Lecture Notes on Geometry of Numbers Martin Henk May 12, 2015 Contents Once upon a time... iii 1 Basic (and some more) facts from Convexity 1 Exercises 13 Notes 15 2 Basic facts about Lattices 17 3 Minkowski's successive minima 31 References 41 i ii Once upon a time... One of the main research directions of number theory around the middle of the 19th century was devoted to the study of general quadratic forms q(x) n in n variables x = (x1; : : : ; xn)| and its values on integral points z 2 Z . In particular, for the class of positive definite quadratic forms, i.e., we can write n×n q(x) = x|Qx with a positive definite matrix Q 2 R , one was interested n in upper bounds on its minimum on Z n f0g, i.e., minz2Znnf0g q(z). Of course, such a bound has to depend on Q and since Acknowledgement. For helpful remarks, corrections and discussions I would like to thank Andr´eApitzsch, Eva Linke, Mar´ıade los Angeles Hern´andez Cifre, Eva Linke,.... iii 1 Basic (and some more) facts from Convexity In this chapter we will briefly collect some basic results and concepts con- cerning convex bodies, which may be considered as the main "continuous" component/ingredient of the Geometry of Numbers. For proofs of the results we will refer to the literature, mainly to the books [2, 5, 6, 7, 9, 11, 13, 14], which are also excellent sources for further information on the state of the art concerning convex bodies, polytopes and their applications. As usual we will start with setting up some basic notation; more specific ones will be introduced when needed. n Let R = fx = (x1; : : : ; xn)| : xi 2 Rg be the n-dimensional Euclidean Pn space, equipped with the standard inner product hx; yi = i=1 xi yi, x; y 2 n R . n For a subset X ⊆ R , let lin X and aff X be its linear and affine hull, respectively, i.e., with respect to inclusions, the smallest linear or affine subspace containing X. If X = ; we set lin X = f0g, and aff X = ;. The dimension of a set X is the dimension of its affine hull and will be denoted n by dim X. By X +Y we mean the vectorwise addition of subsets X; Y ⊆ R , i.e., X + Y = fx + y : x 2 X; y 2 Y g: n The multiplication λ X of a set X ⊆ R with λ 2 R is also declared vec- torwise, i.e., λ X = fλ x : x 2 Xg. We write −X = (−1) X, X − Y = X + (−1) Y and x + Y instead of fxg + Y . n A subset C ⊆ R is called convex, iff for all c1; c2 2 C and for all λ 2 [0; 1] hold λ c1 + (1 − λ) c2 2 C. A convex compact (i.e., bounded and n closed) subset K ⊂ R is called a convex body. The set of all convex bodies n n n in R is denoted by K . K 2 K is called o-symmetric, i.e., symmetric with respect to the origin, if K = −K. The set of all o-symmetric convex bodies n n in R is denoted by Ko . There is a simple one-to-one correspondence between n-dimensional o- n n symmetric convex bodies and norms on R : given a norm j · j : R ! R≥0 1 2 1. Basic (and some more) facts from Convexity n we consider its unit ball fx 2 R : jxj ≤ 1g, which is a o-symmetric convex body by the definition of a norm. For the reverse correspondence we need n the notion of a distance function. For K 2 Ko , dim K = n, the function n |·|K : R ! R≥0 given by jxjK := minfρ 2 R≥0 : x 2 ρ Kg is called the distance function of K. By the convexity and symmetry of K it is easily verified that |·|K is indeed a norm. A well-studied family of o-symmetric convex bodies are the p n unit balls Bn := fx 2 R : jxjp ≤ 1g associated to the p-norms n !1=p X p jxj := jxij ; p 2 [1; 1); and jxj := max fjxijg: p 1 1≤i≤n i=1 In the Euclidean case, i.e., p = 2, we denote the Euclidean norm just by |·| and its unit ball by Bn. 0 −1 1 1 1 1 −1 1 0 0 0 0 0 0 0 −1 1 −1 −1 −1 p = 1 p = 2 p = 4 p = 1 Figure 1.1: Unit balls of p-norms 1 n Bn = fx 2 R : −1 ≤ xi ≤ 1g is the o-symmetric (regular) cube with 1 n Pn edge length 2, and Bn = fx 2 R : i=1 jxij ≤ 1g is called the (regular) crosspolytope. One big advantage of convex sets K is their property that points not belonging to K can easily be separated from K. In order to formulate this n precisely and a bit more generally, we denote for a 2 R n f0g, b 2 R n by H(a; b) = fx 2 R : ha; xi = bg the hyperplane with normal vector a and right hand side b. The associated two closed halfspaces are denoted by n H≤(a; b) and H≥(a; b), respectively, i.e., H≤(a; b) = fx 2 R : ha; xi ≤ bg and H≥(a; b) = H≤(−a; −b). n 1.1 Theorem [Separation Theorem]. Let C1;C2 ⊂ R be convex with C1 \ C2 = ;. i) Then there exists a separating hyperplane H(a; b), i.e., C1 ⊆ H≤(a; b) and C2 ⊆ H≥(a; b). n ii) If C1 is compact, i.e., C1 2 K and C2 is closed then there exists a strictly separating hyperplane H(a; b), i.e., C1 ⊂ H≤(a; b) and C2 ⊆ H≥(a; b) and C1 \ H(a; b) = ; = C2 \ H(a; b). 1. Basic (and some more) facts from Convexity 3 H≤(a; b) H≥(a; b) C2 C1 H(a; b) Figure 1.2: Strictly separating hyperplane of two convex bodies A hyperplane H(a; b) passing through the boundary of a closed convex set C and containing C entirely in one of the two halfspaces, i.e., C \ H(a; b) 6= ; and C ⊆ H≤(a; b), say, is called a supporting hyperplane of C. The intersection of F = C \ H(a; b) is called a face of C, or more precisely, a k-face if dim F = k. If k = 0 then F is called an extreme point of C. Faces themselves are closed convex sets, and the interesection of faces of C is again a face of C, where we regard the empty set as the (−1)-dimensional face of C. The boundary of C, denoted by bd C, is the union of its faces. For a convex body K 2 Kn and a given normal vector (direction) a 2 n R n f0g the supporting hyperplane in direction a is given by H(a; hK (a)) where n hK : R ! R with hK (a) := maxfha; xi : x 2 Kg is called the support function of K. The support function is positive homoge- neous, i.e., hK (λ a) = λ hK (a), λ 2 R≥0, and sub-additive, i.e., hK (x + y) ≤ n hK (x) + hK (y) for all x; y 2 R . Indeed, these two properties characterize n support functions of convex bodies among the real-valued functions on R . A quite general concept in different branches of mathematics is that of n duality and/or polarity. In our setting it is defined as follows: for X ⊆ R , the set ? n X := y 2 R : hx; yi ≤ 1 for all x 2 X is called the polar set of X. As the intersection of convex closed sets con- taining the origin, the polar set is always a convex closed set containing 0. Its actual shape, however, depends on its position relative to the origin. For instance, X? is bounded if and only if 0 is an interior point of X. On the other hand, as stated in the next proposition, its shape be- haves properly with respect to linear transformations and a polar set of a o-symmetric convex body is again an o-symmetric convex body. 1.2 Proposition. n ? − ? i) Let X ⊆ R , and let M 2 GL(n; R). Then (MX) = M |X . Here GL(n; R) denotes the general linear group of all invertible real n × n 4 1. Basic (and some more) facts from Convexity 1=2 ? ? 1=2 0 0 0 0 1=2 −1=2 1 1 1 Figure 1.3: Polar bodies of B2 and 0 + B2 matrices. n ? ? ii) Let K ⊆ R be convex and closed with 0 2 K. Then (K ) = K. n ? n iii) Let K 2 Ko . Then K 2 Ko and jxjK? = maxy2K hx; yi = hK (x). By Proposition 1.2 iii) and H¨older'sinequality for sums, n X n jxk ykj ≤ jxjp jyjp=(p−1) ; x; y 2 R ; p 2 (1; 1); i=1 p−1 with equality if and only if jxij = c jyij for a constant c, the polar bodies of the p-norms are given by p p ? p−1 1 ? 1 1 ? 1 (Bn) = Bn ; 1 < p < 1; and (Bn) = Bn and thus (Bn ) = Bn: Since we are mainly dealing with simple structured bounded convex sets, we define the volume of sets via the Jordan-Peano measure: A subset D ⊂ n n R is called (Jordan-Peano) measurable if its indicator function χD : R ! f0; 1g given by ( 1; x 2 D; χD(x) := 0; x 2= D; is Riemann integrable, i.e., the Riemann integral Z vol(D) := χD(x) dx Rn exists.