scholarly journals The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

2008 ◽  
Vol 51 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Ernie Croot
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Famous Conjecture ◽  
Positive Integers

AbstractHow few three-term arithmetic progressions can a subset S ⊆ ℤN := ℤ/Nℤ have if |S| ≥ υN (that is, S has density at least υ)? Varnavides showed that this number of arithmetic progressions is at least c(υ)N2 for sufficiently large integers N. It is well known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös's famous conjecture about whether a subset T of the naturals where Σn∈T 1/n diverges, has a k-term arithmetic progression for k = 3 (that is, a three-term arithmetic progression).We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density υ as N runs through the primes, and as N runs through the odd positive integers.

scholarly journals Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

2014 ◽  
Vol 57 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Daniel M. Kane ◽  
Scott Duke Kominers
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers

AbstractFor relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1..., un) of the finite arithmetic progression . We derive new lower bounds on Ln that improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp as n → ∞.

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 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 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 Thue-like Sequences and Rainbow Arithmetic Progressions

10.37236/1660 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Jaroslaw Grytczuk
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Lovász Local Lemma ◽  
Fixed Integer ◽  
Local Lemma ◽  
Minimum Number ◽  
Lovasz Local Lemma ◽  
Positive Integers

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.

On sets of integers not containing arithmetic progressions of prescribed length

1974 ◽  
Vol 18 (2) ◽  
pp. 188-193 ◽  
Author(s):  
H. L. Abbott ◽  
A. C. Liu ◽  
J. Riddell
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Free Set ◽  
Positive Integers

Let m, n and l be positive integers satisfying m ≦ n ≦ l ≦ 3. Denote by h(m, n, l) the largest integer with the property that from every n-subset of {1,2, …, m} one can select h(m, n, l) integers no l of which are in arithmetic progression. Let f(n, l) = h(n, n, l) and let g(n, l) = minmh(m, n, l). In what follows, by a P1-free set we shall mean a set of integers not containing an arithmetic progression of length l.

On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions

1999 ◽  
Vol 42 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Tom C. Brown ◽  
Ronald L. Graham ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Infinite Number ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Large Set ◽  
Open Questions ◽  
The Family ◽  
Positive Integers

AbstractAnalogues of van derWaerden’s theorem on arithmetic progressions are considered where the family of all arithmetic progressions, AP, is replaced by some subfamily of AP. Specifically, we want to know for which sets A, of positive integers, the following statement holds: for all positive integers r and k, there exists a positive integer n = w′(k, r) such that for every r-coloring of [1, n] there exists a monochromatic k-term arithmetic progression whose common difference belongs to A. We will call any subset of the positive integers that has the above property large. A set having this property for a specific fixed r will be called r-large. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set {an : n = 1, 2,…} can have . Sufficient conditions for a set to be large are also given. We show that any set containing n-cubes for arbitrarily large n, is a large set. Results involving the connection between the notions of “large” and “2-large” are given. Several open questions and a conjecture are presented.

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

scholarly journals A New Method to Construct Lower Bounds for Van der Waerden Numbers

10.37236/925 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
P. R. Herwig ◽  
M.J.H. Heule ◽  
P. M. Van Lambalgen ◽  
H. Van Maaren
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
New Method ◽  
Natural Numbers

We present the Cyclic Zipper Method, a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers $r$, $k$ and $n$ a Van der Waerden certificate $W(r,k,n)$ is a partition of $\{1, \ldots, n\}$ into $r$ subsets, such that none of them contains an arithmetic progression of length $k$ (or larger). Van der Waerden showed that given $r$ and $k$, a smallest $n$ exists - the Van der Waerden number $W(r,k)$ - for which no certificate $W(r,k,n)$ exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities, two techniques are introduced to visualize certificates.

scholarly journals Monochromatic Hilbert Cubes and Arithmetic Progressions

10.37236/7917 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
József Balogh ◽  
Mikhail Lavrov ◽  
George Shakan ◽  
Adam Zsolt Wagner
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
State Of The Art ◽  
Arithmetic Progressions ◽  
The Other ◽  
Hilbert Cube ◽  
Optimal Result ◽  
Sidon Set

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic affine $k$–cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geqslant 3$ be an integer. Then for every $\epsilon >0$, there is a $c > 0$ such that $$h(k,4) \geqslant \min\{W(\lfloor c k^2\rfloor, 2), 2^{k^{2.5-\epsilon}}\}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that if the Hilbert cube number is close to its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$. We also show the optimal result that for any Sidon set $A \subset \mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = \Omega( |A|^3) .$$

Sign in / Sign up

Export Citation Format

Share Document