Counting Singular Matrices with Primitive Row Vectors

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1 Monatsh. Math. 44, 7 84 (005) DOI 0.007/s Counting Singular Matrices with Primitive Row Vectors By Igor Wigman Tel Aviv University, Israel Communicated by W. M. Schmidt Received March 3, 003; in revised form September 9, 003 Published online June, 004 # Springer-Verlag 004 Abstract. We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n n matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as T!,the number is asymptotic to ðn Þun n ðnþðn Þ n Tn log ðtþ for n 5 3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt. 000 Mathematics Subject Classification: H06 Key words: Singular matrices, primitive vectors, lattices, equidistribution theorem, asymptotics. Introduction.. A basic problem in the geometry of numbers is counting integer matrices with certain additional properties. In this paper we will solve a new counting problem of this kind. Let us consider the set of singular n n matrices with integer entries. We are interested in the question how many among these matrices have primitive row vectors, that is each row is not a nontrivial multiple of an integer vector. We count the matrices according to the maximal allowed Euclidean length of the rows. Without the constraint of primitivity the problem of counting such matrices was solved by Katznelson []. We will find that for n 5 3 a positive proportion of integer singular matrices have all rows primitive. Let PN n ðtþ be the counting function of the set PM n ðtþ of n n singular integer matrices, all of whose rows are primitive and whose Euclidian length is at most T. That is, let PM n ðtþ fm M n ðzþ: detðmþ 0; primitive rows v i Z n ; jv i j 4 Tg; and set PN n ðtþ jpm n ðtþj. In this paper we will determine the asymptotic behaviour of PN n,ast!. Define a similar counting function N n ðtþ, where N n counts n n integer matrices M with rows of length 4T, that is N n ðtþ jm n ðtþj, where M n ðtþ fm M n ðzþ: detðmþ 0; rows v i ; jv i j 4 Tg: ðþ

2 7 I. Wigman Katznelson [] showed that for n 5 3, N n ðtþ ðn Þu n T n n log ðtþþoðt n n Þ; ðnþ where the constant in the O-notation depends only on n. Recall that n!! denotes the product of integers 4n of the same parity as n. The constant u n is given by: 8 n ðþ m n < ðm Þ!! m m! ; n m; u n : n m m! n ðþ m ðmþþ!! ; n m þ : Trivially, PN n ðtþ 4 N n ðtþ, and thus PN n ðtþ T n n log ðtþ. Moreover, PN n ðtþ T n n, since we can consider, for example, only matrices M with primitive rows v i, which satisfy v n v and jv i j 4 T. A random vector in Z n is primitive with a positive probability, that is, the number of primitive vectors whose length is at most T, is T n. The number of such matrices is obviously ðt n Þ n T n n. Combining the observations of this paragraph, we conclude that T n n PN n ðtþ T n n log ðtþ. For n an elementary argument shows that PN ðtþ ðþ T þ OðTÞ: ð3þ Our main result is: Theorem. (i) For n 5 4 we have PN n ðtþ (ii) For n 3 we have ðn Þu n ðnþðn Þ n T n n log ðtþþoðt n n Þ: PN 3 ðtþ u 3 ð3þðþ 3 T6 log ðtþþoðt 6 log log ðtþþ:.. Another way to treat our problem is to consider it as a counting problem of rational points with bounded height on a projective variety, see []. Let V P fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n P n be the projective variety where the determinant n times vanishes. The height of a point P n ðqþ is defined by H 0 ðþ j~j; where ~ is a primitive integral point in Z n representing and jjis the standard Euclidian norm on R n. Now, for Y ðy ;...; Y n ÞV, define the height HðYÞ max 4 i 4 n H0 ðy i Þ: Then PN n ðtþ is the number of points Y V of height HðYÞ 4 T. ðþ

3 Counting Singular Matrices with Primitive Row Vectors We will present now the main idea in the case n 3. If M is an integer singular matrix, then all the rows of M lie in a -dimensional lattice Z 3. Thus we should count triples of vectors lying in a -dimensional sublattice of Z 3 and sum over all such lattices. For a primitive Z 3 we define a -dimensional lattice L fvz 3 : v?g?. Denote the subset of all primitive points in L, PL fprimitive v L g and PL ðtþ fv PL : jvj 4 Tg. Moreover, denote PA fmm 3 ðzþ: rows in PL g, and PA ðtþ fm PA : jrowsj 4 Tg. Thus jpa ðtþjjpl ðtþj 3. For 6 0 the intersection L \ L 0 is a set of integer points on a line, and thus jpa \ PA 0j It can be shown that the contribution of such intersections is negligible, and that PN n ðtþ 0 jpl ðtþj 3 ð4þ jjt where the last sum is over primitive Z 3, such that L is bounded by T (see Section 5). Now PL ðtþ v T log ðtþ ðþjj þ O ð5þ (see Section 3), and summing the cube of the main term of PL ðtþ will give the result. A complication in dimension 3 is that for some of the lattices L in the sum (4), the error term in (5) is asymptotically greater than the main term. Such a phenomenon does not happen for higher dimensions. In order to show that this phenomenon is rare and the contribution of such lattices is negligible, we will use an equidistribution theorem of Wolfgang Schmidt [4] (see Theorem )..4. Contents. We will use some known results of counting integer points in Z, or more generally, counting points of a sublattice of Z n, as well as counting primitive points in such a sublattice. We will give some basic background on lattices in Section and some facts concerning counting lattice points will be given in Section 3. The goal of Sections 4 and 5 is to prove cases (ii) and (i) of Theorem respectively.. Background on Lattices In this section we will give some basic facts which deal with sublattices of Z n. For general background see [5]. Definition. Let be a lattice. A basis of, f ; ;...; m g, such that the product of the lengths of the vectors in it is minimized is called reduced. For such a basis we have: j j...j m jdetðþ: ð6þ A basis that satisfies the last inequality has properties similar to a reduced one. We say that is bounded by T, if it has a reduced basis consisting of vectors of length at most T. Ifak-dimensional lattice has k linearly independent vectors, all

4 74 I. Wigman of length at most T, it follows that this lattice is bounded by ct, for some constant c that depends only on the dimension k. In that case we will treat it just as if it was bounded by T, since it will affect only some constants in our upper bounds not affecting the asymptotic behavior. Also, for a lattice, we will denote N ðtþ jfv: jvj 4 Tgj as well as P ðtþ jfv : jvj 4 T; v primitivegj: An m-dimensional lattice Z n is called primitive if there is no m-dimensional lattice properly containing. In particular, each vector in any basis of a primitive lattice is a primitive vector (the converse is not necessarily true). The orthogonal lattice? of consists of all vectors v Z n, such that v u 0 for all u. It is a primitive integral lattice of dimension n m. If is a primitive lattice, then ð? Þ?. Also, in this case, it was shown in [3] (Chapter, formula (4)), that detðþ detð? Þ ð7þ 3. Counting Lattice Points The goal of this section will be to give some expressions for the number of integer points in a lattice, as well as estimations for the error terms of these expressions, which correspond to primitive lattices. The next lemma is a basic one, which could be found in different variations in the literature, see e.g. [3], Lemma. Lemma. Let R n be an m-dimensional lattice, and let f ;...; m g be a reduced basis for, sorted in increasing order of their norms. Denote j i j i for 4 i 4 m. Let R m be an m-dimensional convex body containing m linearly independent vectors of, then j \ j volðþ volð@þ detðþ þ O... m (the 0 s in the denominator are all except for the greatest one). Let v n be the volume of the standard n-dimensional unit ball, that is ( m m! n m v n ; n m þ : ðþ m ðmþþ!! Lemma. Let Z n be an ðn Þ-dimensional primitive lattice which is bounded by T, for n 5 3. Let f ; ;...; n g, be a reduced basis of. Then: (i) For n 5 4 we have P ðtþ v n ðn Þ detðþ Tn þ O T n : j j...j n j

5 Counting Singular Matrices with Primitive Row Vectors 75 (ii) For n 3 we have P ðtþ v T log T ðþ detðþ þ O : Proof. Since is primitive, every vector is a [possibly trivial] integer multiple of a primitive vector in L, and thus N ðtþ P btc k P T k, hence by Moebius inversion P ðtþ btc T ðkþ N : k k Using the last expression the result of Lemma, where m n, and is the ðn Þ-dimensional ball on the ðn Þ-dimensional hyper-plane spanned by, we get P ðtþ btc k v n T n det k with error term ðtþ given by n vn T T ðkþ þ O n detðþ k k n j j...j n j btc ððkþk ðn Þ þ 0 k ðtþþ k v n ðn Þ detðþ Tn þ ðtþ: ðtþ btc T n O k k n j j...j n j! þ v n T n btc ðkþk ðn Þ : det ðn Þ Thus, since jðnþj 4 for every n N, jðtþj j j...j n j j j...j n j btc T n k k n þ Tn btc n T k k n þ Tn kbtcþ T n ðkþk ðn Þ! We also used here the fact that j j...j n jdetðþ. Now in order to obtain case (i) of the lemma use the convergence of the series P k k for n 5 4. n We use P btc k k log ðtþ in order to prove the other case of the lemma. 4. The Case n 3 In this section we will prove case (ii) of Theorem. The computation of the main term is also valid in the case n 5 4, while for n 3 we should be more delicate in order to obtain the appropriate error term. Thus every lemma which is to be used in Section 5 will be stated for general n in the current section. The main difficulty in this case is that the error term for estimating P (see Lemma, case (ii)) could be asymptotically greater than the main term itself. In!

6 76 I. Wigman order to show that such lattices are rare, and thus their contribution to the error term is asymptotically negligible, we will need an equidistribution result of Schmidt [4] for the space of lattices. Suppose M is a singular matrix. This means that there exists a vector 0 6 Z n, such that all rows of M are orthogonal to. Thus all the rows of M lie in a ðn Þdimensional lattice?. Since multiplying by a constant does not affect this property we can assume that is primitive. Our basic idea is to sum the number of n-tuples of primitive vectors with bounded length lying in, where Z n runs over all such lattices. We will see that we can limit the sum to a finite number of such lattices. Let Z n be primitive. Recall that? denotes the ðn Þ-dimensional orthogonal dual lattice to, that is the primitive ðn Þ-dimensional lattice in Z n which consists of all vectors in Z n which are orthogonal to. Our discussion leads to the following definitions: denote L?, L ðtþ fv L : jvj 4 Tg; PL fprimitive v L g; PL ðtþ PL \ L ðtþ: Given denote by A the set of matrices whose rows lie in L : A fmm n ðzþ: M 0g L L L fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ; ð8þ n times Also denote by PA the subset of matrices in A, whose rows are primitive. Obviously, PA PL PL PL fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n times Denoting by A ðtþ and PA ðtþ the set of matrices in A with (primitive) rows in L of length 4T, we clearly have: and A ðtþ L ðtþ n ; PA ðtþ PL ðtþ n : The next lemma connects between the terms just defined with our problem. Lemma 3 ([], Lemma 4). Let M n ðtþ. Then there is a primitive Z n, such that A is bounded by T and A. Remarks. (i) As it was mentioned, bounded by T means bounded by c n T, where the constant c n > 0 depends only on n. We will see that c n doesn t affect our computations, so we will ignore it everywhere except the computation of the main term. (ii) If A is bounded by T then jj n detða ÞT n n by (6) and (7), and thus jj T n. The converse is not true, since there are primitive vectors Z n, such that T n,buta is not bounded by T. We will call such vectors bad ; they will only have a minor influence on the asymptotics. ð9þ

7 Counting Singular Matrices with Primitive Row Vectors 77 (iii) In every case we will deal with a reduced basis, the vectors will be ordered in increasing order of their norms, unless specified otherwise. We may conclude from Lemma 3, that M n ðtþ S 00 jjt A ðtþ and also n PM n ðtþ [ 00 PA ðtþ: ð0þ jjt n Here and everywhere in this paper, we use S00 and P00 to denote a union=sum over primitive vectors Z n, for which? is T-bounded. Analogously, S0 and P0 will denote a union=sum over primitive vectors not saying anything about the orthogonal dual. It is natural to relate the cardinality of the left side of (0) to the sum of the cardinalities of the right side: PN n ðtþ 00 jpa ðtþj þ 0 ðtþ jjt n 0 jpa ðtþj þ 0 ðtþþ0 3 ðtþ jjt n The factor of is due to the fact that PA PA ; the terms 0 ðtþ and 0 3ðTÞ are the error terms which implied by the intersections of PA for different and the contribution of the so called bad vectors respectively (a bad vector is a primitive vector, with jj T n such that? is not bounded by T), which do not allow us to get an estimate of the primitive vectors contained within it. However j 0 ðtþj 4 j ðtþj and j 0 3 ðtþj 4 j 3ðTÞj,where, 3 are the analogous error terms in the case of the problem solved in [], which were shown to be OðT n n Þ ([], pp 30 33). Thus: PN n ðtþ 0 jpa ðtþj þ OðT n n Þ jjt n 0 jpl ðtþj n þ OðT n n Þ: ðþ jjt n For n 3, we would like to demonstrate how we can achieve the bound for 0, since in our case it is quite simple. Indeed, the only matrices in PA \ PA 0 with! v primitive 6 0 are of the form v v with primitive v Z 3. Conversely, given a primitive v Z 3, its contribution to the sum in () is 8PN v?ðc 3 T Þ, where the factor 8 is the number of all possible signs of the 3 rows. Hence: j 0 ðtþj PN v?ðt Þ 4 N v?ðt Þ T 4 jvj : jvj 4 T jvj 4 T jvj 4 T Now P jvj 4 T jvj where the last equality is due to summation by parts, making use of the fact that jfprimitive v Z 3 with jvj 4 Tgj T 3.Thusj 0 ðtþj T6. At this point we would like to substitute the result of Lemma into (). This is exactly what we are going to do in case n 5 4 (see Section 5). However, for

8 78 I. Wigman n 3, the error term could be asymptotically greater than the main term. In order to overcome this difficulty, we will first reduce the last sum to convenient lattices, that is those with not too big determinant (Lemma 4) and where the norms of vectors in a reduced basis do not differ too much (Lemma 5). Corollary will show that for such lattices the error term is negligible relative to the corresponding main term. We will adapt the following notations: Notations. Foranðn Þ-dimensional lattice Z n, we will denote the main term of ðp ðtþþ n as well as the corresponding error term: v n n cðt; Þ ðdetðþþ n ðn Þ n T n n ; ðt; Þ ðp ðtþþ n cðt; Þ: ðþ Moreover, for a vector Z n denote cðt;þcðt;? Þ; ðt;þðt;? Þ: ð3þ Lemma 4. For any constant A > 0, the following estimate holds: P ðtþ 3 T 6 log log T; T < det ðþ 4 where the sum is over primitive T-bounded lattices Z 3. Proof. We will use here the trivial inequality P ðtþ 4 N ðtþ. Now, from Lemma, N ðtþ detðþ þ O T T detðþ, where ðþ is the shortest vector in a reduced basis of (that is, the shortest nontrivial vector in ). The last inequality is due to being bounded by T, since it implies T detðþ j j T T j j 5 T. We will denote by nðrþ the number of primitive two-dimensional lattices Z 3 with detðþ r and NðtÞ P btc r nðrþ. Then NðtÞ t 3 ; ð4þ since such are determined by a primitive vector orthogonal to it with jj detðþ 4 t; the number of such vectors is t 3. Thus ðp ðtþþ T < detðþ 4 3 T 6 T ðlog TÞ 4 T 6 rd T 6 " A < detðþ 4 e NðtÞ t 3 nðrþ r 3 T T 6 log log T: þ T detðþ 3 ð T NðtÞ T t 4 dt #

9 Counting Singular Matrices with Primitive Row Vectors 79 We used here summation by parts, substituting (4) to get an estimate for NðtÞ in order to obtain the last inequality. This concludes the proof of the lemma. It should be noted, that in addition to what was originally stated, we proved here also the following inequality: cðt; Þ T 6 log ðlog ðtþþ; ð5þ T ðlog ðtþþ A < det ðþ 4 where cðt; Þ is as in (). & We will need the following theorem, which is special case of Theorem 5 from [4]. Theorem. For a 5 let Nða; TÞ be the number of lattices Z 3 with successive minima, f ; g which satisfy 5 a, and dðþ 4 T, then Nða; TÞ const arcsin T 3 þ O a 5 T : a We will use Theorem in order to prove the following lemma: Lemma 5. For any constants A > 0, B >, the following estimate holds: P ðtþ 3 T6 ðlog TÞ B ; detðþ < j j > ðlog TÞB where the sum is over primitive T-bounded lattices Z 3. Proof. We will use summation by parts as well as Theorem to bound the sum. In order to do so we will denote m T ðrþ jf Z 3 ; -dimensional lattice: detðþ r; j j= > ðlog ðtþþ B gj: so that P r 4 t m TðrÞ Nððlog TÞ B ; tþ. Using the trivial inequality P ðtþ 4 N ðtþ detðþ, as in the proof of Lemma 4, we have: ðp ðtþþ 3 detðþ 4 j j > ðlog TÞB T 6 detðþ 4 T T 6 j j > ðlog TÞB r m T ðrþ r 3 ðdetðþþ 3

10 80 I. Wigman T T 6 T 6 " r m T ðrþ r 3 NððlogTÞ B ; tþ t 3 T ðlogtþ A By Theorem, this is T 6 =ðlog TÞ B as required. As in the case of Lemma 4, we proved here also: detðþ < j j > ðlog TÞB cðt; Þ T6 ðlog TÞ B ; ð T þ # NððlogTÞ B ; tþ t 4 dt ð6þ & Lemma 6. Let Z 3 be a -dimensional lattice, with a reduced basis f ; g, such that detðþ 4 and j j 4 ðlog TÞB. Then T log T Proof. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j j j ðj jj jþ Therefore, T log T detðþ B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlog TÞ B detðþ T log T j j j j T log T detðþ : ð7þ T B T B detðþ ðlog TÞ ; A B which concludes the proof of the lemma. & We will always want to choose the constants A and B, for which A B > 0, since in this case the error term of a certain counting function will be asymptotically less than the corresponding main term, as we will notice in the following corollary, which follows immediately from the previous lemma. In fact, we would like to choose constants, that will satisfy A B 5 ; A > 0; B > ð8þ for example, A 6, B, so this error term will not affect the general error term. Using the proof of Lemma 8 below with case (ii) of Lemma and substituting the result of Lemma 6 we obtain: :

11 Corollary. Let Z 3 with j, such that j 4 ðlog TÞB, for constants A, B, which satisfy (8). Then, under the notations of (), ðt;þcðt;þ oðcðt;þþ: B We are now ready to finish the proof of case (ii) of Theorem. Recall () and choose the constants A, B, which satisfy (8). We will ignore the difference between and 4, which is not significant for bounding the error terms as well as computation of the main term, as we will see a bit later. Thus: 0 jpa ðtþj T 0 ðp?ðtþþ 3 T 0 ðp?ðtþþ 3 þ j j 0 ðp?ðtþþ 3 T ðlog TÞA < 0 ðp?ðtþþ 3 þ 0 ðp?ðtþþ 3 þ OðT 6 log ðlog TÞÞ 4 ðlog TÞB j j > ðlog TÞB 0 ðp?ðtþþ 3 þ OðT 6 log ðlog TÞÞ: j j 4 ðlog TÞB We used here Lemmas 4 and 5 (recall that B > because of (8)). Substituting the result of Corollary into the last sum we get: 0 ðp?ðtþþ 3 j j 4 ðlog TÞB j j þ O 4 ðlog TÞB þ O þ O Counting Singular Matrices with Primitive Row Vectors 8 0 ðcðt;þþ þ O B B B 0 ðcðt;þþ j j 4 ðlog TÞB 0 ðcðt;þþ þ OðT 6 Þ 0 ðcðt;þþ þ OðT 6 log ðlog TÞÞ: B T

12 8 I. Wigman We used here Corollary as well as (5) and (6). It should be noted that the usage of Lemmas 4 and 5 as black boxes is not enough in this case, since each summand of the current series is not necessarily less or equal to the corresponding one in these lemmas (because of the error term). Substituting () we obtain (writing rather than 4 in the domain of summation for consistency with ()). PN 3 ðtþ 0 T 6 ðcðt;þþ ðþ 3 0 v 3 jj 3 : ð9þ jjt jjt It remains to compute the inner sum. The same computation holds for n 5 4 and will be used in Section 5, so we will state it for general n. Lemma 7. M 0 v n n jj n u n log ðmþþoðþ: ðnþ ð0þ Proof of Lemma 7. Using Moebius inversion on the set of multiples of each primitive vector separately, we obtain: M 0 v n n jj n M Thus, just as in [], we get: M 0 v n n jj n M k k ðkþk n ðkþk n v n n ð 4 juj 4 bm=kc 4 jxj 4 M=k The reason that the last equality holds is that v n ð ð n dx juj n 4 juj 4 M=k 4 jxj 4 M=k jxj n ð 4 v n n juj n : dx jxj n þ OðÞ: dx 4 jxj 4 M=k jxj nþ jxj 5 dx < ; nþ jxj ðþ ðþ and the fact that the series P k ðkþk n converges absolutely. Computing the last integral in () in polar coordinates we get: M 0 v n n jj n M k ðkþk n u n ðlog ðmþ log kþ u n log ðmþþoðþ; ðnþ Ð with u n vn n S dx, where the last integral is computed in the usual spherical n [polar] Ð coordinates. Thus, using the formula for v n as well as the fact that v n n S dx, yields (). n

13 Counting Singular Matrices with Primitive Row Vectors 83 Thus, M 0 v n n jj n u n log M; ðnþ ð3þ where the error term is O(), which yields (0) and completes the proof of Lemma 7. & Substituting the result of the last lemma in (9) with M c n T, where c n is the constant implied by the -notation in (), will yields case (ii) of Theorem. The error term OðÞ does not bother us, since after multiplying it by const T n n, while substituting it in (), we will get an error term of OðT n n Þ, and adding it to other error terms will not increase an estimate for the general error term of the asymptotics. As mentioned before, the constant c n does not affect the computation, since we substitute it in a logarithm in any case. 5. The Case n 5 4 We will need the following lemma: Lemma 8. Let Z n be a primitive vector, such that its orthogonal dual,? is T-bounded. Let f ; ;...; n g be a reduced basis of?. Then for n 5 4 we have v n n jpa ðtþj n ðn Þjj n T n n T n n þ O n j j n...j n j n j n j n Proof. Due to (9), we have jpa ðtþj ðp?ðtþþ n, and substituting the case (i) of Lemma in the last equality, as well as the fact that detð? Þjj, because of (7), we get: jpa ðtþj v n ðn Þjj Tn þ O T n n j j...j n j ðaþbþ n. We will use the binomial formula for the last expression. The first summand (that is, a n )isjustthe main term in the result of the lemma. Now, since the lattice? is T-bounded, a is asymptotically greater than b (that is a b), and thus the only asymptotically significant summand in the binomial is the second one (that is n a n b). The coefficient n is constant, and thus, by (6), this is the error term we just stated in the lemma. & Notations. In this section we will use the notations in (3) as well. Substituting the result of Lemma 8 into (), we obtain: PN n ðtþ jjt n 0 T n n n ðn Þ v n n n ðn Þjj n T n n þ ðt;þ jjt n 0 v n n : þ OðT n n Þ jj n þ 0 ðtþþoðtn n Þ; ð4þ

14 84 I. Wigman: Counting Singular Matrices with Primitive Row Vectors with 0 ðtþ 0 jðt;þj: ð5þ T n We will prove in the end of the section that 0 ðtþ n. Tn Using (0) with M c n T n, (where c n is the constant implied by the -notation in ()) in (4) will imply PN n ðtþ ðn Þu n ðnþðn Þ n T n n log T þ OðT n n Þ which ends the proof of Theorem (i). Just as in case n 3, the error term OðÞ does not bother us. Thus, PN n ðtþ ðn Þu n n log T þ 0 ðtþ, where 0 ðtþ is the error term of this ðnþðn Þ n Tn asymptotics. Accumulating all the error terms we confronted with and assuming 0 ðtþ n (which we will prove immediately), will imply j 0 ðtþj T Tn n n. Error term. The only error term which is not less or equal to the corresponding error term in [] is 0. However in this case, we can immediately bound it given the results of the work that was already done by Katznelson. Under the notations (3), for n 5 4 we have, due to (5) and Lemma 8: 0 ðtþ 0 0 T n n jðt;þj n j j n...j n j n j n j n : T n T n From the definition of P 0 and P00, it is obvious that P 00 4 P 0 as long as only nonnegative numbers are involved. The inequality 00 T n n n j j n...j n j n j n j n n Tn T n was showed in [] (pages 30 33) in the course of proving that ðtþ T n n, where ðtþ is the corresponding error term in the case of N n ðtþ. & Acknowledgement. This work was carried out as part of the author s M.Sc. thesis at Tel Aviv University, under the supervision of Prof. Zeev Rudnick. The author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Science and Humanities. References [] Franke J, Manin YI, Tschinkel Y (989) Rational points of bounded height on Fano varieties. Invent Math 95: [] Katznelson YR (993) Singular matrices and a uniform bound for congruence groups of SL n ðzþ. Duke Math J 69: 36 [3] Schmidt WM (968) Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height. Duke Math J 35: [4] Schmidt WM (998) The distribution of sublattices of Z m. Monatsh Math 5: 37 8 [5] Siegel CL (988) Lectures on the Geometry of Numbers. Berlin Heidelberg New York: Springer Author s address: I. Wigman, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, igorv@post.tau.ac.il