Number Theory: An Introduction to Mathematics

326 VII The Arithmetic of Quadratic Forms [2] T. Beth, D. Jungnickel and H. Lenz,Design theory, 2nd ed., 2 vols., Cambridge Univ ...

VIII The Geometry of Numbers.................................... It was shown by Hermite (1850) that if f(x)=xtAx is a positive ...

328 VIII The Geometry of Numbers contained in a hyperplane, and 0<λ(C)<∞if and only ifCis bounded and is not contained in ...

1 Minkowski’s Lattice Point Theorem 329 The continued fraction algorithm enables one to find rational approximations to irration ...

330 VIII The Geometry of Numbers Choosing somet′> 2 /δ,wefindanewsetofintegersq 1 ′,...,qm′,p′ 1 ,...,p′nsat- isfying the sam ...

2 Lattices 331 The converse of Proposition 6 is also valid. In fact we will prove a sharper result: Proposition 7IfΛis a discret ...

332 VIII The Geometry of Numbers As forS 1 , it may be seen thatSk+ 1 consists of all positive integer multiples ofμk+ 1. Hence ...

2 Lattices 333 more general definition and, with this warning, believe it will be clear from the context when this occurs. The b ...

334 VIII The Geometry of Numbers Rn=∪ x∈Λ (Π+x), int(Π+x)∩int(Π+x′)=∅ ifx,x′∈Λandx=x′. For any latticeΛ⊆Rn,thesetΛ∗of all vecto ...

3 Proof of the Lattice Point Theorem; Other Results 335 xj−xk=zj−zk∈Λ( 1 ≤j,k≤m+ 1 ). Suppose next thatSis compact andλ(S)=md(Λ) ...

336 VIII The Geometry of Numbers ∫ Π |φ(x)|^2 dx= ∑ w∈Zn |cw|^2 , where cw= ∫ Π φ(x)e−^2 πiw tx dx. But cw= ∫ Π ∑ z∈Zn Ψ(x+z)e−^ ...

3 Proof of the Lattice Point Theorem; Other Results 337 Hence, by Proposition 10, sup x∈Rn φ(x)≥ ∫ Rn Ψ(x)dx. For example, letS⊆ ...

338 VIII The Geometry of Numbers We can chooseε>0 so small that the ballsBε+x 1 andBε+x 2 are disjoint and contained inG.Then ...

3 Proof of the Lattice Point Theorem; Other Results 339 Put yj=(xj−xm+ 1 )/ 2 (j= 1 ,...,m). Then, by construction,yj∈Λ(j= 1 ,.. ...

340 VIII The Geometry of Numbers It follows from Proposition 12 that∆(K)≥ 2 −nλ(K). A conjectured sharpening of Minkowski’s theo ...

3 Proof of the Lattice Point Theorem; Other Results 341 m <n, such that|αjk|≤βfor allj,k, then the system of homogeneous line ...

342 VIII The Geometry of Numbers 4 Voronoi Cells Throughout this section we supposeRnequipped with theEuclidean metric: d(y,z)=‖ ...

4 Voronoi Cells 343 It follows at once from (2) thatV(x 0 )is closed and convex. HenceV(x 0 )is the closure of its nonempty inte ...

344 VIII The Geometry of Numbers In the representation (3) it may be possible to omit some closed half-spacesG ̄xi without affec ...

4 Voronoi Cells 345 ‖x/ 2 ‖<‖z−x/ 2 ‖, i.e.x/ 2 ∈Gz.Since‖x/ 2 ‖=‖x−x/ 2 ‖, it follows thatx/ 2 ∈V(Λ)andxis a facet vector. S ...