scholarly journals A blurred view of Van der Waerden type theorems

2021 ◽  
pp. 1-18
Author(s):  
Vojtech Rödl ◽  
Marcelo Sales
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Higher Dimensional

Abstract 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.

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

scholarly journals Boxes, extended boxes and sets of positive upper density in the Euclidean space

2021 ◽  
pp. 1-21
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ
Keyword(s):  
Euclidean Space ◽  
Arithmetic Progressions ◽  
Density Theorems ◽  
Higher Dimensional ◽  
Banach Density ◽  
Upper Density

Abstract We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2 n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

scholarly journals The Faulhaber Problem on Sums of Powers on Arithmetic Progressions Resolved

2018 ◽  
Vol 10 (2) ◽  
pp. 5
Author(s):  
Do Tan Si
Keyword(s):  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Arithmetic Progressions ◽  
Power Sums ◽  
Sums Of Powers

We prove that all the Faulhaber coefficients of a sum of odd power of elements of an arithmetic progression may simply be calculated from only one of them which is easily calculable from two Bernoulli polynomials as so as from power sums of integers. This gives two simple formulae for calculating them. As for sums related to even powers, they may be calculated simply from those related to the nearest odd one’s.

scholarly journals NONZERO VALUES OF DIRICHLET L-FUNCTIONS IN VERTICAL ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 813-843 ◽  
Author(s):  
GREG MARTIN ◽  
NATHAN NG
Keyword(s):  
Arithmetic Progression ◽  
Positive Constant ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Linearly Independent ◽  
Theoretical Evidence

Let L(s, χ) be a fixed Dirichlet L-function. Given a vertical arithmetic progression of T points on the line ℜs = ½, we show that at least cT/ log T of them are not zeros of L(s, χ) (for some positive constant c). This result provides some theoretical evidence towards the conjecture that all nonnegative ordinates of zeros of Dirichlet L-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of L(s, χ).

scholarly journals Sets of Integers that do not Contain Long Arithmetic Progressions

10.37236/546 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kevin O'Bryant
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions

Combining ideas of Rankin, Elkin, Green & Wolf, we give constructive lower bounds for $r_k(N)$, the largest size of a subset of $\{1,2,\dots,N\}$ that does not contain a $k$-element arithmetic progression: For every $\epsilon>0$, if $N$ is sufficiently large, then $$r_3(N) \geq N \left(\frac{6\cdot 2^{3/4} \sqrt{5}}{e \,\pi^{3/2}}-\epsilon\right) \exp\left({-\sqrt{8\log N}+\tfrac14\log\log N}\right),$$ $$r_k(N) \geq N \, C_k\,\exp\left({-n 2^{(n-1)/2} \sqrt[n]{\log N}+\tfrac{1}{2n}\log\log N}\right),$$ where $C_k>0$ is an unspecified constant, $\log=\log_2$, $\exp(x)=2^x$, and $n=\lceil{\log k}\rceil$. These are currently the best lower bounds for all $k$, and are an improvement over previous lower bounds for all $k\neq4$.

scholarly journals Cesàro averages for Goldbach representations with summands in arithmetic progressions

2021 ◽  
pp. 1-15
Author(s):  
Marco Cantarini ◽  
Alessandro Gambini ◽  
Alessandro Zaccagnini
Keyword(s):  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Counting Function ◽  
Weighted Average ◽  
Arithmetic Progressions ◽  
Von Mangoldt Function ◽  
Cesaro Averages

Let [Formula: see text] be the von Mangoldt function, let [Formula: see text] be an integer and let [Formula: see text] be the counting function for the Goldbach numbers with summands in arithmetic progression modulo a common integer [Formula: see text]. We prove an asymptotic formula for the weighted average, with Cesàro weight of order [Formula: see text], with [Formula: see text], of this function. Our result is uniform in a suitable range for [Formula: see text].

Rational Points in Arithmetic Progressions on y2 = xn + k

2012 ◽  
Vol 55 (1) ◽  
pp. 193-207 ◽  
Author(s):  
Maciej Ulas
Keyword(s):  
Elliptic Curve ◽  
Arithmetic Progression ◽  
Hyperelliptic Curve ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Multiple Roots

AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k's with the property that on curves y2 = xn + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k's such that on curves y2 = xn +k there are six rational points in arithmetic progression.

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