Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

Factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Balances in the Set of Arithmetic Progressions

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included.

ON HIGHER DIMENSIONAL ARITHMETIC PROGRESSIONS IN MEYER SETS

In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.

Algorithms for computing sums of powers of arithmetic progressions by using Eulerian numbers

The sums of powers of arithmetic progressions is of the form a^p+(a+d)^p +(a+2d)^p+\cdots+(a+(n-1)d)^p, where n\geq 1, p is a non-negative integer, and a and d are complex numbers with d\neq 0. This sum can be computed using classical Eulerian numbers \cite{worpitzky1883studien} and general Eulerian numbers \cite{xiong2013general}. This paper gives a new method using classical Eulerian numbers to compute this sum. The existing formula that uses general Eulerian numbers are more algorithmically complex due to more numbers to compute. This paper presents and focuses on two novel algorithms involving both types of Eulerian numbers. This paper gives a comparison to Xiong \textit{et al.}’s result with general Eulerian numbers \cite{xiong2013general}. Moreover, an analysis of theoretical time complexities is presented to show our algorithm is less complex. Various values of p are analyzed in the proposed algorithms to add significance to the results. The experimental results show the proposed algorithm remains around 70\% faster as p increases.

A blurred view of Van der Waerden type theorems

Let $\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\varepsilon>0$ we call a set $\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\varepsilon$ -approximate arithmetic progression if for some a and d, $|x_i-(a+id)|<\varepsilon d$ holds for all $i\in\{0,1\ldots,k-1\}$ . Complementing earlier results of Dumitrescu (2011, J. Comput. Geom.2(1) 16–29), in this paper we study numerical aspects of Van der Waerden, Szemerédi and Furstenberg–Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $\varepsilon$ -approximation.

Popular Differences for Right Isosceles Triangles

For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $\alpha N^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(\alpha^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field $\mathbb{F}_q$ we comment on exponential smallness of subsets of $(\mathbb{F}_q^n)^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing proofs regarding three-term arithmetic progressions.