conjectured growth rate of l(d, k, n, r ) is true if we allow r to be (significantly) larger than k + 1.
We study these problems in a more general setting where we are given an arbitrary lattice from L d and a body K from K d . Similarly to Theorem 1.1 by Bárány et al. [4], our bounds are expressed in terms of the successive minima λ i ( , K ), i ∈ [d].

Covering Lattice Points by Linear Subspaces
First, we prove a new upper bound on the minimum number of k-dimensional linear subspaces that are necessary to cover points in the intersection of a given lattice with a body from K d .

Theorem 2.1 For integers d and k with
We also prove the following lower bound.

Theorem 2.2 For integers d and k with
, then, for every ε ∈ (0, 1), there is a positive integer r = r (d, ε, k) and a set S ⊆ ∩ K of size at such that every k-dimensional linear subspace of R d contains at most r − 1 points from S.
We remark that we can get rid of the ε in the exponent if k = 1 or k = d − 1; for details, see There are examples that show that α cannot be replaced by β in Theorem 2.1. It suffices to consider d = 3, k = 1, and let be the lattice {(x 1 /n, x 2 /2, x 3 /2) ∈ R 3 : x 1 , x 2 , x 3 ∈ Z} for some large positive integer n. Then λ 1 ( , However, it is not difficult to see that we need at least (n) 1-dimensional linear subspaces to cover ∩ B 3 , which is asymptotically larger than β 2 = O(1). On the other hand, α = (λ 1 λ 2 λ 3 ) −1/2 and O(α 2 ) = O(n) 1-dimensional linear subspaces suffice to cover ∩ B 3 . We thus suspect that the lower bound can be improved.
Since λ i (Z d , B d (n)) = 1/n for every i ∈ [d], we can apply Theorem 2.2 with = Z d and K = B d (n) and obtain the following lower bound on l(d, k, n, r ).

Corollary 2.3 Let d and k be integers with
Then, for every ε ∈ (0, 1), there is an r = r (d, ε, k) ∈ N such that for every n ∈ N we have The existence of the set S from Theorem 2.2 is shown by a probabilistic argument. It would be interesting to find, at least for some value 1 < k < d − 1, some fixed r ∈ N, and arbitrarily large n ∈ N, a construction of a subset R of Z d ∩ B d (n) of size d,k n d(d−k)/(d−1) such that every k-dimensional linear subspace contains at most r − 1 points from R. Such constructions are known for k = 1 and k = d − 1; see [5,19].
Since we have l(d, k, n, r ) ≤ (r − 1)g(d, k, n) for every r ∈ N, Theorem 2.1 and Corollary 2.3 give the following almost tight estimates on g(d, k, n). This nearly settles Problem 1.2.

Covering Lattice Points by Affine Subspaces
For affine subspaces, Brass and Knauer [5] considered only the case of covering the d-dimensional n × · · · × n lattice cube by k-dimensional affine subspaces. To our knowledge, the case for general ∈ L d and K ∈ K d was not considered in the literature. We extend the results of Brass and Knauer to covering ∩ K .

Theorem 2.5 For integers d and k with
On the other hand, at least d,k ((λ k+1 · · · λ d ) −1 ) k-dimensional affine subspaces of R d are necessary to cover ∩ K .

Point-Hyperplane Incidences
As an application of Corollary 2.3, we improve the best known lower bounds on the maximum number of point-hyperplane incidences in R d for d ≥ 4. That is, we improve the bounds from Theorem 1.3. To our knowledge, this is the first improvement on the estimates for I(P, H) in the general case during the last 13 years.

Theorem 2.6
For every integer d ≥ 2 and ε ∈ (0, 1), there is an r = r (d, ε) ∈ N such that for all positive integers n and m the following statement is true. There is a set P of n points in R d and an arrangement H of m hyperplanes in R d such that K r,r G(P, H) and We can get rid of the ε in the exponent for d ≤ 3. That is, we have the bounds (mn) 2/3 for d = 2 and (mn) 7/10 for d = 3. For d = 3, our bound is the same as the bound from Theorem 1.3. For larger d, our bounds become stronger. In particular, the exponents in the lower bounds from Theorem 2.6 exceed the exponents from Theorem for d even. However, the bounds are not tight. The exponents in the known bounds for I(P, H) for small values of d are summarized in Table 1.
In the non-diagonal case, when one of n and m is significantly larger that the other, the proof of Theorem 2.6 yields the following stronger bound.

Theorem 2.7
For all integers d and k with 0 ≤ k ≤ d − 2 and for ε ∈ (0, 1), there is an r = r (d, ε, k) ∈ N such that for all positive integers n and m the following statement is true. There is a set P of n points in R d and an arrangement H of m hyperplanes in R d such that K r,r G(P, H) and For example, in the case m = n (3−3ε)/(d+1) considered by Sheffer [20], Theorem 2.7 gives a slightly better bound than I (P, H) ≥ (mn) 1−2/(d+4)−ε ) if we set, for example, k = (d − 1)/4 . 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 n points in R d and m hyperplanes in R d , how fast can we count the incidences between them? We note that the lower bounds from Theorem 2.6 also establish the best known lower bounds for the time complexity of so-called partitioning algorithms [9] for the counting version of Hopcroft's problem; see [5] 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.

Proof of Theorem 2.1
Here we show the upper bound on the minimum number of k-dimensional linear subspaces needed to cover points from a given d-dimensional lattice that are contained in a body K from K d . We first prove Theorem 2.1 in the special case K = B d (Theorem 3.4) and then we extend the result to arbitrary K ∈ K d .

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 [17] and shows a relation between vol(K ), det( ), and the successive minima of ∈ L d and K ∈ K d . Theorem 3.1 (Minkowski's second theorem [17]) Let d be a positive integer. For every ∈ L d and every K ∈ K d , we have A result similar to the first bound from Theorem 3.1 can be obtained if the volume is replaced by the point enumerator; see Henk [12].
We show the following result.
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 q be an integer from , where α 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 O d,k ((λ k · · · λ d ) −1 ) on the number of k-dimensional subspaces of R d needed to cover ∩ B d (Corollary 3.6). This bound is obtained from Theorem 3.2 and Lemma 3.5, which states that, for each s with 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 3.6 then allow us to to assume d ≥ 4 and q ≥ d − k + 2. The latter assumption can be used to obtain two estimates on products of successive minima of and B d (Lemma 3.7).
The proof of Theorem 3.4 is then carried out by induction on d −k, starting with the case d − k = 1, in which we cover ∩ B d by hyperplanes. This initial step is treated essentially in the same way as in [4] and it is derived using the pigeonhole principle and results of Mahler [15] and Banaszczyk [3]. In the resulting covering S of ∩ B d by hyperplanes, the intersection of with a hyperplane from S induces a lattice of lower dimension. We can thus apply the induction hypothesis on ( ∩ H ) ∩ B d for each hyperplane H ∈ S. Using Minkowski's second theorem and Lemma 3.7, we can show that the larger the norm of the normal vector of H is, the sparser ( ∩ H ) ∩ B d is (Corollary 3.9). Then we partition the hyperplanes from S according to the lengths of their normal vectors and we sum the sizes of the coverings of ( ∩ H ) ∩ B d by k-dimensional subspaces for each H ∈ S. Combining Corollary 3.9, Theorem 3.2, and the bounds from Lemma 3.7, we finally show that the total sum is bounded from Now, as the first step towards the proof of Theorem 3.4, we prove Corollary 3.6. To do so, we prove the following lemma which is also used later in the proof of Theorem 2.5.

Lemma 3.5 Let d and s be integers with
Proof If s = 0, then we set p to be the identity on R d and r := 1. Thus we assume s ≥ 1. For . We let 0 = and, for every j = 0, . . . , s − 1, we use the above-defined projection p j for j and define j+1 := p j ( j ) = j ∩ N j+1 . The statement of the lemma is then obtained by setting p : . Such basis exists by the First finiteness theorem (Theorem 3.3). In particular, Let v j := b 1 and let N j+1 be the linear subspace generated by be the expression of z with respect to B and let v be the Euclidean distance between b 1 and N j+1 .
From the definitions of j+1 and B, we have for every i ∈ [d − j − 1]. Using Minkowski's second theorem (Theorem 3.1) twice, the upper bound in (4), and the length of b 1 (3), we obtain Here, the first inequality follows from Theorem 3.1 for j , the second inequality is by (3) and (4), and the last inequality is obtained from Theorem 3.1 for j+1 . Since det( j ) = v · det( j+1 ), we can rewrite this expression as To derive the last inequality, we use the well-known formula The Euclidean distance between z and N j+1 equals |t 1 | · v, which is at most r j , and we see that Using this fact together with the bounds in (4), we obtain Consequently, for N := N s and r := r s , we have s = ∩ N and Proof By Lemma 3.5, there is a positive integer r = r (d, k − 1) and a projection p and such that We consider the set S := lin ({y, b 1 . Then S consists of k-dimensional linear subspaces. By Theorem 3.2, the size of S is at most where the second inequality follows from the assumption λ d ≤ 1, as then The case k = 1 of Theorem 3.4 follows from Theorem 3.2 (and also from Corollary 3.6). The case k = d − 1 was shown by Bárány et al. [4]; see Theorem 1.1. Therefore we may assume d ≥ 4. Corollary 3.6 provides the same bound as Theorem 3.4 also if q = d − k + 1, thus we assume q ≥ d − k + 2 in the rest of the proof.
We use * to denote the dual lattice of . That is, * is the set of vectors y from R d that satisfy x, y ∈ Z for every x ∈ . In the rest of the section, we use μ i to denote λ i ( * , B d ) for every i ∈ [d] and we let be the parameter from the statement of Theorem 3.4. It follows from the results of Mahler [15] and Banaszczyk [3] that holds for every i ∈ [d]. Observe that μ 1 ≥ 1 and α = d,k (μ 1 · · · μ q ) 1/(q−1) by (5) and by the assumption λ d ≤ 1. We also recall that λ 1 ≤ · · · ≤ λ d and μ 1 ≤ · · · ≤ μ d .
We now prove Theorem 3.4 by induction on d − k. The case d − k = 1 is treated similarly as in the proof of Theorem 1.1 by Bárány et al. [4]. Let w 1 , . . . , w d be linearly independent vectors from R d such that w i ∈ * ∩ μ i B d for every i ∈ [d]. The existence of every w i is guaranteed from the definition of μ i .
Let D := D cα . We show that for every x ∈ ∩ B d there exists z ∈ D \ {0} perpendicular to x. Let x be an arbitrary element from ∩ B d . For every y ∈ D + , Every w i is an element of μ i B d and thus the Cauchy-Schwarz inequality implies | x, w i | ≤ μ i . Using a i ≤ cα/μ i , we thus see that | x, y | ≤ q i=1 cα μ i · μ i = qcα. Since y ∈ D + ⊆ * , we have x, y ∈ Z. Therefore x, y attains at most 2qcα+1 values. Since |D + | > 2qcα+1, the pigeonhole principle implies that there exist distinct y 1 ∈ D + and y 2 ∈ D + with x, y 1 = x, y 2 . The For the inductive step, assume that d − k ≥ 2. Consider the set S of hyperplanes in R d that has been constructed in the base of the induction. For every hyperplane H ∈ S, let H be the set ∩ H . Note that H is a lattice of dimension at most d − 1. We now proceed inductively and cover each set H ∩ B d using the inductive hypothesis for H and k. Later, we show that the total number of k-dimensional subspaces used in the covering of the sets H ∩ B d , H ∈ S, is at most O d,k (α d−k ). To do so, we employ the fact that, for every z ∈ D , the larger z is, the fewer k-dimensional subspaces we need to cover H (z) ∩ B d .
Note that q > 2 according to our assumptions q ≥ d − k + 1 and k ≤ d − 2.
Proof The vector z partitions the lattice into layers (L i ) i∈Z , where L i := {x ∈ : x, z = i}. Since z is primitive, there is a basis B of * with a column z (see [21, Lem. 1 of Sect. X.4]). Then B := (B −1 ) is a basis of and thus there is a column v of B with v, z = 1. We have v ∈ L 1 and i · v ∈ L i for every i ∈ Z. Thus every layer L i satisfies L i = i · v + L 0 and, in particular, L 0 is a (d − 1)-dimensional sublattice of . The Euclidean distance between aff(L i ) and aff(L i+1 ) is 1/ z . This is because, on one hand, y := i · z/ z 2 ∈ aff(L i ), y := (i + 1) · z/ z 2 ∈ aff(L i+1 ), and y − y = 1/ z . On the other hand, for all x ∈ aff(L i ) and x ∈ aff(L i+1 ), the Cauchy-Schwarz inequality implies Since H (z) = {x ∈ : x, z = 0}, the lattice H (z) is the layer L 0 of . The affine hull of the closest layer is in the Euclidean distance 1/ z from aff( H (z) ) and it contains a vector v of such that L i = i · v + L 0 for every i ∈ Z d . Thus if B is a basis of H (z) , then B with the column v added is a basis of . The parallelotope formed by the vectors of B and v has volume det( H (z) )/ z . Thus det( H (z) ) = z det( ).
Using Minkowski's second theorem (Theorem 3.1) twice and also the fact det( H (z) ) = z det( ), we have We now show that Since H (z) ⊆ , we have λ i ≥ λ i for every i ∈ [d − q] and thus λ 1 · · · λ d−q ≥ λ 1 · · · λ d−q . For the other inequality, let w 1 , . . . , w d be linearly independent vectors from * such that w i = μ i for every i ∈ [d]. The existence of every vector w i is guaranteed by the definition of μ i . Clearly, every w i is primitive. Let L be the orthogonal complement of lin({w 1 , . . . , w q }) and let L be the (d − q)-dimensional lattice ∩ L. By iterating the proof of (6) for the vectors w 1 , . . . , w q , we obtain where the last equality follows from (5). Since z lies in D , we have z = q i=1 a i w i for some a i ∈ Z and thus L ⊆ H (z) and L ⊆ H (z) . In particular, we have which proves (7). By combining the estimates (6) and (7), we obtain Since d − k + 1 ≤ q and k < r , we have d − q + 2 ≤ r ≤ d − 1. If r = d − 1, then, using the definition of α r , d − k + 1 ≤ q, (8), and (5), we have , which settles the claim since That is z ≤ qcα, and we have From (8) and (9), we obtain From (10), we have Therefore, using the assumption q ≥ d − k + 2, we may apply part (ii) of Lemma 3.7 with i := d − r + 1 and bound the last expression from below by Since H (z) ⊆ , we have λ i ≥ λ i for every i ∈ [d − 1] and thus we can use the obtained lower bound on 1/(λ r +1 · · · λ d−1 ) and derive In particular, since q ≥ d − k + 1, the definition of α r implies where the last inequality follows from (8).
It remains to show that the exponent in the last term is at most (d − k − 1)/((q − 2)(r − k)), as then the rest follows from (5). Using our assumptions d − k + 1 ≤ q and r ≤ d − 2, we have

Corollary 3.9 If z is a point from D and q
k-dimensional linear subspaces of R d .

The rest follows from
Let r 1 and r 2 be two nonnegative real numbers such that r 1 ≤ r 2 . We use Sh(r 1 , r 2 ) to denote the set {x ∈ R d : r 1 ≤ x < r 2 }. That is, Sh(r 1 , r 2 ) is the spherical shell bounded by r 1 and r 2 . The number r 2 − r 1 is the width of Sh(r 1 , r 2 ). Note that Sh(r 1 , r 2 ) is empty if r 1 = r 2 . Observe that if r 1 ≤ · · · ≤ r m are some nonnegative real numbers, then the shells Sh(0, r 1 ), Sh(r 1 , r 2 ), . . . , Sh(r m−1 , r m ) partition the interior of B d (r m ).
For i ∈ [q − 1], we let S i be the set D ∩ Sh(μ i , μ i+1 ). Furthermore, we use S q to denote the set of points from D that are contained in the closure of the spherical shell Sh(μ q , qcα). The sets S 1 , . . . , S q then partition D , as there are no points of * \ {0} in the interior of B d (μ 1 ) from the definition of μ 1 and z ≤ qcα for every z ∈ D. We thus have |D | = |S 1 | + · · · + |S q |.
Let z be an arbitrary element from D . By Corollary 3.9, the set H (z) ∩ B d can be covered with k-dimensional linear subspaces. Since the hyperplanes H (z) with z ∈ D cover ∩ B d , the total number of k-dimensional subspaces needed to cover ∩ B d is at most z∈S i c(z). To finish the proof of Theorem 3.4, we show k (α d−k ). For i = 1, we have |S 1 | ≤ 1, as, by the definitions of μ 1 and μ 2 , the set S 1 contains only points from D that lie in lin({w 1 }) and the only primitive point satisfying these conditions is w 1 . Moreover, every z ∈ S 1 satisfies For i ∈ {2, . . . , q −1}, we further refine every set S i that is determined by a spherical shell of width larger than 1 into sets S 1 i , . . . , S r i i , each determined by a spherical shell of width in (0, 1], for some positive integer r i ≤ μ i+1 . Such refinement exists, as the Euclidean norm of every vector from S i is at most μ i+1 . Similarly, if the width of the spherical shell of S q is larger than 1, we refine S q into sets S 1 q , . . . , S r q q , each determined by a spherical shell of width in (0, 1], for some positive integer r q ≤ qcα. We set S 1 i := S i and r i := 1 if the width of S i is at most 1. For all i ∈ {2, . . . , q} and j ∈ [r i ], let l i, j be the supremum of x taken over all points x from the spherical shell that determines S j i . If the spherical shell is empty, we set l i, j := μ i+1 . We have l i, j ∈ [μ i , μ i+1 ] from the definition of S j i . Since the width of every spherical shell S j i is at most 1 and every z ∈ D satisfies z ≥ μ 1 ≥ 1, every point z from S j i also satisfies l i, j /2 ≤ z ≤ l i, j . From μ 1 ≥ 1, we also have l i, j ≥ 1.
For every i ∈ {2, . . . , q}, the sets S 1 i , . . . , S r i i partition S i and thus we have To simplify the notation, we let for every i ∈ {2, . . . , q} and j ∈ [r i ]. Then for every i ∈ {2, . . . , q}. We show that By Theorem 3.2, we have where the second inequality follows from l i, j ≤ μ i+1 ≤ · · · ≤ μ d and the last inequality from μ 1 ≤ · · · ≤ μ i ≤ l i, j .
In the first case, we assume , as otherwise, to show (12), we may take ) (or less if there are not that many points) and add c i, j−1 to σ for each one of them instead of adding at most c i, j . This will still bound σ from above, as c i, j−1 ≥ c i, j . Thus we obtain where the second inequality follows from l i, j−1 ≥ l i, j − 1 and a i − (a − 1) i ≤ 2 i a i−1 ≤ 2 q a i−1 for a ≥ 1.
In the other case, j = 1 and we apply the same argument as in the first case, so we can assume where the equality follows from l i−1,r i−1 = μ i and the inequality from l i−1,r i−1 ≥ l i,1 − 1.

Now we have
This finishes the proof of Theorem 3.4.

The General Case
Here, we finish the proof of where the first inequality follows from (13) and the second inequality is due to the choice of r . From p ≥ 2, we see that this probability is less than 1/2. The expected size of X is Since |X | ∼ Bi( p d−1 , P), the variance of |X | is p d−1 P(1 − P) < p d−k−ε and Chebyshev's inequality implies Thus there is a set R of size at least contains at most r − 1 points from R.
Let ε ∈ (0, 1) be given. To derive Theorem 2.2, we combine Lemma 4.1 with Lemma 4.2. This is a similar approach as in [4], where the authors derive a lower bound for the case k = d − 1 by combining Lemma 4.1 with a construction found by Erdős in connection with Heilbronn's triangle problem [19].
Let p be the largest prime number that satisfies the assumptions of Lemma 4.1. If such p does not exist, then the statement of the theorem is trivial. By Bertrand's postulate, we have p > (1 − λ d )β/(16d 2 ). We may assume that p ≥ p 0 , where p 0 = p 0 (d, ε, k) is the constant from Lemma 4.2, since otherwise the statement of Theorem 2.2 is trivial.
For k ≥ 2 and t := p d−k−ε /2 , let R = {v 1 , . . . , v t } ⊆ F d−1 p be the set of points from Lemma 4.2. That is, every (k − 1)-dimensional affine subspace of F d−1 p contains at most r − 1 points from R for r := k(d − k + 1)/ε . In particular, every r -tuple of points from R contains k + 1 affinely independent points over the field F p . For k = 1, we can set r := 2 and let R be the whole set F d−1 p of size t := p d−k = p d−1 . Then every r -tuple of points from R contains two affinely independent points over F p .
For i = 1, . . . , t, let u i ∈ Z d be the vector obtained from v i by adding 1 as the last coordinate. From the choice of R, every r -tuple of points from {u 1 , . . . , u t } contains k + 1 points that are linearly independent over the field F p .
By Lemma 4.1, there exist an integer 1 ≤ j i < p and a point w i ∈ Z d for every i ∈ [t] such that u i := j i u i + pw i lies in K . We have u i ≡ j i u i (mod p) for every i ∈ [t] and thus every r -tuple of vectors from S := {u 1 , . . . , u t } ⊆ Z d contains k + 1 linearly independent vectors over the field F p , and hence over R. In other words, every k-dimensional linear subspace of R d contains at most r − 1 points from S.
This completes the proof of Theorem 2.2.

Proof of Theorem 2.5
Let d and k be integers with 1 ≤ k ≤ d − 1 and let ∈ L d and K ∈ K d . We let λ i := λ i ( , K ) for every i ∈ [d] and assume that λ d ≤ 1. First, we observe that it is sufficient to prove the statement only for K = B d , as we can then strengthen the statement to an arbitrary K ∈ K d using John's lemma (Lemma 3.10) analogously as in the proof of Theorem 2.1.
First, we prove the upper bound. That is, we show that ∩ B d can be covered with O d,k ((λ k+1 · · · λ d ) −1 ) k-dimensional affine subspaces of R d . By Lemma 3.5, there is a positive integer r = r (d, k) and a projection p of R d along k vectors b 1 , . . .
For each point z of ∩ N ∩ B d (r ), we define A(z) to be the affine hull of the set For the lower bound, we prove that we need at least d,  (2)) and we observe that where s is the maximum integer j from [k] with λ j ≤ 1. Since λ i ≥ λ i /2 for every i ∈ [k], we have | ∩ L ∩ B d (2)| ≤ O d,k ((λ 1 · · · λ k ) −1 ). For every x ∈ A ∩ B d , we have x − y ≤ x + y ≤ 2 and thus x − y ∈ L ∩ B d (2). It follows that ( ∩ A ∩ B d ) − y ⊆ ∩ L ∩ B d (2) and thus | ∩ A ∩ B d | ≤ O d,k ((λ 1 · · · λ k ) −1 ).
Let A be a collection of k-dimensional affine subspaces of R d that covers ∩ B d . We have |A| ≥ | ∩ B d |/m, where m is the maximum of | ∩ A ∩ B d | taken over all subspaces A from A. We know that m ≤ O d,k ((λ 1 · · · λ k ) −1 ). It is a well-known fact that follows from Minkowski's second theorem (Theorem 3.1) that | ∩ B d | ≥ d,k ((λ 1 · · · λ d ) −1 ). Thus we obtain which finishes the proof of Theorem 2.5.

Proofs of Theorems 2.and 2.7
Assume that we are given integers d and k with 0 ≤ k ≤ d − 2 and let ε be a real number in (0, 1). Let δ = δ(d, ε, k) ∈ (0, 1) be a sufficiently small constant. By (1), there is a positive integer r 1 = r 1 (d, δ, k) and a constant c 1 = c 1 (d, δ, k) such that for every s ∈ N there is a subset P of Z d ∩ B d (s) of size c 1 · s d−k−δ such that every k-dimensional affine subspace of R d contains at most r 1 − 1 points from P. In the case k = 0, we can clearly obtain the stronger bound c 1 · s d . By Corollary 2.3, there is a positive integer r 2 = r 2 (d, δ, k) and a constant c 2 = c 2 (d, δ, k) such that for every t ∈ N there is a subset N of Z d ∩ B d (t) of size c 2 · t d(k+1−δ)/(d−1) such that every (d − k − 1)-dimensional linear subspace contains at most r 2 − 1 points from N . In particular, every 1-dimensional linear subspace contains at most r 2 − 1 points from N and thus there is a set N ⊆ N of size |N | = |N |/(r 2 − 1) = c 2 · t d(k+1−δ)/(d−1) /(r 2 − 1) containing only primitive vectors. We note that for k = 0 we can apply Theorem 1.1 instead of Corollary 2.3 and obtain the stronger bound |N | = c 2 · t d/(d−1) /(r 2 − 1). We let H be the set of hyperplanes in R d with normal vectors from N such that every hyperplane from H contains at least one point of P.
We show that the graph G(P, H) does not contain K r 1 ,r 2 . If there is an r 2 -tuple of hyperplanes from H with a nonempty intersection, then these hyperplanes have distinct normal vectors that span a linear subspace of dimension at least d − k by the choice of N . The intersection of these hyperplanes is thus an affine subspace of dimension at most k. From the definition of P, it contains at most r 1 −1 points from P.
Then we have |P| = n. For every p ∈ P and z ∈ N , we have p, z ∈ Z and | p, z | ≤ p z ≤ st by the Cauchy-Schwarz inequality. Thus every point z from N is a normal vector to at most 2st + 1 ≤ 3st hyperplanes from H. It follows that From the definition of H, the number of incidences between P and H is at least where the last inequality holds for δ sufficiently small with respect to d, ε, and k. This finishes the proof of Theorem 2.7.
This completes the proof of Theorem 2.6. For d ≤ 3, we have k = 0 and thus we can get rid of the ε in the exponent by applying the stronger bounds on m and n.
Remark An upper bound similar to (2) holds in a much more general setting, where we bound the maximum number of edges in K r,r -free semi-algebraic bipartite graphs G = (P ∪ Q, E) in (R d , R d ) with bounded description complexity t (see [10] for definitions). Fox et al. [10] showed that the maximum number of edges in such graphs with |P| = n and |Q| = m is at most O d,ε,r,t (mn) 1−1/(d+1)+ε + m + n for an arbitrarily small constant ε > 0. Theorem 2.6 provides the best known lower bound for this problem, as every incidence graph G(P, H) of P and H in R d is a semialgebraic graph in (R d , R d ) with bounded description complexity.