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Parametric Geometry of Numbers and Exponents of Diophantine Approximation

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Parametric Geometry of Numbers and Exponents of Diophantine Approximation U.U.D.M. Project Report 2019:29 Parametric Geometry of Numbers and Exponents of Diophantine Approximation Erik Landstedt Examensarbete i matematik, 30 hp Handledare: Andreas Strömbergsson Examinator: Denis Gaidashev Juni 2019 Department of Mathematics Uppsala University Parametric Geometry of Numbers and Exponents of Diophantine Approximation Erik Landstedt June 2019 2 Acknowledgments I would like to express my gratitude to my supervisor, Professor Andreas Str¨ombergsson, for his guidence and enthusiasm about this project. Your feedback has truly been helpful together with our discussions. 3 4 Contents 1 Introduction 7 2 Preliminaries 9 2.1 Lattices . .9 2.2 Convex Bodies and Minkowski's First Theorem . 12 2.3 Minkowski's Second Convex Body Theorem . 17 2.3.1 Gauge Functions . 17 2.3.2 Successive Minima . 18 2.3.3 Statement of Minkowski's Second Theorem . 19 3 Rigid Systems and Convex Bodies 25 3.1 Convex Bodies and Alternating k-forms . 28 4 The Approximation Theorem of Schmidt and Summerer 35 4.1 Appendix: Translation between Roy's article and the Article by Schmidt and Summer 41 5 Roy's Contribution 45 6 Exponents of Diophantine Approximation 67 6.1 Khinchin's Transference Principle . 69 6.2 A Consequence of Roy's Contribution . 74 5 6 Chapter 1 Introduction Diophantine approximation is the part of number theory that studies how well an arbitrary real number can be approximated by a rational number. The famous Dirichlet approximation theorem asserts that, provided a real number θ and a positive integer n, then there are integers α and β such that 1 jαθ − βj < n where 1 ≤ α ≤ n. It follows that if θ is irrational, then there are infinitely many rational numbers α/β such that α 1 θ − ≤ : β β2 A more general idea is to consider an arbitrary unit vector u 2 Rn and construct a rational codimension one subspace that approximately contains u: Define τ(u) to be the supremum of η > 0 such that the two inequalities jjxjj ≤ Q and jx · uj ≤ Q−η admit a nonzero solution x 2 Zn for arbitrarily large values of Q. Then τ(u) is an example of an exponent of Diophantine approximation. There are other exponents as well, and the approximat- ing subspace does not necessarily need to have codimension 1. The present work is a survey of Roy's article: On Schmidt and Summerer Parametric Geometry of Numbers, [15], where the author develops a framework which allows many types of Diophantine exponents to be understood in a new light. The building blocks of Roy's article are the results from Schmidt and Summerer, [16] and [17], where they construct certain one parameter families of convex bodies. The asymptotical behavior of the successive minima of these bodies for large parameter values is directly connected to a certain family of Diophantine exponents. Schmidt 7 and Summerer show that the successive minima can be approximated with bounded difference by certain functions which are amenable to analysis. Roy constructs his own class of approximating functions, called rigid systems. He also proves that, conversely, given any such rigid system, there exists a point in Rn whose corresponding successive minima approximate the rigid system with a bounded difference. The discussion of this result will form the main part of the present thesis. Chapter two is dedicated to preliminaries where concepts such as convex bodies and successive minima are presented. The necessary foundations from the geometry of numbers are also intro- duced here such as Minkowski's second convex body theorem. In Chapter three we build up Roy's theory regarding rigid systems, starting with the construction of a certain type of convex bodies, in order to be able to present the Approximation Theorem of Schmidt and Summerer. This theorem is presented in Chapter four together with a translation between Schmidt and Summerer's notation and Roy's. In Chapter five we focus on Roy's result regarding the existence of a rigid system for any point in Rn. Finally, in Chapter 6, we discuss some of the foundations of Diophantine exponents and explain the connection between these and Roy's result. 8 Chapter 2 Preliminaries 2.1 Lattices The main objective in this introductory section will be the study of lattices and we will introduce some fundamental concepts and results from the geometry of numbers. The content can be found in [19]. n Definition 1. For any linearly independent vectors e1; :::; em in R (m ≤ n), the additive subgroup n of (R ; +) generated by e1; :::; em is called a lattice of dimension (or rank) m. We will denote lattices with Λ or Γ. This means that a lattice of rank m is an abelian group of the form m M Λ = Zek k=1 and one immediate example is (Z × Z; +). This is a lattice of rank 2 in R2 and the geometric interpretation can be seen in Figure 2.1. Intuitively we feel that a lattice should be a discrete subset of Rn and this is true together with the converse. It is possible to show that an additive subgroup of Rn is a lattice if and only if it is discrete. 2 Example 2. Take e1 = (1; 1) and e2 = (0; 1) in R . Then the additive group Λ1 = (fα1e1 + α2e2 : α1; α2 2 Zg; +) 2 2 is a discrete subgroup of R and hence a lattice. Notice that Λ1 in fact will be Z . In the same n way it can be noted that if fe1; :::; eng is a basis in R , then ( n ) ! X Λ2 = αkek : αk 2 Z ; + k=1 will be a lattice in Rn. 4 9 Figure 2.1: This is an example of a lattice of rank 2 in R2. Hence it is of full rank. Definition 3. Given a lattice Λ of full rank in Rn, any set of the form ( n ) X ajej : a1; :::; an 2 [0; 1) j=1 n where fe1; :::; eng is a set of generators of Λ, is called a fundamental domain of R =Λ. Let Λ be a lattice of rank n in Rn. Then the quotient group Rn=Λ is isomorphic to the n-dimensional n ∼ torus, which we will denote by T . This follows from the fact that R=Z = S where S denotes the circle group. The latter follows using the map ' : R ! Z defined by '(x) = e2πix; ∼ which is clearly a surjective homeomorphism and its kernel is Z. Then R=Z = S follows from the n n ∼ first group isomorphism theorem. Then it is natural to define : R ! T = S × · · · × S via n ! X 2πia1 2πian ajej := (e ; :::; e ) j=1 10 n ∼ n n and the isomorphism R =Λ = T follows analogously. The volume of a subset D of R will be denoted by vol(D): Z vol(D) := dx1:::dxn: D It is also necessary to define a volume measure on Tn and to do this we let π : Rn ! Rn=Λ be the canonical homomorphism and put for any smooth subset X ⊂ Rn=Λ vol(X) := vol(T \ π−1(X)); (2.1) where T is a fundamental domain of Λ. This volume measure will indeed be well-defined and this statement is formulated as the following proposition: Proposition 4. The volume measure, vol, on T defined above is well-defined. Proof. To prove this, let T and T 0 be two fundamental domains for Tn. The goal is to show that vol(T \ X) = vol(T 0 \ X) (2.2) where X is any Lebesgue measurable subset of Rn such that we for all x 2 X and λ 2 Λ have x + λ 2 X. We will need the following lemma: Lemma 5. For every x 2 Rn there exists a unique λ 2 Λ such that x 2 λ + T , where T is a fundamental domain of Rn=Λ. Proof. With n X x = ckek k=1 where c1; :::; cn 2 R we can take n X λ = bckcek: k=1 Then it follows that every element of Rn=Λ has a unique representation in T . Lemma 5 above now gives that n [ 0 R = (T + λ) λ2Λ and hence [ T \ X = (T \ X \ (T 0 + λ)) λ2Λ 11 and since these sets are pairwise disjoint it follows that X vol(T \ X) = vol (T \ X \ (T 0 + λ)) : λ2Λ In a similar way one gets X vol(T 0 \ X) = vol (T \ X \ (T 0 + λ)) λ2Λ and the equality in (2:2) is obtained. 2.2 Convex Bodies and Minkowski's First Theorem Convex bodies play a fundamental role in the geometry of numbers and we will see that many theorems require the relevant set to be e.g. convex. What is normally called Minkowski's first theorem asserts that if we have a convex body of a certain size, then it must contain a point of a given lattice. The precise statement and proof will be discussed below. One of the main objects that are studied in both [16] and in [15] are convex bodies but these are also used in many other areas of mathematics such as optimization theory and geometry. Definition 6. Let S be a subset of Rn. Then S is called convex if for any x; y 2 S and λ 2 [0; 1] we have λx + (1 − λ)y 2 S. The expression λx + (1 − λ)y (with λ 2 [0; 1]) is often referred to as a convex combination. Moreover, S is called a convex body if S is a compact, convex and symmetric region that contains a neighborhood of the origin.
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