On values of linear and quadratic forms at integral points

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Let N be the subgroup of SL(n, IR) consisting of all upper triangular matriceswith 1’s on the diagonal and for any τ > 0 let N τ denote the subsetof N consisting of all those matrices for which every off-diagonal entryis of absolute value at most τ. Also let K denote the subgroup ofSL(n, IR) consisting of orthogonal matrices of determinant 1. A set of theform KD σ N τ = {kdu | k ∈ K, d ∈ D σ , u ∈ N τ } is called a Siegel set. Itcan be verified that any Siegel set is of finite Haar measure in SL(n, IR). Itis a crucial fact that for suitable values of σ and τ the corresponding Siegelset is a fundamental domain for SL(n, Z) in SL(n, IR), namely we have thefollowing.Theorem 2.1 (cf. [21], Ch. X). Let F = KD 2/√3N 1/2 . ThenSL(n, IR) = F (SL(n, Z)).Concerning lattice points this implies that if e 1 is the column vector inwhich the first row entry is 1 and the others are 0 then the set F e 1 = {ge 1 |g ∈ F } contains a nonzero point of any lattice Λ ∈ L 1 ; (though the argumentcould be applied to other integral points in the place of e 1 , it does not lead toany useful information when the point is not a multiple of e 1 ). Now, F e 1 canbe seen to be the complement of {0} in the closed ball of radius (4/3) n−14 ,namely the maximum possible value for the first entry of any element ofD√2/ 3. This establishes the result of Hermite mentioned earlier, that a ballcentered at 0 contains a nonzero point of any lattice from L 1 if its radiusexceeds (4/3) n−14 ; for a closed ball conclusion holds for radius (4/3) n−14 aswell. Since any positive definite quadratic form Q on IR n is equivalent tothe quadratic form Q 0 given by the square of the usual norm (namely thereexists a g ∈ GL(n, IR) such that Q(v) = Q 0 (gv) for all v ∈ IR n ) this impliesthe following.Corollary 2.2. Let Q be a positive definite quadratic form on IR n . Thenthere exists x ∈ Z n such that Q(x) ≤ (4/3) (n−1)/2 d(Q) 1/n , where d(Q) denotesthe discriminant of Q.3 A theorem of Howe and MooreIn this section we recall a theorem of Howe and Moore, specialised to the caseof the homogeneous space SL(n, IR)/SL(n, Z), and discuss its implications4
to lattices in IR n . As before we shall denote by m the normalised SL(n, IR)-invariant measure on L 1 = SL(n, IR)/SL(n, Z).Theorem 3.1 (Howe and Moore; see [15]). Let {g i } be a divergentsequence in SL(n, IR) (namely with no limit point in SL(n, IR)). Then forany square-integrable functions f and φ on L 1 we have∫f(g i x)φ(x) dm →L 1∫f dmL 1 ∫φ dm,L 1 as i → ∞.This implies the following corollary; it may be mentioned here that thecorollary generalises a theorem proved in a number-theoretic context byW.M. Schmidt, (see [23]); the result of Schmidt is the particular case of thecorollary when the sequence {g i } is chosen to be {diag (i, i, . . . , i, i −n+1 )}.Corollary 3.2. Let {g i } be a divergent sequence in SL(n, IR). Then foralmost every lattice Λ ∈ L 1 the set of lattices {g i Λ} is dense in L 1 .For convenience we shall include here a proof of the corollary. Beforegoing over to the proof let me however note the following. In the corollary‘almost every’ is meant in the sense that the set of lattices Λ ∈ L 1 for whichthe statement does not hold (for a given sequence {g i } as in the hypothesis)is a set of m-measure 0, where m is the SL(n, IR)-invariant measure on L 1 .In the set of all lattices (not only from L 1 ) one can also interpret ‘almostevery’ to mean a lattice generated by n linearly independent vectors v 1 , . . . , v nwhich may be chosen from a set of full Lebesgue measure. It can be seenthat the corollary means also that for almost every lattice Λ in this sense{g i Λ} is dense in the set of lattices whose discriminant coincides with thatof Λ (we shall however not go into the details of this).Proof of the Corollary: Firstly we observe that as a consequence of the theorem,for any measurable subset E of L 1 such that m(E) > 0 we havem( ⋃ i g−1 i (E)) = 1; for if E is such a set and f and φ are the characteristicfunctions of E and ⋃ i g−1 i (E) respectively then we have f(g i x)φ(x) =f(g i x) for all x ∈ L 1 , for all i. Hence by the theorem ( ∫ f dm)( ∫ ∫φ dm) =f(gi x) dm(x) = ∫ f dm and since ∫ f dm = m(E) > 0 this implies thatm( ⋃ i g−1 i (E)) = ∫ φ dm = 1, as claimed. Now let {Ω j } ∞ j=1 be a countablebasis for the topology of L 1 . Then for all j we have m(Ω j ) > 0 and hencem( ⋃ i g−1 i (Ω j )) = 1. Put Y = ⋂ j⋃i g−1 i (Ω j ). Then m(Y ) = 1 and for anyΛ ∈ Y , {g i Λ} is dense in L 1 .5
Page 6 and 7: The corollary implies in particular
Page 8 and 9: 5 The Oppenheim conjectureWe have s
Page 10 and 11: 6 Complements to the Oppenheim conj
Page 12 and 13: 7 Simultaneous solution of quadrati
Page 14 and 15: |e t x 3 | = |x 1 x 3 |/ɛ ≤ ɛ a
Page 16 and 17: 9 Comments in conclusionThe study o
Page 18: [17] D. Kleinbock and G.A. Margulis

On values of linear and quadratic forms at integral points

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