The classical lattice covering problem asks for the most economical way to cover space by overlapping Euclidean balls centered at points of a lattice. To make this precise, given a lattice , normalized so that it has covolume one, define its covering density, denoted , to be the minimal volume of a closed Euclidean ball , for which Define
Similarly, let , where denotes the set of compact convex subsets of with nonempty interior. We define the -covering density of , denoted , to be the minimal volume of a dilate such that , and define
The collection of lattices of covolume one in can be identified with the quotient , via the map
This identification endows with a natural probability measure; namely, there is a unique -invariant Borel probability measure on . We will refer to this measure as the Haar-Siegel measure and denote it by . In this paper we give new bounds on and on the -typical value of .
There is so that for any and any ,
This improves on the best previous bound of which was proved by Rogers [Rog59]. We note that for the case that is the Euclidean ball, Rogers obtained [Rog59], and this was extended to certain symmetric convex bodies by Gritzmann [Gri85]. This bound is better than what we obtain here.
There are positive constants such that for any , any , and any
We remark that the constants appearing in the statement of Theorem 1.2 can be explicitly estimated.
Setting in (1.4), we see that
There is a constant such that for any sequence , the Haar-Siegel probability that tends to 1 exponentially fast, as .
This resolves a question of Strömbergsson, who showed in [Str12] that the conclusion holds with and , for an explicit number .
We introduce two quantities which describe the growth rate of the Haar-Siegel typical covering density. Let
Prior to our results it was not known whether and are finite, i.e., whether the typical behavior of the covering density is polynomial. It would be interesting to know whether our upper bound on and can be improved.
1.1. Simultaneous covering and packing
We describe another application of Theorem 1.2, improving a result of Butler [But72]. To state it, define the -packing density of , denoted , to be the maximal volume of a dilate such that the translates are disjoint. Then we have:
There is such that for all and all , there is such that
This improves a previous upper bound of proved by Butler [But72]. The proof follows by observing that (a) a dilate of volume is with high probability packing for a Haar-Siegel random (by Siegel’s theorem [Sie45]), and that (b) a dilate of volume is with high probability covering for a Haar-Siegel random (by Theorem 1.2). The union bound then shows that with high probability both events hold simultaneously, completing the proof. We leave the details to the reader.
1.2. Ingredients of the proof
Our proof of Theorem 1.2 utilizes some lower bounds on the cardinality of discrete Kakeya sets (see §2.4). Specifically, relying on a result of Kopparty, Lev, Saraf, and Sudan [KLSS11], we obtain a new lower bound on the size of a discrete -Kakeya set of rank 2, see Corollary 2.9. What is important for us is that the dependence of this bound on the parameter is linear.
We also use a variant of the Hecke correspondence to analyze the properties of a -typical lattice. Namely, we show in §2.1 that for parameters , if one draws a Haar-Siegel random lattice , and then replaces it by a lattice uniformly drawn from those containing as a sub-lattice of index , and with a prescribed quotient group , then (properly rescaled) is also Haar-Siegel random. Our construction is inspired by a similar construction which was investigated by Erez, Litsyn and Zamir [ELZ05] in the information theory literature.
We are grateful to Uri Erez, Swastik Kopparty, and Alex Samorodnitsky for useful discussions. The authors gratefully acknowledge the support of grants ISF 2919/19, ISF 1791/17, BSF 2016256, the Simons Collaboration on Algorithms and Geometry, a Simons Investigator Award, and by the National Science Foundation (NSF) under Grant No. CCF-1814524.
2.1. Space of lattices and Haar-Siegel measure
Recall from the introduction that . This space is endowed with the quotient topology and hence with the Borel -algebra arising from this topology. The group acts naturally on lattices via the linear action of matrices on , or equivalently, by left translations on the quotient . The measure is the unique Borel probability measure on which is invariant under this action. From generalities on coset spaces of Lie groups, such a measure exists and is unique, see e.g., [Rag72]. We will also consider a slightly more general space, namely for each we write for the collection of lattices of covolume in . The obvious rescaling isomorphism commutes with the -action, and thus there is a unique -invariant measure on , and we will denote it by . We will refer to any of the measures as the Haar-Siegel measure.
For a prime and an integer , associate to each lattice the finite collection of lattices in which contain as a sub-lattice, and for which the quotient is isomorphic to . Note that these lattices are of covolume . The assignment is a particular case of the so-called Hecke correspondence (see e.g., [COU01]). The following useful observation is well-known, we include a proof for completeness.
For each as above, let . Then for each , i.e., each continuous compactly supported real valued function on ,
In other words, choosing randomly according to Haar-Siegel measure on is the same as choosing randomly according to Haar-Siegel measure on , and then choosing uniformly in .
The right-hand side of (2.1) describes a positive continuous linear functional on , and hence, by the Riesz representation theorem, is equal to for some Radon measure on . Taking a monotone increasing sequence of compactly supported functions tending everywhere to 1, we see that (2.1) holds for the function , and from this it follows that is a probability measure. By the uniqueness property of Haar-Siegel measure, in order to show that it suffices to show that is invariant under left-multiplication by any . From the definition of we see that , and so the invariance of follows from the following computation:
We now interpret this in terms of the discrete Grassmannian, as follows. For a prime let denote the field with elements. For , let denote the collection of subspaces of dimension in , or equivalently, the rank- additive subgroups of . We can identify with the residues , and thus identify with the quotient . We have a natural reduction mod homomorphism , which sends each coordinate of to its class modulo . Any element gives rise to a sub-lattice , which contains as a subgroup of index , and with isomorphic as an abelian group to . Similarly, for any we have . This shows that for any lattice we have
We have shown:
Choosing according to is the same as choosing according to , then choosing uniformly and setting
We can state Proposition 2.2 in more concrete terms as follows. Choose a random lattice distributed according to , choose generators of , so that the parallelepiped
is a fundamental domain for . Define the discrete ‘net’
These are coset representatives for the inclusion . Choose elements
from the uniform distribution over linearly independent (as elements of) -tuples. Then the lattice is a random lattice distributed according to .
2.2. Some bounds of Rogers and Schmidt
We now recall some fundamental results of Rogers and Schmidt. For a lattice let be the quotient torus, let be the Haar probability measure on , and let be the quotient map. Let denote the Lebesgue measure on . For a Borel measurable subset , and a lattice , let
equivalently, is the density of points in not covered by . Also let
There is a positive constant such that for all , for every Borel measurable with
Using the Markov inequality, this immediately implies the following:
With the same notation and assumptions, for any ,
2.3. From half covering to full covering.
Here we show the standard fact (cf. [Rog59, Lemma 4] or [HLR09]) that if a convex body covers half the space, then dilating it by a factor 2 covers all of space. Notice that this translates to a factor in volume, as a result of which we will only use this lemma for very small bodies.
Let and let be a lattice in . Suppose that
Then we have
we have that for any ,
Therefore, there are so that , or equivalently, there are so that
The claim now follows from . ∎
2.4. Lower bound on the size of a discrete -Kakeya set
Now let be a power of a prime, let denote the field with elements and for a line , let denote the affine line through parallel to . A subset is called a Kakeya set if for every there is such that that is, contains a line in every direction. For , is called an -Kakeya set if
that is contains a line in at least an -proportion of directions. Extending this notion to higher dimensions, let and . Then a set is called a Kakeya set of rank if for any there is such that , and an -Kakeya set of rank if
In this subsection we will derive lower bounds on the size of an -Kakeya set of rank . Our main observation is that the possible sizes of an -Kakeya set and a -Kakeya set are related as follows.
Let . Assume is an -Kakeya set of rank , then there exists a -Kakeya set of rank with cardinality
Fix and . For , denote
Let be an element of , that is an invertible matrix with entries in . For we clearly have if and only if . Let be a finite subset of and consider
Recall that by definition of an -Kakeya set of rank , we have that . Consequently, our claim will follow once we show that if
We will prove this using a standard probabilistic argument. Define a probability space by drawing elements of , uniformly and independently. Fix as in (2.6) and for each , denote by the event that . The events are i.i.d., since the are. Therefore
Since acts transitively on ,
It therefore follows that
This implies that there exists a subset that satisfies (2.7). ∎
Let . If is a -Kakeya set of rank then
The bound in Lemma 2.7 is quite tight for large , but is loose for . We now leverage Lemma 2.6 to obtain a much sharper bound for small . It replaces the exponential (in , with fixed) dependence on with a linear dependence. We remark that the bound (2.10) will not be used in this paper, and is included for future reference.
Let . If is an -Kakeya set of rank , then
and for we also have that
We first claim that if is an -Kakeya set of rank , then for any
To see this, let , and assume for contradiction that is an -Kakeya set of rank with cardinality smaller than the right-hand side of (2.11). By Lemma 2.6, this implies that there must exist a -Kakeya set of rank with cardinality , which contradicts Lemma 2.7.
Next, we use (2.11) to show that if is an -Kakeya set of rank , it must satisfy (2.9). For , this follows immediately from Lemma 2.7. We may therefore assume without loss of generality that . Let and note that for all in this range
We will need the following consequence:
If is an -Kakeya set of rank 2 then
If satisfies then the set
For we have that
Thus (i) is an immediate consequence of (2.9). Assertion (ii) follows from (i) by setting . ∎
3. Proof of Theorem 1.2
Let be a prime number satisfying
We define a probability space as follows. Let be a random lattice chosen according to , and be randomly chosen from the uniform distribution on , independently of . Define the lattice and note that . By Proposition 2.2, we have that is distributed according to . Therefore, the left-hand side of (1.4), which we are trying to bound from above, is equal to
Let be the dilate of of volume
Applying Corollary 2.4 with we have
where . Here we used that (where is as defined in (2.4)) which holds assuming the constant is chosen small enough. From now on, we fix an for which
and we show that when choosing , with probability at least , for to be chosen below, we have .
so that . Let be as in (2.3), and let
The Haar measure on the torus satisfies that
and applying this with taken to be we find that there is such that
Recall that we have an identification of with by reducing mod and then dividing by , and a further identification of with . With these identifications in mind we view as a subset of , and define
This implies via Corollary 2.9(ii), applied with
that with probability at least over the choice of , it holds that for all , intersects . Recalling that , this equivalently says that
Together with (3.7), this implies that
To summarize, Proposition 2.2 shows that with all but probability (due to the choice of and ), we have and hence . Using our choices (3.1), (3.2) and (3.6) we see that for appropriate choices of constants we have (1.4).
Appendix A Proof of Lemma 2.7
The case is precisely [KLSS11, Theorem 1]. The general case (as in Lemma 2.7) follows from minor modifications to their proof. For the reader’s convenience, we include the full proof here, much of it taken verbatim from [KLSS11].
We start with some necessary background. Let denote the set of non-negative integers. For an -tuple , we define and if then . Any polynomial in variables over some field can be expanded in the form
for some polynomials over in variables. We refer to as the Hasse derivative of of order . It is easy to see that and that for , . Moreover, if denotes the homogeneous part of , then
For a nonzero polynomial in variables over a field , we define its multiplicity of zero at some point , denoted , as the largest such that for all with . Alternatively, it is the largest for which we can write
for some . We sometimes also say that vanishes at with multiplicity .
We will use the following relatively straightforward lemmas.
Lemma A.1 ([Dkss13, Lemma 5]).
Let be an integer. For any nonzero polynomial in variables over a field , , and , it holds that
Lemma A.2 ([Dkss13, Proposition 10]).
Let and be integers, and a field. If a finite set satisfies , then there exists a nonzero polynomial over in variables of degree at most , vanishing at every point of with multiplicity at least .
Lemma A.3 ([Klss11, Lemma 14]).
Let be integers, and a nonzero polynomial in variables over a field . Suppose that . Then for any ,
where we view as a polynomial in the formal variables .
Finally, we will need a multiplicity version of the standard Schwartz-Zippel lemma [DKSS13].
Lemma A.4 ([Dkss13, Lemma 2.7]).
Let be an integer, a nonzero polynomial in variables over a field , and a finite set. Then