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A Metrical Theorem in Geometry of Numbers

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A Metrical Theorem in Geometry of Numbers A METRICAL THEOREM IN GEOMETRY OF NUMBERS BY WOLFGANG SCHMIDT Introduction. If S is a pointset in Rn, n>l, then we write L(S) for the number of lattice-points in S. Here, and throughout this paper, a lattice-point is a point with integral coordinates. If 5 is a Borel set of finite volume V(S), one would expect that L(S) is of about the same order of mag- nitude as V(S). Hence we define the "discrepancy" D(S) by (1) D(S) = \ L(S)V(S)-> - 1\ . As a companion for 7(5), we introduce P(S), the number of primitive lattice-points in 5. (A lattice-point is primitive, if its coordinates are relatively prime.) We put (2) E(S) = | P(5)r(«)F(5)-> - 11 . Next, let $ be a family of Borel sets with finite volumes, such that (i) If SQ$, FG*, then either SQT or TQS. (ii) There exist SQQ with arbitrarily large V(S). Finally, throughout this paper, \^(s), 5^0, should be a positive, nondecreas- ing function, such that /^(s)-1^ exists. Theorem 1. Suppose n>2. Then for almost every linear transformation A ("almost every" in the sense of the n2-dimensional euclidean metric induced by matrix-representation for A), (3) D(AS) = 0(V~li2 log V ^^(log V)), (4) E(AS) = 0(V~V2 log V ^''(log V)) for SQ$. More explicitly, for almost every A there exist constants Ci(^4), Ci(A), such that D(AS) fg a(A)V-^2 log F^/2(log V), whenever V(S)^Ci(A) and SQ<f>. In Ri our results are a little weaker: Received by the editors March 13, 1959. 516 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A METRICAL THEOREM IN GEOMETRY OF NUMBERS 517 Theorem 2. Let re = 2. Then for almost every linear transformation A (5) DiAS) = OiV-1'2 log2 V yfrll2ilogV)), (6) E(AS) = OiV'1'2 log2 V if>1',(logV)) for SE<P- Theorem 1 and Theorem 2 yield the Corollary. Suppose S is a Borel set with infinite volume in Rn, re > 1. Then PiAS) is infinite for almost all A. This corollary was first proved by C. A. Rogers [8] in case re>2. In fact, for w>2 our results will follow very easily from methods of Rogers [8] and Cassels [2]. The proof of Theorem 2 is more difficult. It was shown by Lekkerkerker [5], that the corollary does not hold in Pi. Therefore Theorem 1 has no analogon in Ru For the proof of Theorem 2 we shall need the 2-dimensional case of Theorem 3. Let S be a Lebesgue-measurable set in Rn with characteristic function piX) and volume V. By ||Xi, - • • , Xf\\ we denote the absolute value of the nXn-determinant, whose row-vectors are Xi, • • • , X„. Then f ■■ ■ jpiXf) ■• • p(Z„)x(||Zi, • • •, Xn\\)dXi■ ■ -dXnS n2'f xiOdtV"-1 for every non-negative, nonincreasing function x(0, defined for t^O, whose integral foXil)dt converges. Probably, it would be possible to prove Theorem 3, with re2n replaced by a smaller constant if re> 2, using Steiner symmetrization and the methods of §§22-26 in [l]. Our proof will be shorter. We shall first prove Theorem 1, then Theorem 3, finally Theorem 2. 1. We have to use the invariant measure piA) over the space of matrices A of determinant 1, defined by Siegel [10] and further developed in [6; 8] and [9]. uiA) is normalized such that fduiA) = 1, where P is a fundamental region of matrices with respect to the subgroup of unimodular matrices. We mention the following equations, proved in the papers cited above: If S is a Borel set not containing the origin 0, with characteristic function piX), volume V, then (7) f £ PiAg)duiA)= V, J F 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 518 WOLFGANGSCHMIDT [June (8) f Zl . S.. ]P(Ag)dp(A)= Vt(n)~K Jp L primitive J The summation is over lattice-points g. Further, if n>2, (9) f EL gU.gl ]P(Agi)p(Agi)dp(A)= V2, J p Lhn. indep.J and (10) f EL .JUK'. -- \(Agi)P(Agi)dp(A)= V2t(n)-\ J f Lim. mdep. primitive J We observe, again for n>2, Jf f ET/1'/2 l_nn. dep.J \(Agi)p(Agi)dp(A) = ^rflZ2Z\ . g.. 1 P(Apg)p(Aqg)dn(A) 2 J f _>,_ LpnmitiveJ =SE E f El" . *.. ~\P(Aqg)dp(A) _^o pjipiai.i Jf LpnmitiveJ <VZZ E \q\~"<czV, 9^0 3»;lplsl8l where c_ is an absolute constant. Using this result, as well as (7), (8), (9) and (10), we obtain f ( E p(Ag)~ f) dn(A)< c3V and f (Z\ .g.. ]p(Ag) - Van)-^dp(A) < c3V. J f \ LpnmitiveJ / The second inequality was proved in [8] in the same way. We also mention the following integral identity, established by Rogers [8]: (11) f a(A)dA= c.(ra)f "n_1j f o-(vllnA)dp(A)\dv. Here, C is the cone of matrices A which satisfy \AQF for some X^ 1. dA is the euclidean volume element in the «2-dimensional space of matrices, and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1960] A METRICAL THEOREM IN GEOMETRY OF NUMBERS 519 oiA) a matrix-function integrable with respect to dA. c^n) is a constant de- pending on re only. We obtain immediately (12) f(\ZpiAg) - VlUlh1)dA < cbin)vf v-~2dv< Cl(n)V, (13) [(Zi . S.. ~\p(Ag)- Fr(re)iU|hYcP4 < ct(n)V, J c\ LpnmitiveJ / where ||^4|| is the absolute value of the determinant of A. 2. Lemma 1. To every set $ satisfying (i), (ii), there exists a set >IO$, which satisfies (i), (ii) and (iii) To every real number F = 0 there exists a SE^ satisfying F(5) = V. Proof. Write «(<£) for the set of numbers FSgO such that there exists a SE® with F(S) = V. Then (iii) states that «(f) consists of every F^O. We first show: There exists a X3$ satisfying (i), (ii) and (iiia) a(X) is closed. For let V be a limit point of ai®). We first assume there exist Vi ^ V* • • • , VjE<xi$), lim Vj=V. Then there exist Si2S23 • • • , SjE$, ViSf) = Vj. We take S = SiV) = 0,1, Sj. Then 5 is again a Borel set, and F(5) - F. If, however, there exists no sequence Fi= F2^ • • • of the required form, then there exists a sequence Fi= F2^, • • • , VjEoti®), lim Fy= V and we can proceed similarly, taking S = \JfLi Sj. It we take x as the union of ® and the sets 5 just constructed, then X satisfies all required conditions. It remains to show that there exists a^3X, satisfying (i), (ii), and (iii). Suppose FG«(X). Then there existO) FiG«(X), F2G«(X), Vi < V< V2,such that no point in the open interval (Fi, Vf) belongs to a(X). There exist Si, S2 with ViSi) = Vi, ViS2)= Vt. We write St-Si for the set of all XES2, XESi and iS2-Si)t for the set of all XES*-Si with \X\ St. Then F(5iW(52 —Si)/) is a continuous function of t which equals F(5i) when t = 0 and approaches F(52) when t —* oo. Hence there exists a to such that 7(SAJ(S,-Si)„) = V. We introduce SiV) =SiVJiSi-Sf)t, and take ^ to be the union of X and all sets 5(F). 3. According to Lemma 1, we may assume that $ satisfies (i), (ii) and (iii). Hence for every positive integer N, we may pick a SE® with F(5) =N and denote it by SiN). Write Pjv(-X")for the characteristic function of SiN) and N^NiiX) for the characteristic function of SiNf)—SiNf). Finally, we introduce, fol- lowing Cassels [2], i1) We may assume that the null set belongs to X, such that 0G«(X). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 520 WOLFGANGSCHMIDT [June Rn(a) = IZpN(Ag)- n\\a\\-\ 9 MA) = ET . '. ~\pn(AS)- Nt(n)-iA\\-\ LpnmitiveJ nMA) = E s1P*Mg)- w* - tfOlMlh. a NiSNt(A)= Ef . g. ]NiPN2(Ai)- W* - Ni)t;(n)-iA\\-\ LpnmitiveJ Lemma 2. Let P be a positive integer and Kt the set of all pairs of integers Nu Ni of the type 0 fg Ni <Nt fg 2T, Ni = u2*, N2 = (u +1)2', for integers u and t^O. Then (14) E f NlRNi(A)dA fg c6(n)(T+1)2T. (Ni,N2)€KT Jc Proof. (12) yields f Nll&t(A)dA fg ch(n)(Ni - Ni). J c Each value of Ni-Ni = 2' (Ofg/fgT) occurs 2T-' times. Hence E (Ni - Ni) = (T + l)2r. iNi,Nz)eKT Lemma 3. For all sufficiently large T there is a subset BT of C of measure (15) f dA gcs(n)4r*(T log 2- 1), J BT such that (16) Rn(A) fg 7(7 + 1)2V(r log 2 - 1) for ewery A7fg2r and all AQC, but A QBT. Proof. If we take BT to be the set consisting of all AQC for which it is not true that (16a) E NiRl,(A) ^(P+ l)2THP log 2 - 1), (Ni,N2)eKT then (15) follows immediately from Lemma 2.
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