Problem 11 Ist \(a_{1}, \ldots, a_{p} \in \... [FREE SOLUTION] (2024)

Chapter 3: Problem 11

Ist $a_{1}, \ldots, a_{p} \in \mathbb{R}^{p}$ eine Basis des $\mathbb{R}^{p}$,so heißen $\Gamma:=\mathbb{Z} a_{1} \oplus \ldots \oplus \mathbb{Z} a_{p}$ einGitter im $\mathbb{R}^{p}, a_{1}, \ldots, a_{p}$ eine $\mathbb{Z}$-Basis von$\Gamma$ und $$ P:=\left\\{\lambda_{1} a_{1}+\ldots \lambda_{p} a_{p}: 0 \leq \lambda_{j}<1,j=1, \ldots, p\right\\} $$ ein Fundamentalparallelotop von $\Gamma . P$ ist ein Vertretersystem derNebenklassen aus $\mathbb{R}^{p} / \Gamma$. a) $\lambda^{p}(P)$ hat unabhängig von der Wahl der $\mathbb{Z}$-Basis von$\Gamma$ stets denselben Wert, und dieser ist gleich$\left|\operatorname{det}\left(a_{1}, \ldots, a_{p}\right)\right|$ b) Für $R \rightarrow \infty$ gilt $$ |\\{x \in \Gamma:\|x\| \leqR\\}|=\frac{\lambda^{p}\left(K_{1}(0)\right)}{\lambda^{p}(P)}R^{p}+O\left(R^{p-1}\right). $$ (Zur Erinnerung: Sind $f, g:\left[a, \infty\left[\rightarrow\mathbb{C}\right.\right.$ zwei Funktionen, so bedeutet ${ }_{\text {" }}f(t)=O(g(t))$ für $t \rightarrow \infty$ " definitionsgemäß, daß $|f(t)| \leqC|g(t)|$ für alle $t \geq t_{0}$ mit geeignetem $C>0, t_{0} \geq a .$.) c) Es seien $M \in \mathfrak{L}^{p}$ und $\lambda^{p}(M \cap(M+g))=0$ für alle$g \in \Gamma, g \neq 0 .$ Dann ist $\lambda^{p}(M) \leq \lambda^{p}(P)$. d) Ist $K \subset \mathbb{R}^{p}$ eine kompakte Menge mit $\lambda^{p}(K) \geq\lambda^{p}(P)$, so gibt es $x, y \in K, x \neq y$ mit $x-y \in \Gamma$ (H.F.BLICHFELD (1914)) e) Aussage d) wird schon für $p=1$ falsch, wenn „kompakt" durch,abgeschlossen" ersetzt wird. f) Es sei $C \subset \mathbb{R}^{p}$ eine kompakte, konvexe und bez. 0symmetrische (d.h $x \in C \Longrightarrow-x \in C$ ) Menge mit$\lambda^{p}(C) \geq 2^{p} \lambda^{p}(P) .$ Dann gibt es ein $x \in C \cap\Gamma$ mit $x \neq 0$ (Gitterpunktsatz von $\mathrm{H}$. MINKOWSKI (1896)).(Hinweis: d).)

Short Answer

Expert verified

To solve this exercise, you would have to verify each sub-point. Some of these require mathematical proofs, while others are simply statements to be understood in the context of the given problem. These will test your understanding and knowledge of concepts related to lattices, volumes of multi-dimensional parallelepipeds, proofs of theorems, and their applications in the case of higher dimensional Euclidean spaces.

Step by step solution

Identify the Basis and Create the Lattice $\Gamma$

Identify the basis vectors $ a_{1}, a_{2}, ..., a_{p} $ and define the lattice $ \Gamma $ as the set of all integer linear combinations of these vectors.

Define the Fundamentalparallelotop $ P $

Define $ P $ as the set of all linear combinations of the basis vectors, where the coefficients $ \lambda_{j} $ are between 0 and 1.

Analyze the Independent Value of $ \lambda^{p}(P) $

Understand that the volume of the Fundamentalparallelotop or its Lebesgue measure $ \lambda^{p}(P) $, remains the same irrespective of the $ \mathbb{Z} $-basis chosen from $ \Gamma $. And this value corresponds to the absolute value of the determinant of the matrix formed by basis vectors, $ |det(a_{1}, a_{2}, ..., a_{p})| $.

Verify the Asymptotic Relation

Verify the statement that the cardinality of the set of all lattice points within a ball of radius $ R $, behaves asymptotically as the ratio of the volume of the ball to that of the Fundamentalparallelotop multiplied with $ R^{p} $, as $ R $ goes to infinity.

Compare Measure of $ M $ and $ P $

If $ M $ is a measurable set in $ \mathbb{R}^{p} $, such that for any non-zero lattice point $ g $, the intersection of $ M $ and its translated set $ M+g $ has a null measure, then the measure of $ M $ must be less than or equal to that of $ P $.

Blichfeld's Theorem

Examine Blichfeld's Theorem in this context. It states that if $ K $ is a compact subset of $ \mathbb{R}^{p} $ such that its measure is larger than or equal to that of $ P $, then there exist $ x,y $ in $ K $, with $ x \neq y $, for which $ x-y $ is a lattice point.

Counterexample for Compactness

Realize that replacing 'compact' with 'closed' in the above step results in a contradiction.

Minkowski's Theorem

Understand Minkowski's Theorem, which asserts that if $ C $ is a compact, convex, and symmetric subset of $ \mathbb{R}^{p} $, such that its measure is larger than or equal to twice the $ p $-th power of 2 times the measure of $ P $, there exists a non-zero lattice point in $ C $.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lattice in R^p

In mathematics, particularly in geometry and group theory, a lattice in R^p is a discrete subgroup of R^p formed by points with an integer structure. In simpler terms, imagine a grid that extends infinitely in every direction in R^p space. We construct this grid (or lattice) by taking linear combinations of basis vectors, where the coefficients are integers.

Mathematically, if you have basis vectors a₁, a₂, ..., a_p in R^p, then the lattice Gamma is defined as the set containing all possible combinations of the form k₁a₁ + k₂a₂ + ... + k_pa_p, where each k_i is an integer. This concept is central in understanding the geometry of numbers and has numerous applications in various fields like cryptography and crystallography.

Fundamental parallelotope

The fundamental parallelotope can be visualized as a multi-dimensional generalization of a parallelogram, which is specific to a lattice. It is formed by considering all the points reached by linear combinations of the basis vectors of a lattice, where the coefficients are restricted to the interval from 0 to 1. It serves as the basic building block, replicating to fill the entire space.

In the provided exercise, the fundamental parallelotope P of lattice Gamma is described. The importance of this shape lies in its use as a representative of the lattice's equivalence classes in R^p/Gamma. It is also a critical tool in computing volumes in lattices since its volume, given by the Lebesgue measure lambda^p(P), is invariant under the choice of basis for the lattice and equal to the absolute value of the determinant formed by those basis vectors.

Asymptotic behavior

The term asymptotic behavior refers to the behavior of functions as the argument grows large. In the context of lattices, this concept is used to describe how the number of lattice points within a certain region grows as the size of that region expands.

For the problem at hand, the relation given pertains to the count of lattice points that lie within a sphere of radius R. As R tends to infinity, the number of such lattice points relates to the volume of the sphere divided by the volume of the fundamental parallelotope, multiplied by R^p, with an error term that grows at a lower rate of O(R^p-1). This asymptotic formula is crucial for number theory and geometric analysis.

Measure theory

At its core, measure theory is a branch of mathematical analysis that studies ways to assign a volume, length, or area—which we call a 'measure'—to different subsets of a given space. The Lebesgue measure is one of the most important types of measures and is especially adept at handling irregular shapes in R^p, which makes it essential for the modern treatment of integration and probability.

In the exercise, we use the concept of Lebesgue measure denoted by lambda^p, to discuss properties of sets in R^p, such as the lattice Gamma, the fundamental parallelotope P, and various other shapes. This measure is key to establishing the volume of these sets and understanding theorems related to lattices, such as Minkowski's theorem.

Minkowski's theorem

Finally, Minkowski's theorem is a cornerstone of the geometry of numbers. This theorem provides a bridge between the algebraic structure of lattices and the geometric properties of sets in Euclidean space. It asserts that for any lattice Gamma in R^p and any compact, convex set C that is symmetric about the origin and has a volume greater than 2^p times the volume of the fundamental parallelotope of the lattice, there must be at least one lattice point in C other than the origin.

The theorem has significant implications in various areas such as diophantine approximation, number theory, and optimization. In our exercise, Minkowski's theorem is applied in step f) to show that under certain volume conditions, a compact convex set contains a non-zero lattice point, illustrating the deep connection between geometry and the arithmetic of lattices.

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