Notes on lattice points of zonotopes and lattice-face polytopes

Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.Comment: 16 pages, incorporated referee remarks, corrected proof of Theorem 1.2, added new co-autho

Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde

Remarks on the plus-minus weighted Davenport constant

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1, i.e., there is a non-empty subset $I \subset \{1,..., \ell\}$ such that $\sum_{i \in I} a_i g_i =0$ for $a_i \in \{+1,-1\}$. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups

On the weak order of Coxeter groups

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).Comment: 37 pages, submitte

균질공간에서의 동역학과 디오판틴 근사

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2022. 8. 임선희.Dynamics of group actions on homogeneous spaces, which is referred to as "homogeneous dynamics", has a lot of connections to number theory. These connections have been intensively and extensively studied over the past decades, and have produced various and abundant number-theoretic results. In this thesis, we focus on the relationship between homogeneous dynamics and Diophantine approximation, and consider the following three objects in Diophantine approximation: Dirichlet non-improvable affine forms, badly approximable affine forms, and weighted singular vectors. We improve equidistribution results in homogeneous dynamics in terms of weak L1 estimates, and establish local ubiquity systems for Dirichlet non-improvable affine forms using Transference Principle in Diophantine approximation. These developments imply zero-infinite phenomena for Hausdorff measures of Dirichlet non-improvable affine forms. Next, we establish an effective version of entropy rigidity, which implies the effective upper bound of Hausdorff dimension of badly approximable affine forms by constructing "well-behaved" σ-algebras and certain invariant measures with large entropy. We further characterize full Hausdorff-dimensionality of badly approximable affine forms for fixed matrix by a Diophantine condition of singularity on average. We also consider Diophantine approximation over global function fields and have similar results in this setting. Finally, we improve lattice point counting in geometry of numbers, which arises from the fractal structure of weighted singular vectors. Combining the improvement and the shadowing property in homogeneous dynamics, we obtain the sharp lower bound of Hausdorff dimension of weighted singular vectors.균질공간에서 군 작용의 동역학을 의미하는 균질동역학은 정수론과 많은 연결관계가 있다. 이러한 연결관계는 지난 수십 년간 광범위하고 집중적으로 연구되었으며 다양한 정수론 결과를 제공하였다. 본 학위 논문에서는 균질동역학과 디오판틴 근사의 관계에 대해 살펴보고 다음과 같은 디오판틴 근사에서의 세가지 대상에 대해 알아볼 것이다: 디리끌레 향상 불가능 아핀형식, 나쁜 근사를 가지는 아핀형식, 가중치를 가지는 특이 벡터. 우선 우리는 약한 L1 측정을 통해 균질동역학에서의 동등분포 결과를 향상시키고 디오판틴 근사에서의 전이원리를 이용하여 디리끌레 향상 불가능 아핀형식에 대한 국소 편재 체계를 구축한다. 이러한 연구를 바탕으로 디리끌레 향상 불가능 아핀형식의 하우스도르프 측도에 대한 0 − ∞ 현상을 규명한다. 다음으로 엔트로피 강직성의 효과적인 표현을 건설하는데 이를 이용하여 잘 행동하는 시그마 대수를 건설하고 큰 엔트로피를 가지는 불변측도를 건설함으로써 나쁜 근사를 가지는 아핀형식의 하우스도르프 차원의 효과적인 상계를 얻는다. 뿐만 아니라 나쁜 근사를 가지는 아핀형식이 최대차원을 갖기 위한 필요충분조건으로 평균적 특이성을 보인다. 또한 대역적 함수체 위에서의 디오판틴 근사를 생각하고 비슷한 결과를 얻는다. 마지막으로 가중치를 가지는 특이 벡터의 프랙탈 구조와 관련된 수의 기하학의 격자점 셈을 발전시키고 균질동역학의 투영 성질을 이용하여 가중치를 가지는 특이 벡터의 하우스도르프 차원의 하계를 얻는다.Abstract i 1 Introduction 1 1.1 Dirichlet non-improvable affine forms 3 1.2 Badly approximable affine forms 7 1.3 Badly approximable affine forms over global function field 10 1.4 Weighted singular vectors 12 2 Equidistribution and Ubiquitous system 16 2.1 Preliminaries 16 2.1.1 Hausdorff measure and auxiliary lemmas 16 2.1.2 Homogeneous dynamics 17 2.2 Equidistribution and Weak-L1 estimate 21 2.3 Application to Diophantine approximation 23 2.3.1 Successive minima function 23 2.3.2 The number of covering balls 24 2.4 Local ubiquitous system 27 2.4.1 Historical Remarks 27 2.4.2 Transference Principle on Diophantine approximation 28 2.4.3 Mass distributions on Ψϵ-approximable matrices 30 2.4.4 Establishing the local ubiquity 33 3 Entropy rigidity and Best approximation vectors 50 3.1 General entropy theory 50 3.2 Entropy on homogeneous spaces 53 3.2.1 General setup 53 3.2.2 Construction of a-descending, subordinate algebra and its entropy properties 55 3.2.3 Effective variational principle 67 3.3 Preliminaries for the upper bound 70 3.3.1 Dimensions with quasinorms 71 3.3.2 Correspondence with dynamics 72 3.3.3 Covering counting lemma 73 3.4 Upper bound for Hausdorff dimension of BadA(ϵ) 76 3.4.1 Constructing measure with entropy lower bound 77 3.4.2 The proof of Theorem 1.2.2 82 3.5 Upper bound for Hausdorff dimension of Badb(ϵ) 87 3.5.1 Constructing measure with entropy lower bound 87 3.5.2 Effective equidistribution and the proof of Theorem 1.2.1 96 3.6 Characterization of singular on average property and Dimension esitimates 103 3.6.1 Best approximations 103 3.6.2 Characterization of singular on average property 104 3.6.3 Modified Bugeaud-Laurent sequence 108 3.6.4 Dimension estimates 115 4 Diophantine approximation over global function fields 117 4.1 Background material for the lower bound 117 4.1.1 On global function fields 117 4.1.2 On the geometry of numbers and Dirichlet’s theorem 119 4.1.3 Best approximation sequences with weights 121 4.1.4 Transference theorems with weights 125 4.2 Characterisation of singular on average property 130 4.3 Full Hausdorff dimension for singular on average matrices 133 4.3.1 Modified Bugeaud-Zhang sequences 133 4.3.2 Lower bound on the Hausdorff dimension of BadA(ϵ) 141 4.3.3 Proof that Assertion (2) implies Assertion (1) in Theorem 1.3.1 148 4.4 Background material for the upper bound 149 4.4.1 Homogeneous dynamics 149 4.4.2 Dani correspondence 152 4.4.3 Entropy, partition construction, and effective variational principle 155 4.5 Upper bound on the Hausdorff dimension of BadA(ϵ) 167 4.5.1 Constructing measures with large entropy 167 4.5.2 Effective upper bound on dimH BadA(ϵ) 175 5 Weighted singular vectors 183 5.1 Fractal sutructure and Hausdorff dimension 183 5.1.1 Fractal structure 183 5.1.2 Self-affine structure and lower bound 184 5.2 Counting lattice points in convex sets 193 5.2.1 Preliminaries for lattice point counting 193 5.2.2 Lattice point counting in Rd+1 195 5.3 Lower bound 206 5.3.1 Construction of the fractal set 206 5.3.2 The lower bound calculation 212 Abstract (in Korean) 225 Acknowledgement (in Korean) 226박

On globally sparse Ramsey graphs

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available