Critical loci of convex domains in the plane
aa r X i v : . [ m a t h . M G ] J a n Critical loci of convex domains in the plane
Dmitry Kleinbock, Anurag Rao, Srinivasan SathiamurthyJanuary 13, 2021
Abstract
Let K be a bounded convex domain in R symmetric about the origin. The criticallocus of K is defined to be the (non-empty compact) set of lattices Λ in R of smallestpossible covolume such that Λ ∩ K = { } . These are classical objects in geometryof numbers; yet all previously known examples of critical loci were either finite setsor finite unions of closed curves. In this paper we give a new construction which, inparticular, furnishes examples of domains having critical locus of arbitrary Hausdorffdimension between and . Counting lattice points in convex symmetric domains is a classical problem, which datesback to Gauss and belongs to geometry of numbers , the branch of number theory thatstudies number-theoretical problems by the use of geometric methods. Geometry of num-bers in its proper sense was pioneered by Minkowski, see [Mi] or the books [Ca, GL] for acomprehensive introduction to the subject.A lattice Λ in R n is the set of all integer linear combinations of n linearly independentvectors v , . . . , v n ∈ R n ; in other words Λ = Z v + · · · + Z v n = g Z n , (1.1)where g ∈ GL n ( Z ) is the matrix with column vectors v , . . . , v n ∈ R n . The most naturalexample to consider is the grid of points Z n ⊂ R n , which is generated by the standardbasis of R n and corresponds to g = I n , the n × n identity matrix. Note that g Z n = Z n ifand only if g ∈ GL n ( Z ) , where the latter stands for the group of invertible n × n matriceswith integer entries. Consequently, the space X n of lattices in R n is isomorphic, as a GL n ( R ) -space, to the quotient space GL n ( R ) / GL n ( Z ) .For any Λ ∈ X n of the form (1.1) we define its covolume d (Λ) as the volume of theparallelepiped spanned by v , . . . , v n . Clearly it does not depend on the choice of thegenerating set and is equal to the absolute value of the determinant of g .Let now K be a bounded convex domain in R n symmetric about the origin; denote by V ( K ) the volume of K . Definition 1.1.
A lattice Λ ⊂ R n is called K - admissible if Λ ∩ K = { } . To K as abovewe associate a real number ∆( K ) , called the critical determinant of K , given by ∆( K ) := inf { d (Λ) : Λ is K -admissible } (1.2)Perhaps one the most fundamental results in geometry of numbers is Minkowski’sConvex Body Theorem, see e.g. [Ca, §III.2.2], which states that for K as above, any latticein R n with covolume less than V ( K ) / n must contain a point of K distinct from . In1ther words, for any such K the critical determinant ∆( K ) is positive, and, moreover, onehas ∆( K ) ≥ V ( K )2 n . This motivates the problem of exhibiting K -admissible lattices with the smallest co-volume; those are called K -critical. Definition 1.2.
A lattice Λ in R n is called K - critical if it is a K -admissible lattice with d (Λ) = ∆( K ) . The set of K -critical lattices is denoted by L ( K ) and is called the criticallocus of K .Since admissibility is preserved by taking limits, this set is non-empty by a sequentialcompactness argument due to Mahler, which can be found e.g. in [Ca, §V.4.2]. The criticallocus L ( K ) can be thought of as a subset of X n , and we will endow it with the topologyinduced from GL n ( R ) / GL n ( Z ) . With this topology it is compact, again by Mahler’sargument.It is worthwhile to point out that K -critical lattices are in one-to-one correspondencewith the densest lattice packings of R n by translates of K . Indeed, suppose that Λ is K -admissible. Then it is easy to see that the collection of sets { v + K : v ∈ Λ } is pairwise disjoint. And, conversely, if such a collection of sets is pairwise disjoint for some Λ , then Λ must be K -admissible. Minimizing the covolume of Λ over all admissible latticesthus corresponds to maximizing the relative area covered by the collection of the abovesets. Another motivation for studying critical loci of convex symmetric domains comesfrom Diophantine approximation, see §4 for more detail.From now on let us restrict our attention to the case n = 2 . Critical loci of planardomains have been systematically studied by Mahler in a series of papers written in the1940s, see Section 2 for a list of examples. Perhaps the most straightforward is the casewhen K = D := { ( x, y ) : x + y < } is the unit disc: then one has ∆( D ) = √ / , and L ( D ) = (cid:26) k (cid:20) / √ / (cid:21) Z : k ∈ SO(2) (cid:27) , (1.3)which is homeomorphic to S .However all examples of sets L ( K ) previously constructed by Maher and others wereeither finite sets or finite unions of closed curves, and there does not exist a precise de-scription of compact subsets of X which can arise as critical loci of convex symmetricbounded domains of R . (The situation for n > is even further from being understood.)For instance, after studying Examples 2.2–2.5 below, one may pose the following naturalquestion: Is it possible for some K to have critical locus homeomorphic to a Cantor set? We give a construction that shows that this is indeed possible. In particular, we prove
Theorem 1.3.
Each non-empty closed subset of L ( D ) , where D ⊂ R is the unit disc, isthe critical locus of some convex symmetric domain K ⊃ D .
2n particular, one can choose a closed subset of L ( D ) homeomorphic to the middle-thirdCantor set, or to any other closed fractal subset of [0 , .More generally, instead of D we can take any irreducible strictly convex symmetricdomain. The concept of irreducibility, introduced by Mahler in [Ma1], is as follows: Definition 1.4.
A convex symmetric bounded domain K in R is said to be irreducible if each convex symmetric bounded H $ K has ∆( H ) < ∆( K ) . K is said to be reducible if it is not irreducible, that is, if there exists H $ K with ∆( H ) = ∆( K ) .Here and hereafter H, K will always denote convex symmetric bounded domains in R . We remark that Mahler actually defines a stronger notion of irreducibility; namely,according to his definition K is irreducible if all bounded symmetric star domains S $ K have ∆( S ) < ∆( K ) . However for convex domains K it is equivalent to the one given inDefinition 1.4; this follows from [Ma2, Theorem 1], see a footnote to Lemma 2.8 below.Among the examples in Section 2 below, the square [ − . , the disc D and their imagesunder linear transformations are irreducible. A far more interesting example of an irre-ducible domain is the curvilinear octagon described in [Ma3, Section 12], see Remark 3.3.Mahler studied irreducible domains and proved a number of fundamental results illus-trating their importance. For example, the theorem stated below shows that irreducibledomains are ubiquitous in the following sense: Theorem 1.5 ([Ma2], Theorem ) . Every K ⊂ R contains an irreducible H with ∆( H ) = ∆( K ) . In Section 2, capitalizing on this and other results of Mahler, we prove
Theorem 1.6. If K is irreducible but not a parallelogram, then there is a continuous map φ : [0 , → X which descends to a homeomorphism S → L ( K ) . In the case of a parallelogram, as made explicit in Example 2.3, the critical locus istopologically the wedge of two circles.
Corollary 1.7. If K is not a parallelogram, L ( K ) is a closed subset of an embedded circle L ( H ) , the critical locus of some irreducible non-parallelogram H . This simple corollary of Theorems 1.5 and 1.6 is proved in Section 2, and our maintheorem is a partial converse to it:
Theorem 1.8. If H ⊂ R is strictly convex and irreducible, so that L ( H ) is an embeddedcircle inside X , then each non-empty closed subset of L ( H ) is the critical locus of somedomain K ⊃ H . In view of the irreducibility of D , Theorem 1.3 is clearly a special case of Theorem1.8. Note that the above theorem does not hold for parallelograms. Indeed, if H isa parallelogram, one can consider any subset of L ( H ) homeomorphic to the transverseintersection of two line segments; clearly it could never be embedded in a circle, andtherefore, as Corollary 1.7 would imply, could not be a critical locus of any K . At theend of Section 3 we discuss the further possibility of the strict convexity assumption beingremoved for non-parallelogram irreducible domains.As for the rest of the article, Section 2 gives the basic theorems on irreducible domainsand critical lattices and examples to motivate the study, and Section 3 contains the proofof Theorem 1.8. The strict convexity of H means that the line joining two distinct points on ∂H cannot intersect ∂H at any other points. Preliminaries and examples
The following fundamental and intuitive theorem on admissible and critical lattices is takenfrom [Ca, §V.8.3] where one may also find a formal proof. It shows that critical latticesare realised by inscribed parallelograms in the domains.
Theorem 2.1.
Let Λ be K -critical, and let C be the boundary of K . Then one can findthree pairs of points ± p , ± p , ± p of the lattice on C . Moreover these three points can bechosen such that p + p = p (2.1) and any two vectors among p , p , p form a basis of Λ . Conversely, if p , p , p satisfying (2.1) are on C , then the lattice generated by p and p is K -admissible. Furthermore no additional (excluding the six above) point p of Λ ison C unless K is a parallelogram. In light of this theorem, we may discuss the critical loci for the following examples ofdomains K ⊂ R . Example 2.2.
As was mentioned in the introduction, when K = D , the unit disc, onehas that ∆( D ) = √ / , and the set of critical lattices is given by (1.3). pqq − p Points p and q generate a critical lattice.All other critical lattices are obtained by rotating this one. More precisely, the map φ : R → X given by φ ( t ) := (cid:20) cos t − sin t sin t cos t (cid:21) (cid:20) / √ / (cid:21) Z descends to a homeomorphism R / ( π Z ) ≃ L ( D ) .This is an example of an irreducible domain. For if H is a subset of D with ∆( H ) =∆( D ) , then all D -critical lattices are H -admissible and thus also H -critical. Then the firstpart of Theorem 2.1 shows that each D -critical lattice φ ( t ) above must contain three pairsof points on ∂H . Since H ⊂ D , those three pairs of points must coincide exactly with theset φ ( t ) ∩ ∂D , thus showing that the boundaries of H and D coincide. Example 2.3.
When K is the square with side-length and sides perpendicular to thecoordinate axes, Theorem 2.1 again shows that the critical lattices are given by (cid:26)(cid:20) t (cid:21) Z : t ∈ R (cid:27) [ (cid:26)(cid:20) t (cid:21) Z : t ∈ R (cid:27) , (2.2)4hich is topologically a wedge of two circles. In this case too, K is irreducible by the sameapplication of Theorem 2.1 as in the case of the unit disc. Shearing the standard lattice along each axis gives rise to all the critical lattices.
Example 2.4.
When K is a hexagon, there is exactly one critical lattice. This lattice isspanned by the two vectors on the midpoints of adjacent sides. Moreover, in this case onehas V ( K ) = 4∆( K ) . This follows as a corollary of Theorem 2.1, and a proof can be foundin [Ca, §V.8.4]. It is also known [Ma3, Theorem 5] that when K is a n -gon with n ≥ ,there can only be a finite number of critical lattices. pqq − p The lattice generated by p and q is the only critical lattice. Example 2.5. If K is the unit ball of the L p norm on R ( < p < ∞ , p = 2 ), then,depending on p , the critical locus comprises either of one or two lattices. For a historicalaccount and a proof of this (Minkowski’s conjecture), see the paper [GGM]. Remark 2.6. If H = gK for some g ∈ GL ( R ) , then the critical loci of H and K arerelated by L ( H ) = g L ( K ) . Also, irreducibility is preserved under transformation by g .We now collect the results needed to make Theorem 1.6 more precise. Moreover, theparameterization of the critical locus arising will be crucial to our result. Lemma 2.7 ([Ma2], Lemma ) . Suppose that K is not a parallelogram, Λ is K -critical,and let p i : i = 1 , . . . , be the points of Λ contained in C = ∂K , labelled in a counter-clockwise order. Let A i denote open segment of C between p i and p i +1 . If Λ ′ is another K -critical lattice distinct from Λ , then each A i contains exactly one point of Λ ′ . Lemma 2.8 ([Ma2], Lemma ) . Assume K is not a parallelogram and is irreducible.Then for each p ∈ C = ∂K there is exactly one critical lattice of K containing p . Even though the statement of Lemma 2.8 is taken almost verbatim from Mahler’s paper, our meaningis changed in light of the different definitions of irreducibility. Logically speaking, one can use [Ma2,Theorem 1] to show that the two definitions of irreducibility are equivalent. roof of Theorem 1.6. Fix Λ , any critical lattice for K , and let C denote the boundaryof K . Let p i denote the six points of C ∩ Λ oriented counter-clockwise. By convexity,one can parameterize the closed segment of C between p and p by a continuous map p : [0 , → R .From Lemma 2.8 above, we see that each p ( t ) belongs to a unique K -critical lattice Λ t . Let q ( t ) denote the point of Λ t ∩ C coming after p ( t ) going counter-clockwise inangle. We claim that the function q is continuous. If not, then for some t we would find aneighborhood U of q ( t ) with a converging sequence t n → t but q ( t n ) / ∈ U . Continuity of p implies p ( t n ) → p ( t ) . Without loss of generality, we can assume t n < t so that applyingLemma 2.7 to a lattice Λ containing a point in U ∩ C between q ( t ) and each q ( t n ) gives acontradiction. Thus we can set φ ( t ) = [ p ( t ) q ( t )] Z , which, in light of Lemma 2.7, descends to the required homeomorphism. Proof of Corollary 1.7.
Suppose that K is not a parallelogram. By Theorem 1.5, K con-tains an irreducible H with ∆( K ) = ∆( H ) . It follows that we have the containment L ( K ) ⊂ L ( H ) . Moreover we claim that H cannot be a parallelogram.For if this were the case, as made explicit in Example 2.3 above, ∆( H ) would have toequal V ( H )4 . However, we also have that V ( H ) < V ( K ) and that V ( K )4 ≤ ∆( K ) . Togetherthese give ∆( H ) = V ( H )4 < V ( K )4 ≤ ∆( K ) , a contradiction.Since H is not a parallelogram, Theorem 1.6 applies and we are done. Remark 2.9.
Mahler proved (cf. Theorem in [Ma3]) that for such K , the boundary C is a C submanifold of R . Thus the preceding proof can be modified to show that L ( K ) is a C submanifold of the space of lattices. The proof of our theorem will be based on the following simple observation:
Lemma 3.1. If H ⊂ K and one of the H -critical lattices is also K -admissible, then L ( K ) is exactly the set of K -admissible lattices in L ( H ) .Proof. Since H ⊂ K , we have that ∆( H ) ≤ ∆( H ) . Further, the existence of an H -criticaland K -admissible lattice shows that we have the equality ∆( H ) = ∆( K ) .First, let Λ be K -critical. It is then also H -admissible with d (Λ) = ∆( K ) = ∆( H ) .Thus it is H -critical and, by definition, also K -admissible.For the other containment, let Λ ∈ L ( H ) be K -admissible. Since ∆( H ) = ∆( K ) , Λ must in fact be K -critical. Proof of Theorem 1.8.
Take H , not a parallelogram and irreducible, and say Z ⊂ L ( H ) isnon-empty and compact. Let Λ ∈ Z . Let C denote the boundary of H , and, as before,let us label the six points of Λ ∩ C as p , . . . , p (ordered by angle). Let p ( t ) denote acontinuous parameterization of the segment C from p to p by the interval [0 , (seediagram (3.1) below), let Λ t denote the unique critical lattice containing p ( t ) , and let q ( t ) Λ t ∩ C coming after p ( t ) (going counter-clockwise). p (0) p (1) = q (0) q (1) p ( a i ) p ( b i ) An illustration when H is a disc. The union of H and two curvilinear regions. (3.1)We use the parameterization φ ( t ) := [ p ( t ) q ( t )] Z for L ( H ) . We can now assume thatour non-empty, closed set is the image under φ of some compact Q ⊂ [0 , with { , } ⊂ Q .Set [0 , r Q = G ( a i , b i ) . Now, for each i , consider p ( a i ) , p ( b i ) . In light of Remark . , we can find tangent lines L , L to C at these two points. Let T i be the curvilinear triangle bounded by the threecurves L , L and C i = { p ( t ) : a i ≤ t ≤ b i } . We define K i to be the union H ∪ T i ∪ ( − T i ) as is illustrated in Diagram (3.1) below. Note that the lines L , L are not parallel; oneargument is that − L is also a tangent line to C by symmetry and strict convexity wouldprevent L from being parallel to either of these. Moreover, strict convexity also impliesthat the curve segment C i is contained in K i .We now define K to be the union S K i . Clearly K is a bounded, open, convex,symmetric domain containing H . Moreover, K and H share the boundary points p ( t ) , q ( t ) , q ( t ) − p ( t ) for each t ∈ Q , which shows that the H -critical lattice φ (0) is K -admissible.Lemma 3.1 thus applies to show that L ( K ) is exactly the set of K -admissible latticesof L ( H ) . For any t ∈ Q , the H -critical lattice φ ( t ) is K -admissible by Theorem 2.1. Onthe other hand, for t / ∈ Q , φ ( t ) is not K -admissible since C i ⊂ K . This ends the proof. Remark 3.2.
Note the construction of above can be modified to ensure that K is C bysmoothing an edge of each curvilinear triangle. Remark 3.3.
Our method does not work for non-parallelogram H which are not strictlyconvex, and it even seems unlikely that the theorem holds in that generality. The interestedreader is encouraged to examine the following irreducible domain, explicitly described in[Ma3, Section ]. It is central to a conjecture, going back to Reinhardt [Re], concerningdomains H that have minimal packing density area( H )∆( H ) .7 he curvilinear octagon with one of its critical lattices. Reinhardt proved that such a domain exists and proposed this curvilinear octagon as theunique candidate. See [H] for a historical account along with some partial results.
Given a non-decreasing function ψ : R + → (0 , , say that a real number α is ψ -Dirichlet-improvable if the system of inequalities | αq − p | ≤ ψ ( T ) and | q | ≤ T (4.1)has a nonzero solution ( p, q ) ∈ Z for all sufficiently large T .Dirichlet’s Theorem, see[S, Theorem II.1E], asserts that the system (4.1) always has a nonzero integer solutionif ψ ( T ) = 1 /T . On the other hand, it is known [Mo, DS] that the choice ψ ( T ) = c/T with c < produces a null set of ψ -Dirichlet-improvable numbers. A precise criterion forthis set to have zero/full measure has been recently obtained by the first-named authorand Wadleigh [KWa]. It was also shown in that paper that the property of being ψ -Dirichlet-improvable can be equivalently phrased in terms of dynamics on the space X of unimodular lattices in R . Namely, take K to be the unit ball with respect to thesupremum norm k · k on R n , and let g t := (cid:20) e t e − t (cid:21) ; then α is Dirichlet-improvable if andonly if the one-parameter trajectory (cid:26) g t (cid:20) α (cid:21) Z : t ≥ (cid:27) ⊂ X (4.2)eventually stays away from a certain family (determined by ψ ) of shrinking neighborhoodsof the critical locus L ( K ) described in Example 2.3. See [KWa, Proposition 4.5] for aprecise statement, and [Da, KM, KWe] for other instances of the correspondence betweenDiophantine approximation and dynamics, usually referred to as the Dani Correspondence.Recently, in [AD, KR] a generalization of the property of being ψ -Dirichlet-improvablewas introduced with the supremum norm replaced by an arbitrary norm ν on R . Similarlyto the above, that property can be restated in terms of the trajectory (4.2) eventuallystaying away from a family of shrinking neighborhoods of the critical locus L ( K ) , where K is the unit ball with respect to ν (see [KR, Proposition 2.1]). This has been one of themotivating factors for the study of critical loci undertaken in the present paper. This collaboration was made possible by the MIT PRIMES program. The authors wouldlike to commend the dedication and enthusiasm of the organizers and thank Tanya Kho-vanova in particular for valuable comments and guidance. The second named author wassupported by NSF grants DMS-1600814 and DMS-1900560.8 eferences [AD] N. Andersen and W. Duke,
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Dmitry Kleinbock, Brandeis University, Waltham MA 02454-9110 , [email protected] Anurag Rao, Brandeis University, Waltham MA 02454-9110 , [email protected] Srinivasan Sathiamurthy, Lexington MA 02420 , [email protected]@gmail.com