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Mahler's Work on the Geometry of Numbers

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Mahler's Work on the Geometry of Numbers MAHLER’S WORK ON THE GEOMETRY OF NUMBERS JAN-HENDRIK EVERTSE Mahler has written many papers on the geometry of numbers. Ar- guably, his most influential achievements in this area are his compact- ness theorem for lattices, his work on star bodies and their critical lat- tices, and his estimates for the successive minima of reciprocal convex bodies and compound convex bodies. We give a, by far not complete, overview of Mahler’s work on these topics and their impact. 1. Compactness theorem, star bodies and their critical lattices Many problems in the geometry of numbers are about whether a particular n-dimensional body contains a non-zero point from a given lattice, and quite often one can show that this is true as long as the determinant of the lattice is below a critical value depending on the given body. Mahler intensively studied such problems for so-called star bodies. Before mentioning some of his results, we start with recalling some definitions. We follow [22]. Let n 2 be an integer that we fix henceforth. A distance function ≥ on Rn is a function F : Rn R such that: → (i) F (x) 0 for all x Rn and F (x) > 0 for at least one x; ≥ ∈ (ii) F (tx)= t F (x) for x Rn and t R; | |· ∈ ∈ (iii) F is continuous. A (symmetric) star body in Rn is a set of the shape arXiv:1806.00356v1 [math.HO] 31 May 2018 = x Rn : F (x) 1 , S { ∈ ≤ } where F is a distance function. We call the star body with distance S function F . The boundary of is x Rn : F (x)=1 , and the S { ∈ } 2010 Mathematics Subject Classification: 11H06, 11H16, 11H60. Keywords and Phrases: star bodies, critical lattices, compound convex bodies, successive minima, transference principles. June 4, 2018. 1 2 J.-H. EVERTSE interior of is x Rn : F (x) < 1 . The set is bounded, if and only S { ∈ } S if F (x) > 0 whenever x =0. The star bodies contain as a subclass the 6 symmetric convex bodies, which correspond to the distance functions F satisfying in addition to (i),(ii),(iii) the triangle inequality F (x + y) ≤ F (x)+ F (y) for x, y R. n ∈ Let Λ = z a : z ,...,z Z be a lattice in Rn with basis { i=1 i i 1 n ∈ } a1,..., an .P We define its determinant by d(Λ) := det(a1,..., an) . { } | | Let be a star body. We call Λ -admissible if 0 is the only point of S S Λ in the interior of . The star body is called of finite type if it has S S admissible lattices, and of infinite type otherwise. Bounded star bodies are necessarily of finite type, but conversely, star bodies of finite type do not have to be bounded. For instance, let := x =(x ,...,x ) S { 1 n ∈ Rn : x x 1 . Take a totally real number field K of degree n, | 1 ··· n| ≤ } denote by O its ring of integers, and let α α(i) (i = 1,...,n) be K 7→ the embeddings of K in R. Then (α(1),...,α(n)): α O is an { ∈ K} -admissible lattice. S Assume henceforth that is a star body of finite type. Then we can S define its determinant, ∆( ) := inf d(Λ): Λ admissible lattice for . S { S} Thus, if Λ is any lattice in Rn with d(Λ) < ∆( ), then contains a non- S S zero point from Λ. The quantity ∆( ) cannot be too small. From the S Minkowski-Hlawka theorem (proved by Hlawka [8] and earlier stated without proof by Minkowski) it follows that ∆( ) > (2ζ(n))−1V ( ), ∞ S S where ζ(n)= k−n and V ( ) is the volume (n-dimensional Lebes- k=1 S gue measure)P of . S We call Λ a critical lattice for if Λ is -admissible and d(Λ) = ∆( ). S S S In a series of papers [17, 18, 19, 20, 21] Mahler studied star bodies in R2, proved that they have critical lattices, and computed their determinant in various instances. Later, Mahler picked up the study of star bodies of arbitrary dimension [22]. We recall Theorem 8 from this paper, which is Mahler’s central result on star bodies. Theorem 1.1. Let be a star body in Rn of finite type. Then has S S at least one critical lattice. The main tool is a compactness result for lattices, also due to Mahler. We say that a sequence of lattices Λ ∞ in Rn converges if we { m}m=1 MAHLER’S WORK ON THE GEOMETRY OF NUMBERS 3 can choose a basis am,1,..., am,n of Λm for m = 1, 2,... such that aj := limm→∞ am,j exists for j = 1,...,n and a1,..., an are linearly independent. We call the lattice Λ with basis a1,..., an the limit of the sequence Λ ∞ ; it can be shown that this limit, if it exists, is unique. { m}m=1 Denote by x the Euclidean norm of x Rn. The following result, k k ∈ which became known as Mahler’s compactness theorem or Mahler’s selection theorem and turned out to be a valuable tool at various places other than the geometry of numbers, is Theorem 2 from [22]. Theorem 1.2. Let ρ > 0, C > 0. Then any infinite collection of lattices Λ in Rn such that min x : x Λ 0 ρ and d(Λ) C {k k ∈ \{ }} ≥ ≤ has an infinite convergent subsequence. We recall the quick deduction of Theorem 1.1. Proof of Theorem 1.1. By the definition of ∆( ), there is an infinite S sequence Λ ∞ of -admissible lattices such that ∆( ) d(Λ ) { m}m=1 S S ≤ m ≤ ∆( )+1/m for m =1, 2,.... Since 0 is an interior point of ,there is S S ρ> 0 such that x Rn : x ρ . hence x ρ for every non- { ∈ k k ≤ } ⊆ S k k ≥ zero x Λ and every m 1. Further, the sequence d(Λ ) is clearly ∈ m ≥ { m } bounded. So by Theorem 1.2, Λ has a convergent subsequence. { m} After reindexing, we may write this sequence as Λ ∞ and denote { m}m=1 its limit by Λ. We show that Λ is a critical lattice for . S Choose bases am,1,..., am,n of Λm for m =1, 2,... and a1,..., an of Λ such that a a for j =1,...,n. Clearly d(Λ) = lim →∞ d(Λ )= m,j → j m m ∆( ). To prove that Λ is -admissible, take a non-zero x Λ and S S 0 ∈ assume it is in the interior of . Then there is ǫ> 0 such that all x Rn S n ∈ with x x0 < ǫ are in the interior of . Write x0 = i=1 ziai with k − k n S zi Z, and then xm = ziam,i for m 1, so that xPm Λm 0 . ∈ i=1 ≥ ∈ \{ } For m sufficiently large,Pxm x0 < ǫ, hence xm is in the interior of , k − k S which is however impossible since Λ is -admissible. This completes m S the proof. In [22], Mahler made a further study of the critical lattices of n- dimensional star bodies. Among other things he proved [22, Theorem 11] that if is any bounded n-dimensional star body and Λ a critical S lattice for , then there are n linearly independent points of Λ lying S on the boundary of . If P ,...,P are such points, then the 2n points S 1 n P ,..., P lie on the boundary of . A simple consequence of this ± 1 ± n S 4 J.-H. EVERTSE is, that any lattice of determinant equal to ∆( ) has a non-zero point S either in the interior or on the boundary of . Mahler showed further S [22, Corollary on p. 165] that for any integer m n there exist an n- ≥ dimensional star body and a critical lattice Λ of having precisely S S 2m points on the boundary of . S In an other series of papers on n-dimensional star bodies [23] Mahler introduced the notions of reducible and irreducible star bodies. A star body is called reducible if there is a star body ′ which is strictly S S contained in and for which ∆( ′)=∆( ), and otherwise irreducible. S S S An unbounded star body of finite type is called boundedly reducible if S there is a bounded star body ′ contained in such that ∆( ′)=∆( ). S S S S Mahler gave criteria for star bodies being (boundedly) reducible and deduced some Diophantine approximation results. To give a flavour we mention one of these results [23, Theorem P, p. 628]: Theorem 1.3. There is a positive constant γ such that if β1, β2 are any real numbers and Q is any number > 1, then there are integers v1, v2, v3, not all 0, such that v v (β v + β v + v ) 1, | 1 2 1 1 2 2 3 | ≤ 7 x Q, x Q, β v + β v + v γQ−2. | 1| ≤ | 2| ≤ | 1 1 2 2 3| ≤ Idea of proof. Let be the set of x = (x , x , x ) R3 given by S 1 2 3 ∈ x x x 1. By a result of Davenport [3], is a finite type star body | 1 2 3| ≤ S and has determinant ∆( )=7. Mahler [23, Theorem M, p. 527] proved S that is in fact boundedly reducible, which implies that there is r> 0 S ′ such that the star body given by x x x 1 and max ≤ ≤ x r S | 1 2 3| ≤ 1 i 3 | i| ≤ also has determinant 7. Now let Λ be the lattice consisting of the points rQ−1v , rQ−1v , 7r−2Q2(β v +β v +v ) with v , v , v Z.
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