An Introduction to the Geometry of Numbers

Prologue.- I. Lattices.- 1. Introduction.- 2. Bases and sublattices.- 3. Lattices under linear transformation.- 4. Forms and lattices.- 5. The polar lattice.- II. Reduction.- 1. Introduction.- 2. The basic process.- 3. Definite quadratic forms.- 4. Indefinite quadratic forms.- 5. Binary cubic forms.- 6. Other forms.- III. Theorems of BLICHFELDT and MINKOWSKI.- 1. Introduction.- 2. BLICHFELDT'S and MINKOWSKI'S theorems.- 3. Generalisations to non-negative functions.- 4. Characterisation of lattices.- 5. Lattice constants.- 6. A method of MORDELL.- 7. Representation of integers by quadratic forms.- IV. Distance functions.- 1. Introduction.- 2. General distance-functions.- 3. Convex sets.- 4. Distance functions and lattices.- V. MAHLER'S compactness theorem.- 1. Introduction.- 2. Linear transformations.- 3. Convergence of lattices.- 4. Compactness for lattices.- 5. Critical lattices.- 6. Bounded star-bodies.- 7. Reducibility.- 8. Convex bodies.- 9. Spheres.- 10. Applications to diophantine approximation.- VI. The theorem of MINKOWSKI-HLAWKA.- 1. Introduction.- 2. Sublattices of prime index.- 3. The Minkowski-Hlawka theorem.- 4. SCHMIDT'S theorems.- 5. A conjecture of ROGERS W.- 6. Unbounded star-bodies.- VII. The quotient space.- 1. Introduction.- 2. General properties.- 3. The sum theorem.- VIII. Successive minima.- 1. Introduction.- 2. Spheres.- 3. General distance-functions.- 4. Convex sets.- 5. Polar convex bodies.- IX. Packings.- 1. Introduction.- 2. Sets with V(L) = 2n?(L).- 3. VORONOI'S results.- 4. Preparatory lemmas.- 5. FEJES TÓTh'S theorem.- 6. Cylinders.- 7. Packing of spheres.- 8. The product of n linear forms.- X. Automorphs.- 1. Introduction.- 2. Special forms.- 3. A method of MORDELL.- 4. Existence of automorphs.- 5. Isolation theorems.- 6.Applications of isolation.- 7. An infinity of solutions.- 8. Local methods.- XI. Inhomogeneous problems.- 1. Introduction.- 2. Convex sets.- 3. Transference theorems for convex sets.- 4. The product of n linear forms.- References.