Lattices and the Geometry of Numbers

In this paper, we discuss the properties of lattices and their application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has an application in various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis, etc. This paper gives an elementary introduction to the field of the geometry of numbers. In this paper, we shall first give a broad overview of the concept of lattice and then discuss the geometrical properties it has and its applications.

Lattices and the Geometry of Numbers

In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper gives an elementary introduction to the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications.

Complexity of linear relaxations in integer programming

For a setXof integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with Xis called the relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$rc(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of Xwithout using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding$${{\,\mathrm{rc}\,}}(X)$$rc(X)and its variant$${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rcQ(X), restricting the descriptions of Xto rational polyhedra. As our main results we show that$${{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rc(X)=rcQ(X)when: (a)Xis at most four-dimensional, (b)Xrepresents every residue class in$$(\mathbb {Z}/2\mathbb {Z})^d$$(Z/2Z)d, (c) the convex hull of Xcontains an interior integer point, or (d) the lattice-width of Xis above a certain threshold. Additionally,$${{\,\mathrm{rc}\,}}(X)$$rc(X)can be algorithmically computed when Xis at most three-dimensional, orXsatisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on$${{\,\mathrm{rc}\,}}(X)$$rc(X)in terms of the dimension of X.

Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> be a semisimple linear algebraic group defined over rational numbers, <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{K} $\end{document}</tex-math></inline-formula> be a maximal compact subgroup of its real points and <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> be an arithmetic lattice. One can associate a probability measure <inline-formula><tex-math id="M4">\begin{document}$ \mu_{ \mathrm{H}} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> for each subgroup <inline-formula><tex-math id="M6">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M7">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> defined over <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Q} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. As G acts on <inline-formula><tex-math id="M9">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> from the right, we can push forward this measure by elements from <inline-formula><tex-math id="M10">\begin{document}$ \mathrm{G} $\end{document}</tex-math></inline-formula>. By pushing down these measures to <inline-formula><tex-math id="M11">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula>, we call them homogeneous. It is a natural question to ask what are the possible weak-<inline-formula><tex-math id="M12">\begin{document}$ * $\end{document}</tex-math></inline-formula> limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of <inline-formula><tex-math id="M13">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general <inline-formula><tex-math id="M15">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on <inline-formula><tex-math id="M16">\begin{document}$ {\text{SL}}_n $\end{document}</tex-math></inline-formula> proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. <b>193</b> words.</p>

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 &lt; \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.