Lattice deformations in the Heisenberg group
aa r X i v : . [ m a t h . N T ] A p r LATTICE DEFORMATIONS IN THE HEISENBERGGROUP
JAYADEV S. ATHREYA AND IOANNIS KONSTANTOULAS
Abstract.
The space of deformations of the integer Heisenberggroup under the action of Aut( H ( R )) is a homogeneous space for anon-reductive group. We analyze its structure as a measurable dy-namical system and obtain mean and variance estimates for Heisen-berg lattice point counting in measurable subsets of R ; in partic-ular, we obtain a random Minkowski-type theorem. Unlike theEuclidean case, we show there are necessary geometric conditionson the sets that satisfy effective variance bounds. Introduction
Minkowski’s theorem in the geometry of numbers shows that in anysufficiently large convex centrally symmetric open set in R n there arenon-zero integral points. The asymptotic count of the number of suchpoints is well understood in terms of the volume of the set, but theoptimal error term is hard to obtain and can depend sensitively on theregularity of the boundary.It is an old idea that to understand a particular instance of a compli-cated system, it is beneficial to understand its typical behavior. Fromthe classical viewpoint of the ‘metric theory of equidistribution’, Z n isone of many lattices in R n , and their average behavior is easier to graspthan the individual point counting stories each lattice has to tell. Inthe more modern conception of homogeneous dynamics, Z n is a pointin the space of unimodular lattices in R n , a finite volume homogeneousspace of SL( n, R ). Thus, we can formulate questions about the averagecount of lattice points and their variance for a given set, with respectto the Haar probability measure on this space.The first steps in that program were taken by Siegel [9] who con-sidered averaged lattice point counting over unimodular lattices andgave a mean value formula. Subsequently, Rogers [7] studied highermoments of functions on the space of lattices and obtained a variancebound in R n with n ≥ R n with unbounded finite volumes, almost all lattices have the expected number of lattice points with explicitdiscrepancy bounds. The hardest part in Schmidt’s work was the case n = 2 where most of Rogers’s identities were not applicable: there wasno variance bound in R to rely upon and he had to work ‘by hand’using classical estimates from analytic number theory and the actionof SL(2 , Z ) on pairs of integer vectors.The issue of a variance bound was also treated by Randol [6] who ob-tained Rogers-type variance estimates for primitive lattice point count-ing in disks using the spectral decomposition of L c (SL(2 , R ) / SL(2 , Z )).Using Randol’s results and a deeper analysis of the Siegel operator (de-fined in [9]) for SL(2 , R ), Athreya-Margulis obtained in [1] a varianceestimate for primitive lattice point counting allowing arbitrary Borelsets in R .The results above provide significant information on the average be-havior of Euclidean lattices. A natural subsequent question that arisesinvolves lattices in non-abelian groups. The recent work of Garg, Nevoand Taylor [3] addresses the lattice point count of Z n +1 in large cen-tered balls in certain norms homogeneous with respect to the dilationsof the (2 n + 1) − dimensional Heisenberg group.The present work provides variance bounds for random lattices in thecase of the 3-dimensional Heisenberg group. We will see that even inthis modest excursion outside the abelian world, the average behaviorof lattices is much more erratic than the Euclidean case. In particular,we show in Proposition 5.1 that very simple sets S ⊂ R have no usefulvariance bound for the number of primitive points of a random Heisen-berg lattice in S . On the other hand, we also provide a natural classof sets for which optimal variance bounds exist and discuss extensions(see eg. Corollary 4.14).Our main technique is to realize the space of Heisenberg lattices asa fiber bundle over the space of Euclidean lattices, use the action ofthe automorphism group of H ( R ) on it and relate it to the action ofSL(2 , R ) on the space of Euclidean lattices. Since the space of Heisen-berg lattices embeds into the space X of Euclidean lattices in R , ourresults can also be seen as looking at orbits in X of certain lowerdimensional subgroups of SL(3 , R ).1.1. Heisenberg group.
The real Heisenberg group H ( R ) has R asthe underlying manifold with the smooth addition law (see Section 2for details) rst + r ′ s ′ t ′ = r + r ′ s + r ′ t + t ′ + rs ′ − sr ′ . ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 3
With this law Z becomes a discrete subgroup denoted by H ( Z ) whichwill be taken to have co-volume 1 with respect to the Lebesgue (Haar)measure on R . Let X H be the orbit of Z under the action of theconnected component of the identity of the automorphism group of H ( R ) preserving volume Aut +1 ( H ( R )) (this is a connected Lie groupwhose structure we describe in Section 2). This set has the structureof a finite volume homogeneous space with the projected Haar measurefrom Aut +1 ( H ( R )).Our work involves the following class of sets: consider a Borel set A ⊂ R of finite measure greater than 1. The epsilon-plate over A atlevel z is the set A zǫ := ( z, ,
0) + A × [0 , ǫ ) ⊂ R . A point l = ( m, n, k ) in a Heisenberg lattice g Z is called primitive ifgcd( m, n ) = 1 (this is the correct analogue of primitivity in H ( Z ) aswe shall see in Section 2).The main result provides the following average deviation bound forprimitive lattice point count. Theorem 1.1 (See Corollary 4.13) . Let µ H be the projected Haar mea-sure on the space of Heisenberg lattices X H that are deformations ofthe standard lattice H ( Z ) and m the Lebesgue measure in R . Suppose < ǫ < . We have (1.1) µ H (cid:18) Λ ∈ X H : (cid:12)(cid:12)(cid:12)(cid:12) A zǫ ∩ Λ prim ) − m ( A zǫ ) ζ (2) (cid:12)(cid:12)(cid:12)(cid:12) > r p m ( A zǫ ) (cid:19) ≤ Cr . where C is an absolute constant. Combining these results with standard analytic manipulations we getvariance bounds for sets built up from a moderate number of plates.However, there is a gap between the sets for which we get bounds andthose for which we prove there is no such bound. This reflects thelimitations of our knowledge of Euclidean lattice point distribution in R . Despite this, regarding features of Heisenberg lattices that comegenuinely from the action of the Heisenberg group, the results of sec-tions 3 and 4 provide a comprehensive picture and one that generalizesto higher dimensional Heisenberg groups.In subsequent work we plan to treat those higher dimensional groupscombining an analysis of lattice point distribution of symplectic Eu-clidean lattices in R n and the semi-direct product structure of thecorresponding Aut( H ). We hope that the present paper will also serveas an accessible introduction to the more technical results to follow.In the next section we provide all the relevant definitions and illus-trate the differences between Euclidean and Heisenberg lattices. JAYADEV S. ATHREYA AND IOANNIS KONSTANTOULAS The space of Heisenberg lattices
In this expository section we describe the structure of the space ofHeisenberg lattices.
Definition 2.1.
The three dimensional real Heisenberg group H ( R )is defined to be the group with underlying set R (written as columnvectors) and addition law rst + r ′ s ′ t ′ = r + r ′ s + r ′ t + t ′ + rs ′ − sr ′ . The integer Heisenberg group H ( Z ) is the discrete subgroup H ( R ) ∩ Z . Remark . The standard symplectic form appears in the additionformula for H ( R ). Any other bilinear form would give rise to a Heisen-berg group whose structure would be determined by the antisymmetricpart of the form. See [2] for the reduction and more general Heisenberggroups. Proposition 2.3.
Lebesgue measure in R is a Haar measure for H ( R ) .The group H ( Z ) is a lattice in H ( R ) with a fundamental domain [0 , .The group of automorphisms of H ( R ) that preserve volume and orien-tation is the group Aut +1 ( R ) =: Aut consisting of matrices of the form a b xc d y ≃ SL (2 , R ) ⋉ R where g := (cid:18) a bc d (cid:19) ∈ SL (2 , R ) and ~v := (cid:18) xy (cid:19) ∈ R . We abbreviate these elements by (cid:18) g ~v (cid:19) . The group Aut +1 ( R ) acts on H ( R ) by (2.1) (cid:18) g ~v (cid:19) · rst = g ∗ (cid:18) rs (cid:19) t − ~v t · g ∗ · (cid:18) rs (cid:19) ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 5 where g ∗ = ( g − ) t is the inverse transpose. In terms of matrix multi-plication, the action is (cid:18) g ~v (cid:19) ∗ rst . Proof.
All the assertions can be found in [2, I.2 - I.3]. The actionthere is of the transpose of the group we have by simple matrix-columnmultiplication. Our group acts by taking inverse transpose (landingus in the group used by Auslander) and performing matrix-columnmultiplication. (cid:3)
Remark . The choice of group representation for Aut +1 may seemodd; we made this choice because we need to take quotients on the rightand it is easier to see the fiber bundle structure of the automorphismgroup from the upper semidirect product when we take right quotients.Whenever no confusion can arise, we freely switch to the lower one andits matrix action on column vectors rather than passing through theinverse transpose.We turn to our main object of study: Definition 2.5.
Normalize the Haar (Lebesgue) measure m of H ( R )so that the co-volume of H ( Z ) is 1. The space of H ( Z )-deformationsis defined to be the orbit Aut +1 ( R ) · H ( Z ). It can be identified with thequotient X H = Aut +1 ( R ) / Aut +1 ( Z )where the last group is the group of Z -points of the automorphismgroup (this is the stabilizer of H ( Z )).This is essentially the space of all Heisenberg lattices of co-volume 1.A full description of all lattices in H ( R ) is given in [2, I.2]. Note thatnot all lattices in H ( R ) are isomorphic as groups to H ( Z ); those thatare, are isomorphic through an ambient automorphism of H ( R ). Thiscrucial rigidity result, among other important facts about nilpotentgroups, can be found in [5].The next proposition shows how X H is related to the space of uni-modular lattices in R denoted by X . Proposition 2.6.
The space X H has an equivariant fiber bundle struc-ture over X with compact fiber over g SL (2 , Z ) equal to R /g Z . JAYADEV S. ATHREYA AND IOANNIS KONSTANTOULAS
Proof.
Letting an arbitrary g H = (cid:18) g x (cid:19) act on (cid:18) γ δ (cid:19) with γ ∈ SL(2 , Z ), δ ∈ Z , we get (cid:18) g x (cid:19) (cid:18) γ δ (cid:19) = (cid:18) gγ gδ + x (cid:19) . Projection onto the first factor gives the base point modulo SL(2 , Z )and the pullback from that point ranges over x + g Z . Equivariancefollows from the same computation and after choosing a local trivial-ization at the identity coset of X others follow by the homogeneousstructure of X H . The transition maps are given by the correspondingtoral isomorphisms that take R /g Z to R /g ′ Z . (cid:3) Corollary 2.7.
Let µ H be the unique probability measure on X H in-variant under Aut. Then dµ H ( g H ) = dµ E ( g ) × dµ g ( x ) where dµ E ( g ) is the projection of Haar measure on the space of Euclidean lattices X that gives measure to SL (2 , Z ) and dµ g ( x ) the probability Haarmeasure on the toral fiber.Proof. The product measure is translation invariant by the action ofAut and since the base points are in SL(2 , R ), all fibers must have thesame volume. Fubini’s theorem then gives the result. (cid:3) Remark . Note how X H ֒ → X as topological spaces. This inclusionidentifies X H with a subset of Euclidean lattices, with the followingcaveat: when we consider the two spaces not simply as topologicalspaces but as orbit spaces with a common action on R , we need tomodify the inclusion as X H ֒ → X ∗ where the last space is given by theinverse transpose automorphism of SL(3 , R ).3. Lattice points in measurable sets
Let A ⊂ H ( R ) be measurable. Pick a Heisenberg lattice L at randomfrom X H using the Haar measure. What can we say about A ∩ L ? Itturns out that the answer has two parts: one involves the projection of L in the R plane orthogonal to the center of H ( R ) and the other thevertical distribution over each lattice point in the projection. Thereare no ‘slanted lines’ in the central direction for Heisenberg lattices.The corresponding question for Euclidean lattices has a very elegantanswer that we will use extensively. The following result is one of themain points of [1]: ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 7
Theorem 3.1 ([1, Theorem 2.2]) . Let n ≥ . There exists C n > such that if A ⊂ R n has m ( A ) > , (3.1) µ E (Λ ∈ X n : A ∩ Λ = ∅ ) ≤ C n m ( A ) . Here µ E is the probability Haar measure on Euclidean n -lattices X n and m is Lebesgue measure. In fact, the computation in [1, Section4.2] implies the following stronger statement for Euclidean lattices in R : Theorem 3.2.
Let A ⊂ R with m ( A ) > . Then (3.2) µ E (cid:18) Λ ∈ X : (cid:12)(cid:12)(cid:12)(cid:12) A ∩ Λ prim ) − m ( A ) ζ (2) (cid:12)(cid:12)(cid:12)(cid:12) > r p m ( A ) (cid:19) ≤ Cr . In order to derive these theorems, the authors made extensive use ofthe
Theta transform of compactly supported functions.
Definition 3.3.
Let L = g Z n be a Euclidean lattice in R n . Given afunction φ in L ( R n ), the theta transform isΘ φ ( L ) = X λ ∈ L prim f ( λ )where λ ranges over primitive points in L . When φ = χ A , we writeΘ φ = Θ A .We next give a version of the theta transform adapted to our needs. Definition 3.4.
Let L = g Z be a Heisenberg lattice. An element of L is called primitive if λ = g ( k, l, m ) with the greatest common divisorgcd( k, l ) = 1. This definition corresponds precisely to the requirementthat there is no point gγ ∈ g Z such that g ( k, l, m ) = gγ ( k ′ , ′ l, m ′ ) with( k ′ , ′ l, m ′ ) ∈ Z . Definition 3.5.
Given a function φ in L ( R ) letΘ H φ ( L ) = X λ ∈ L prim f ( λ )where λ ranges over primitive points in L . The operator Θ H : φ → Θ H φ is called the nil-theta transform of φ . When φ = χ A , we writeΘ H φ = Θ H A .For characteristic functions of Borel sets, the Theta transform hasthe following properties that we will use repeatedly: JAYADEV S. ATHREYA AND IOANNIS KONSTANTOULAS
Proposition 3.6.
Let φ = χ A with A a bounded Borel set such thatthere exists a neighborhood U of R containing such that A ∩ ( U × R ) = ∅ . Then the Theta transform Θ H φ is a bounded Borel function withcompact support on X H .Proof. The Borel property is clear. For a Heisenberg lattice L to in-tersect A , its projection in R must intersect A = π flat ( A ), i.e. wemust have Θ χ A = 0. A being a bounded set that does not meet aneighborhood of the origin shows ([4, Chapter XIII, Par.1]) that Θ χ A has compact support C in X .Therefore the closure of the set of Heisenberg lattices giving a non-zero Θ H A is a subset of a torus bundle over the compact set C andtherefore is compact.Finally, whenever φ ≤ ψ we have Θ H φ ≤ Θ H ψ so if ψ is a continuousfunction whose support is compact and contains A , we have Θ H A ≤ Θ H ψ . Since Θ H ψ is continuous (continuity is proven as in loc. cit.)with compact support, it is bounded, and therefore so is Θ H A . (cid:3) The nil-theta transform of characteristic functions counts latticepoints in sets like its Euclidean counterpart. However, a result asuniform as theorem 3.1 cannot hold in the Heisenberg setting:
Proposition 3.7.
Let π flat be the projection in H ( R ) to the first twocoordinates. Let A ⊂ H ( R ) have positive measure. The following in-equality holds: (3.3) µ H ( L ∈ X H : A ∩ L = ∅ ) ≥ µ E (Λ ∈ X : π flat ( A ) ∩ Λ = ∅ ) . Proof.
The action (2.1) transforms the flat part of the lattice by anelement of SL(2 , R ) and then translates the third component of eachlattice point accordingly along the central direction. Suppose the pro-jection of A does not intersect a lattice Λ. Then A cannot intersect anylift of Λ, since lifts are determined by values of the third coordinateover the flat part. Thus the entire torus of lattices in X H over Λ misses A , giving the inequality. (cid:3) In particular, we can increase the measure of A indefinitely keeping π flat ( A ) fixed; in the extreme cases of Theorem 3.1 (which are attained),for some C > µ H (Λ ∈ X : π flat ( A ) ∩ Λ = ∅ ) ∼ Cm ( π flat ( A )) . We see that even if m ( A ) itself becomes large, the probability of miss-ing a Heisenberg lattice remains bounded below by the inverse of themeasure of the projection. ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 9
The origin of this discrepancy is in principle easy to understand:the space of Heisenberg lattices is a very thin subset of the space ofEuclidean 3 -lattices; in particular, all members of X H project to 2-lattices in the flat plane, so they never tilt along the central direction.This phenomenon can be illustrated by the following extreme example: Example . For δ > N ≥ T ( δ, N ) to be thecylindrical punctured tube T ( δ, N ) = ( D (0 , δ ) \ (0 , × [ − N, N ]in R . Consider a compact subset C ⊂ X of measure µ E ( C ) = 1 − ǫ .Then by the compactness criterion on the space of lattices, all latticesin C have vectors of length at least 2 δ = √ ǫ . Then µ H ( L ∈ X H : L ∩ T ( δ, N ) = ∅ ) ≥ µ E (Λ ∈ X : Λ ∩ D (0 , δ ) = ∅ ) ≥ µ E ( C ) = 1 − ǫ. On the other hand, the measure of T ( δ, N ) can become arbitrarily largeno matter how small δ is, by increasing N .This is in stark contrast to the situation for Euclidean 3-lattices:starting from Z , we can reach the tube with an infinitesimally smallshear; the points (0 , , n ) ∈ Z will immediately tilt to intersect T ( δ, N ).The following question arises from this discussion: given a set A in H ( R ) and knowledge of the statistics of lattice points on π flat ( A ), howcan we deduce the statistics of Heisenberg lattice points on A ? Thenext two sections are devoted to the answer.4. Level distribution of lattices
Definition 4.1.
Let A ⊂ R be measurable with positive measure, ǫ > z := (0 , , z ) with z ∈ R . Let A zǫ = z + A × [0 , ǫ )be the level- z ǫ -plate over A . Consider a Heisenberg lattice L = g H H ( Z ). The set { L ∩ A zǫ ) : ǫ ∈ (0 , } is called the z -level distribution of L ∈ X H . From now on we assume0 < ǫ < Remark . The definition above captures significant information forcharacterizing lattice statistics given the statistics of the projection,because lattices above a given Euclidean lattice only differ in the ver-tical deviation of lattice points from the integer lattice. As we will see,the level distribution is determined modulo Z so we can take z ∈ [0 , Proposition 4.3 ([1, Proof of Theorem 2.2]) . For A ⊂ R of positivemeasure m ( A ) = a , we have the variance estimate (4.1) (cid:13)(cid:13)(cid:13)(cid:13) Θ A − aζ (2) (cid:13)(cid:13)(cid:13)(cid:13) ≤ a. We now begin the study of moments of Θ H A zǫ . The following propo-sition gives the Siegel formula. Since some intermediate formulas willbe used in the L estimate, we give a quick proof. Proposition 4.4. (4.2) Z X H | Θ H A zǫ ( L ) | dµ H ( L ) = m ( A ) ǫζ (2) . Proof.
We have χ A zǫ = χ A χ z +[0 ,ǫ ) . If g H = (cid:18) g ~x (cid:19) , writeΘ H A zǫ ( g H H ( Z ) prim ) = X ( ~m,k ) ∈ H ( Z ) prim χ A ( g ∗ ~m ) χ z +[0 ,ǫ ) ( k − ~x t · g ∗ · ~m )which splits asΘ H A zǫ ( g H Z ) = X ~m ∈ Z χ A ( g ∗ ~m ) X k ∈ Z χ z +[0 ,ǫ ) ( k − ~x t · g ∗ · ~m )= X ~m ∈ Z χ A ( g ∗ ~m ) X k ∈ Z χ [0 ,ǫ ) ( k − z − ~x t · g ∗ · ~m )= X ~m ∈ Z χ A ( g ∗ ~m ) ( { z + ~x t · g ∗ · ~m } < ǫ ) . (4.3)We want to integrate the last expression over the fiber first, then overthe base, using Corollary 2.7. Z X H Θ H A zǫ ( g H H ( Z ) prim ) dµ H ( g H )= Z X X ~m ∈ Z χ A ( g ∗ ~m ) (cid:18)Z R /g Z ( { z + ~x t · g ∗ · ~m } < ǫ ) dx (cid:19) dµ E ( g ) . (4.4)Now perform the change of variables ~x = g~u in the inner integral anduse det g = 1 to simplify to Z R / Z ( { z + ~u t · ~m } < ǫ ) du. Since SL(2 , Z ) is transitive on primitive vectors, all the toral integralsmust have the same value which we now compute. ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 11
The map S ( u ) = z + u m + u m (mod g Z ) is measure preserving,so splitting the integral over the torus in (4.4), we get Z R / Z Z R / Z ( S − ([0 , ǫ )) du du = ǫ. The outer integral in (4.4) is Z X Θ A (Λ) dµ E (Λ) = m ( A ) ζ (2)using the Siegel formula for Euclidean lattices and our normalizationof X . (cid:3) Now we need some preparations for dealing with the second momentof Θ H A zǫ . Definition 4.5.
Fix an integer D . Consider the space of pairs ofcoprime integer vectors(4.5) M D = { m = ( ~m, ~n ) ∈ Z × Z : det m = D } . The diagonal action of SL(2 , Z ), γ ( ~m, ~n ) = ( γ ~m, γ~n ) leaves invariantthe determinant of m , so it stabilizes each M D . The space of SL(2 , Z )orbits of the set M D ⊂ Z × Z of determinant D pairs is denotedby M O D = SL(2 , Z ) \ M D . Definition 4.6.
Let ~m = ( m , m ) ∈ Z and z ∈ R . Define the map S ~m : T → T by S ~m ( u , u ) = z + m u + m u (mod 1) . Denote the pullback of [0 , ǫ ) by S into T by C ~m ( ǫ ) := S − ~m ([0 , ǫ ]) ⊂ T . Finally, for ~m , ~n in Z , define the correlation of Cor ~m,~n ( ǫ, z ) = µ R / Z ( C ~m ( ǫ ) ∩ C ~n ( ǫ ))where µ R / Z is the probability Haar measure on the 2-torus.Next, we turn to the second moment of Θ H A zǫ . Proposition 4.7.
We have the following identity: (4.6) Z X H | Θ H A zǫ ( L ) | dµ H ( L ) = Z X X ~m,~n ∈ Z prim χ A ( g ∗ ~m ) χ A ( g ∗ ~n ) Cor ~m,~n ( ǫ, z ) dµ E ( g ) . Proof.
Using notation as in Proposition 4.4 and the expression (4.3)for the integrand, expand the square: | Θ H A zǫ ( L ) | = X ~m, ~n χ A ( g ∗ ~m ) χ A ( g ∗ ~n ) · ( { z + ~x t g ∗ ~m } < ǫ ) ( { z + ~x t g ∗ ~n } < ǫ ) . Let w ~m,~n ( g, ~x ) = ( { z + ~x t g ∗ ~m } < ǫ ) ( { z + ~x t g ∗ ~n } < ǫ ) . Once again integrating and changing variables, we get Z R /g Z w ~m,~n ( g, ~x ) dx = Z R / Z ( { z + ~u t ~m } < ǫ ) ( { z + ~u t ~n } < ǫ ) du. Since this expression is invariant under g → gγ , the result follows fromthe definition of C ~m ( ǫ ). (cid:3) Remark . The correlation
Cor ~m,~n ( ǫ, z ) is a ‘correction factor’ thatweighs the double sum in (4.6). Therefore, the study now reduces tothe distribution of the values of Cor ~m,~n ( ǫ, z ). Whenever m = γ n arein the same SL(2 , Z )-orbit, the corresponding correlations must be thesame. Thus one can write Cor ~m,~n ( ǫ, z ) = Cor O ( ǫ, z ) for the commonvalue of the correlation over an orbit O ∈ M O D . The idea is to look fora simple pair m in the orbit where the correlation is easy to compute.The next lemma describes the structure of the above set; the proofcan be found in [8, Lemma 6] or worked out by hand. Lemma 4.9.
The structure of M O D is as follows: (1) If D = 0 , M O D has two elements represented by ± (cid:18) (cid:19) . (2) If D = ± , M O D is a singleton represented by the identity matrix(diag ( − , respectively). (3) If | D | > , then M O D consists of φ ( D ) distinct orbits, representedby (cid:18) D k (cid:19) where k runs through a complete set of residuesmodulo D . Using this lemma, we can show:
Proposition 4.10.
The correlation
Cor O ( ǫ, z ) is equal to ǫ when O ∈ M D with D = 0 and equal to ǫ when D = 0 and O = +1 , when O = − .Proof. When D = 0 with equal parity, choose m = (cid:18) (cid:19) ; then Cor O ( ǫ, z ) = S − ([0 , ǫ )) = ǫ ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 13 as in Proposition 4.4. For opposite parity, the two intervals [0 , ǫ )+ Z and( − ǫ, Z are disjoint so their pullback under S has empty intersection.When D = 0, for any (cid:18) D k (cid:19) compute Z R / Z ( { z + ~u t ~m } < ǫ ) ( { z + ~u t ~n } < ǫ ) du = Z R / Z Z R / Z ( { z + Du + ku } < ǫ ) ( { z + u } < ǫ ) du du = Z R / Z { z + u } < ǫ ) Z R / Z ( { z + Du + ku } < ǫ ) du du ;the inner integrand is the pullback of [0 , ǫ ) by a (measure preserving)affine map of the form Du + τ giving integral ǫ independent of u , andthen the remaining factor contributes another ǫ . (cid:3) We now come to the main result of this section, the analog of The-orem 4.3 for Heisenberg lattices.
Theorem 4.11.
Let A zǫ be an epsilon-plate over A of measure m ( A ) > . We have (4.7) (cid:13)(cid:13)(cid:13)(cid:13) Θ H A zǫ − ǫ m ( A ) ζ (2) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ m ( A ) ζ (2) + 20 ǫ m ( A ) . Proof.
Write (cid:13)(cid:13)(cid:13)(cid:13) Θ H A zǫ − ǫ m ( A ) ζ (2) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) Θ H A zǫ (cid:13)(cid:13) − (cid:18) ǫ m ( A ) ζ (2) (cid:19) and use (4.6) to expand (cid:13)(cid:13) Θ H A zǫ (cid:13)(cid:13) = Z X X ~m,~n ∈ Z χ A ( g ∗ ~m ) χ A ( g ∗ ~n ) Cor ~m,~n ( ǫ, z ) dµ H ( g );by Proposition 4.10 the integrand on the right hand side becomes ǫ X m ∈ M , + χ A ( g ∗ ~m ) χ A ( g ∗ ~n ) + ǫ X m ∈ M D , D =0 χ A ( g ∗ ~m ) χ A ( g ∗ ~n );now add and subtract the sum over M weighted by ǫ to get( ǫ − ǫ ) X m ∈ M , + χ A ( g ∗ ~m ) χ A ( g ∗ ~m ) + ǫ X m ∈ M χ A ( g ∗ ~m ) χ A ( g ∗ ~n )for the integrand; recall det m = 0 and parity preservation implies ~m = ~n . Integrating the two parts separately, we get (cid:13)(cid:13) Θ H A zǫ (cid:13)(cid:13) = ( ǫ − ǫ ) Z X Θ χ A ( L ) dµ E ( L ) + ǫ Z X | Θ A ( L ) | dµ E ( L ) which simplifies to (cid:13)(cid:13) Θ H A zǫ (cid:13)(cid:13) = ( ǫ − ǫ ) Z X Θ χ A ( L ) dµ E ( L ) + ǫ k Θ A k = ( ǫ − ǫ ) m ( A ) ζ (2) + ǫ k Θ A k using χ A = χ A and the Siegel formula for Euclidean 2-lattices (recallour normalization of X ). Now we subtract the constant term andabsorb it into the second summand to get(4.8) (cid:13)(cid:13)(cid:13)(cid:13) Θ H A zǫ − ǫ m ( A ) ζ (2) (cid:13)(cid:13)(cid:13)(cid:13) = ( ǫ − ǫ ) m ( A ) ζ (2) + ǫ k Θ A − m ( A ) ζ (2) k . Applying Theorem 4.3 and gathering ǫ terms together, we obtain(4.9) (cid:13)(cid:13)(cid:13)(cid:13) Θ H A zǫ − ǫ m ( A ) ζ (2) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ m ( A ) ζ (2) + 20 ǫ m ( A ) . (cid:3) Theorem 4.11 immediately implies the analogues of Theorems 3.1and 3.2 for the plate distribution.
Corollary 4.12.
We have the following bound on the probability of aHeisenberg lattice missing a plate A zǫ : (4.10) µ H (Λ ∈ X H : A zǫ ∩ Λ = ∅ ) ≤ Cm ( A zǫ ) . Corollary 4.13.
The average discrepancy of lattice points in a plate A zǫ satisfies the Chebyshev inequality (4.11) µ H (cid:18) Λ ∈ X H : (cid:12)(cid:12)(cid:12)(cid:12) A zǫ ∩ Λ prim ) − m ( A zǫ ) ζ (2) (cid:12)(cid:12)(cid:12)(cid:12) > r p m ( A zǫ ) (cid:19) ≤ Cr . From this information, we can obtain discrepancy estimates fromsets built out of plates in a controlled number of steps. We illustratethis with the example of a stout cylinder:
Corollary 4.14.
Let C = A × I with m ( A ) > and | I | ≤ m ( A ) − δ .Then (4.12) µ H (cid:18) Λ ∈ X H : (cid:12)(cid:12)(cid:12)(cid:12) C ∩ Λ prim ) − m ( C ) ζ (2) (cid:12)(cid:12)(cid:12)(cid:12) > rm ( C ) − δ (cid:19) ≤ Cr . Proof.
Split C into at most m ( A ) − δ plates P i of height at most 1;since these sets are disjoint, we can splitΘ H S − m ( S ) ζ (2) = X P i (cid:18) Θ H P i − m ( P i ) ζ (2) (cid:19) ; ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 15 then apply the Cauchy-Schwartz inequality to (4.7) after taking squareroots to get (cid:13)(cid:13)(cid:13)(cid:13) Θ H S − m ( S ) ζ (2) (cid:13)(cid:13)(cid:13)(cid:13) ≤ m ( S ) − δ . Then proceed as in the proof of Chebychev’s inequality with the ap-propriate exponents. (cid:3)
Remark . The result above is presumably not optimal; we ex-pect that good discrepancy estimates will hold for approximately cu-bical cylinders. However, this requires higher moment analogues of 4.3(which are not known) and a corresponding treatment of higher corre-lations in 4.6. In any case, these sets come close to the limit of sets weshould expect to satisfy good deviation estimates as the next sectiondemonstrates. 5.
High discrepancy sets
Here we construct a wide variety of sets for which an estimate likethe one in Corollary 4.13 cannot hold; the argument is an extension ofExample 3.8. The point of the following proposition is to show thatfirstly high discrepancy sets need not be anchored to any specific pointlike the origin, and secondly do not need to have an approximatelycylindrical shape (although they will be built out of cylinders).
Proposition 5.1.
Let B (0 , R ) be a large ball in R centered at theorigin and fix ǫ > . Let Z ◦ = ( Z + B (0 , ǫ )) ∩ B (0 , R ) be a thickening of the standard lattice inside the fixed ball. There existBorel sets S ⊂ R such that: (1) S has arbitrarily large measure, (2) for any cylinder C and ǫ ′ > , m ( C △ S ) > ǫ ′ m ( S ) − ǫ ′ , (3) π flat ( S ) = A is an arbitrary Borel set of positive measure in B (0 , R ) \ Z ◦ and (4) the inequality µ H (Λ ∈ X H : S ∩ Λ = ∅ ) ≥ δ holds, where δ depends only on ǫ .Proof. Let U be a neighborhood of the identity in SL(2 , R ) such that( U Z ) ∩ B (0 , R ) ⊂ Z ◦ and µ E ( U ) = δ its measure. Generally to meet requirements 1, 3 and 4 we can pick an arbitraryBorel A ⊂ B (0 , R ) \ Z ◦ of positive measure, partition it into finitelyor countably many disjoint Borel sets A i of positive measure and overeach set erect a cylinder C i = A i × I i of any desired height to form S so that X i | I i | m ( A i ) = m ( S )is arbitrarily large and distributed in an arbitrary way over the bases A i .To meet the second requirement, observe that all that matters inthe computation is the measures of the A i and the placements of thecylinders over them. Thus we can visualize the A i as disjoint horizontalintervals on the real line arranged in decreasing size and the I i asvertical intervals at prescribed level and of prescribed height.Choose a large finite partition of A into k = m ( A ) N parts so thateach A i has measure approximately i m ( A ) and choose segments I =[1 , I = [2 , I = [4 ,
8] and so on. Then m ( S ) ∼ km ( A ) and bycontrolling k we can make m ( S ) as large as possible. Note that the topof the cylinder over A i is at t i ∼ i +1 .A cylinder that will minimize the difference C △ S will necessarilyhave base inside A , so we may assume that C = ( S j ∈ J A j ) × I where theunion is over some sub-collection of the partition. Assume C containsprecisely l of the k cylinders, with l ≫ C is based on at least l and at most ( l + 2) of the A i . Let r + 1be the smallest index so that the cylinder over A r +1 is contained in C . In order to minimize the difference, the optimal cylinder has to bebased on all A j for j = r + 1 , · · · , r + l (for each j > r + 1 skipped, thegains from removing C j are offset by the fact that indices are shiftedat least one place, doubling the cost of the cylinder over A r ; this beatsthe gain because the measure of A r is at least twice the measure of A j ).For this configuration, the height of the cylinder is t r + l − t r = 2 r +1 − r and the base has measure m ( A )2 − r (1 − − l ); thus its measure is m ( A )2 l .In contrast, the measure of S ∩ C is within ( l ± m ( A ). Therefore, m ( S △ C ) ≥ m ( A )(2 l − l − . If l ≥ log ( k ) we get our result. If l ≤ log ( k ), then m ( S △ C ) ≥ m ( S ) − m ( C ∩ S ) ≥ m ( A )( k − l )which again gives the result.Finally, by the discussion preceding Example 3.8 we know that µ H (Λ ∈ X H : S ∩ Λ = ∅ ) ≥ µ E (Λ ∈ X : A ∩ Λ = ∅ ) ≥ µ E ( U ) = δ. ATTICE DEFORMATIONS IN THE HEISENBERG GROUP 17