scholarly journals A Note on the Number of $(k,l)$-Sum-Free Sets

10.37236/1508 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tomasz Schoen
Keyword(s):  
Natural Number ◽  
Positive Integers ◽  
Free Sets

A set $A\subseteq {\bf N}$ is $(k,\ell)$-sum-free, for $k,\ell\in {\bf N}$, $k>\ell$, if it contains no solutions to the equation $x_1+\dots+x_k=y_1+\dots+y_{\ell}$. Let $\rho=\rho (k-\ell)$ be the smallest natural number not dividing $k-\ell$, and let $r=r_n$, $0\le r < \rho$, be such that $r\equiv n \pmod {\rho }$. The main result of this note says that if $(k-\ell)/\ell$ is small in terms of $\rho$, then the number of $(k,\ell)$-sum-free subsets of $[1,n]$ is equal to $(\varphi(\rho)+\varphi_r(\rho)+o(1)) 2^{\lfloor n/\rho \rfloor}$, where $\varphi_r(x)$ denotes the number of positive integers $m\le r$ relatively prime to $x$ and $\varphi(x)=\varphi_x(x)$.

On the number of sum-free sets in a segment of positive integers

2002 ◽  
Vol 12 (4) ◽  
Author(s):  
K. G. Omelyanov ◽  
A. A. Sapozhenko
Keyword(s):  
Basic Research ◽  
Basic Research Grant ◽  
Positive Integers ◽  
Free Sets ◽  
Research Grant

AbstractA set A of integers is called sum-free if a + b ∉ A for any a, b ∈ A. For an arbitrary Ɛ > 0, let ssThis research was supported by the Russian Foundation for Basic Research, grant 01-01-00266.

Two consistency results on set mappings

2000 ◽  
Vol 65 (1) ◽  
pp. 333-338 ◽  
Author(s):  
Péter Komjáth ◽  
Saharon Shelah
Keyword(s):  
Natural Number ◽  
Consistency Results ◽  
Set Mappings ◽  
Free Sets

AbstractIt is consistent that there is a set mapping from the four-tuples of ωn into the finite subsets with no free subsets of size tn for some natural number tn. For any n < ω it is consistent that there is a set mapping from the pairs of ωn into the finite subsets with no infinite free sets. For any n < ω it is consistent that there is a set mapping from the pairs of ωn into ωn with no uncountable free sets.

On a Waring-Goldbach problem involving squares and cubes

2019 ◽  
Vol 69 (6) ◽  
pp. 1249-1262
Author(s):  
Yuhui Liu
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Goldbach Problem ◽  
Positive Integers

AbstractLet R(n) denote the number of representations of a natural number n as the sum of two squares and four cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most $\begin{array}{}O(N^{\frac{1}{4} + \varepsilon})\end{array}$ positive integers not exceeding N.

Almost Odd Random Sum-Free Sets

1998 ◽  
Vol 7 (1) ◽  
pp. 27-32 ◽  
Author(s):  
NEIL J. CALKIN ◽  
P. J. CAMERON
Keyword(s):  
Random Sum ◽  
Positive Probability ◽  
Finite Sum ◽  
Free Set ◽  
Positive Integers ◽  
Free Sets

We show that if S1 is a strongly complete sum-free set of positive integers, and if S0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S0 and is contained in S0∪S1. As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.

On Product Partitions of Integers

1991 ◽  
Vol 34 (4) ◽  
pp. 474-479 ◽  
Author(s):  
V. C. Harris ◽  
M. V. Subbarao
Keyword(s):  
Natural Number ◽  
Generating Function ◽  
Partitions Of Integers ◽  
Positive Integers

AbstractLet p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

scholarly journals Maximal (k, l)-free sets in ℤ/pℤ are arithmetic progressions

2002 ◽  
Vol 65 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Alain Plagne
Keyword(s):  
Arithmetic Progressions ◽  
Recent Issue ◽  
Complete Determination ◽  
Free Set ◽  
Positive Integers ◽  
Free Sets

Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.

scholarly journals Modular Schur Numbers

10.37236/2374 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Chappelon ◽  
María Pastora Revuelta Marchena ◽  
María Isabel Sanz Domínguez
Keyword(s):  
Positive Integers ◽  
Free Sets

For any positive integers $l$ and $m$, a set of integers is said to be (weakly) $l$-sum-free modulo $m$ if it contains no (pairwise distinct) elements $x_1,x_2,\ldots,x_l,y$ satisfying the congruence $x_1+\ldots+x_l\equiv y\bmod{m}$. It is proved that, for any positive integers $k$ and $l$, there exists a largest integer $n$ for which the set of the first $n$ positive integers $\{1,2,\ldots,n\}$ admits a partition into $k$ (weakly) $l$-sum-free sets modulo $m$. This number is called the generalized (weak) Schur number modulo $m$, associated with $k$ and $l$. In this paper, for all positive integers $k$ and $l$, the exact value of these modular Schur numbers are determined for $m=1$, $2$ and $3$.

scholarly journals Maximum $k$-Sum $\mathbf{n}$-Free Sets of the 2-Dimensional Integer Lattice

10.37236/8895 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ilkyoo Choi ◽  
Ringi Kim ◽  
Boram Park
Keyword(s):  
Positive Integer ◽  
Lattice Point ◽  
Higher Dimensions ◽  
Maximum Density ◽  
Integer Lattice ◽  
Free Set ◽  
Integer Lattice Point ◽  
Positive Integers ◽  
Free Sets

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements $\mathbf{a}_1, \ldots, \mathbf{a}_k$ in $S$ satisfying $\mathbf{a}_1+\cdots+\mathbf{a}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions. 

scholarly journals ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS

2014 ◽  
Vol 57 (3) ◽  
pp. 681-692
Author(s):  
JÖRG BRÜDERN
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Asymptotic Relation ◽  
Circle Method ◽  
Precise Definition ◽  
Singular Series ◽  
Mean Square ◽  
Formula Group ◽  
Image Position ◽  
Positive Integers

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation (1)$ \begin{equation*} r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty). \end{equation*} $ Here ${\mathfrak s}$(n) is the singular series associated with sums of a square and three fifth powers, see (13) below for a precise definition. The main purpose of this note is to confirm (1) in mean square.

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