scholarly journals Lattice Coverings by Congruent Translation Balls Using Translation-like Bisector Surfaces in Nil Geometry

KoG ◽  
2019 ◽  
pp. 6-17
Author(s):  
Angéla Vránics ◽  
Jenö Szirmai
Keyword(s):  
Upper Bound ◽  
Projective Model ◽  
Upper Estimate

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geometries. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given translation tetrahedron. A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius and the volume of the circumscribed translation sphere of a given translation tetrahedron. The found minimal upper bound density of the translation ball coverings $\Delta \approx 1.42783$. In our work we will use for computations and visualizations the projective model of Nil described by E. Molnár in [6].

scholarly journals A Subexponential Upper Bound for van der Waerden Numbers $W(3,k)$

10.37236/9704 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Tomasz Schoen
Keyword(s):  
Arithmetic Progression ◽  
Upper Bound ◽  
Absolute Constant ◽  
Upper Estimate

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$

scholarly journals The Distributional Asymptotics Mod 1 of (logb n)

2019 ◽  
Vol 14 (1) ◽  
pp. 105-122
Author(s):  
Chuang Xu
Keyword(s):  
Rate Of Convergence ◽  
Upper Bound ◽  
Rates Of Convergence ◽  
Limit Set ◽  
Exponential Distributions ◽  
Kantorovich Metric ◽  
Kolmogorov Metric ◽  
Omega Limit Set ◽  
Upper Estimate

AbstractThis paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

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