Sum of Elements in Finite Sidon Sets

2020 ◽  
pp. 1-11
Author(s):  
Yuchen Ding
Keyword(s):  
Cardinal Number ◽  
Sidon Sets ◽  
Sidon Set

[Formula: see text] is called a Sidon set if [Formula: see text] are all distinct for any [Formula: see text]. Let [Formula: see text] be the largest cardinal number of such [Formula: see text]. We are interested in the sum of elements in the Sidon set [Formula: see text]. In this paper, we prove that for any [Formula: see text], [Formula: see text] where [Formula: see text] is a Sidon set and [Formula: see text].

scholarly journals Central Sidon and central ∧p sets

1972 ◽  
Vol 14 (1) ◽  
pp. 62-74 ◽  
Author(s):  
Willard A. Parker
Keyword(s):  
Upper Bound ◽  
Sidon Sets ◽  
Sidon Set

Central Sidon sets and central ∧p sets are defined and equivalent characterizations are given. It is shown that a central Sidon set with an upper bound on the degrees of its elements is a ∧p set (1 < p < ∞ ). The bound on the degrees is shown to be necessary by an example.

Separating functions for V-Sidon sets

1972 ◽  
Vol 71 (1) ◽  
pp. 39-42
Author(s):  
E. Galanis
Keyword(s):  
Compact Set ◽  
Sidon Sets ◽  
Sidon Set ◽  
Tensor Norm

AbstractWe show that if E is a V-Sidon set and K a disjoint compact set then there exists a function of low tensor norm separating them.

scholarly journals Non-tall compact groups admit infinite Sidon sets

1977 ◽  
Vol 23 (4) ◽  
pp. 467-475 ◽  
Author(s):  
M. F. Hutchinson
Keyword(s):  
Compact Group ◽  
Lie Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Compact Lie Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Irreducible Unitary Representations

AbstractRiesz polynomials are employed to give a sufficient condition for a non-abelian compact group G to have an infinite uniformly approximable Sidon set. This condition is satisfied if G admits infinitely many pairwise inequivalent continuous irreducible unitary representations of the same degree. Consequently a compact Lie group admits an infinite Sidon set if and only if it is not semi-simple.

scholarly journals The Relationship Between ϵ-Kronecker Sets and Sidon Sets

2016 ◽  
Vol 59 (3) ◽  
pp. 521-527 ◽  
Author(s):  
Kathryn Hare ◽  
L. Thomas Ramsey
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Dual Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Arithmetic Properties ◽  
The Fourier Transform ◽  
The Relationship ◽  
Discrete Abelian Group

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

scholarly journals An improved upper bound for the size of the multiplicative 3-Sidon sets

2019 ◽  
Vol 15 (08) ◽  
pp. 1721-1729
Author(s):  
Péter Pál Pach
Keyword(s):  
Upper Bound ◽  
Sidon Sets ◽  
Sidon Set

We say that a set is a multiplicative 3-Sidon set if the equation [Formula: see text] does not have a solution consisting of distinct elements taken from this set. In this paper, we show that the size of a multiplicative 3-Sidon subset of [Formula: see text] is at most [Formula: see text], which improves the previously known best bound [Formula: see text].

Expected Norms of Zero-One Polynomials

2008 ◽  
Vol 51 (4) ◽  
pp. 497-507 ◽  
Author(s):  
Peter Borwein ◽  
Kwok-Kwong Stephen Choi ◽  
Idris Mercer
Keyword(s):  
Unit Circle ◽  
Sidon Sets ◽  
Sidon Set ◽  
Almost All

AbstractLet = {a0 + a1z + · · · + an–1zn–1 : aj ∈ ﹛0, 1﹜¯﹜, whose elements are called zero- one polynomials and correspond naturally to the 2n subsets of [n] := ﹛0, 1, … , n – 1﹜. We also let = ﹛α(z) ∈ : α(1) = m﹜, whose elements correspond to the subsets of [n] of size m, and let , whose elements are the zero-one polynomials of degree exactly n.Many researchers have studied norms of polynomials with restricted coefficients. Using ‖α‖p to denote the usual Lp norm of α on the unit circle, one easily sees that α(z) = a0+a1z+· · ·+aNzN ∈ ℝ[z] satisfies and , where .If α(z) ∈ , say α(z) = zβ1 + · · · + zβm where β1 < · · · < βm, then ck is the number of times k appears as a difference βi – βj . The condition that α ∈ satisfies ck ∈ ﹛0, 1﹜ for 1 ≤ k ≤ n – 1 is thus equivalent to the condition that ﹛β1, … , βm﹜ is a Sidon set (meaning all differences of pairs of elements are distinct).In this paper, we find the average of over α ∈ , α ∈ , and α ∈ . We further show that our expression for the average of over yields a new proof of the known result: if m = o(n1/4) and B(n,m) denotes the number of Sidon sets of size m in [n], then almost all subsets of [n] of size m are Sidon, in the sense that .

scholarly journals Sobre sucesiones de Sidon

2019 ◽  
Vol 4 (3) ◽  
pp. 19
Author(s):  
Adrian Infante
Keyword(s):  
Real Numbers ◽  
Sidon Sets ◽  
Sidon Set ◽  
Substantial Modification ◽  
Palabras Clave

Estudiamos los subconjuntos de números reales con la propiedad de que todas las sumas de dos elementos son distintos, es decir que si 𝑎𝑖 + 𝑎𝑗= 𝑎𝑖′+ 𝑎𝑗′ entonces se verifica la igualdad {𝑎′}. A estos conjuntos los llamaremos conjuntos de Sidon. El problema es saber cuál es el mayor número de elementos que puede tener un conjunto de Sidon 𝑖, 𝑎𝑗} = {𝑎𝑖′, 𝑎𝑗 en el intervalo [1, 𝑁]. Presentamos ejemplos que evidencian la necesidad de conocer el tamaño del intervalo [1, 𝑁] donde se va a ubicar el conjunto de Sidon para saber el tamaño 𝐹(𝑁) del conjunto de Sidon. Ruzsa I. Z. (1998) demostró la existencia de una sucesión infinita de Sidon tal que su tamaño 𝐵(𝑁)> 𝑁√2−1+𝑜(1). En este trabajo rehacemos detalladamente la demostración de Ruzsa, introduciendo en la prueba una modificación sustancial, al sustituir las sucesiones {log 𝑝} por la sucesión de los argumentos de los enteros de Gauss 𝑎 + 𝑖𝑏 = 𝑝 con 0 < 𝑎 < 𝑏, 𝑎 y 𝑏 enteros y 𝑝 primo.     Palabras clave: Conjuntos de Sidon, Sumas de dos elementos. 𝑝, 𝑝𝑟𝑖𝑚𝑜   Abstract We study the sub-sets of real numbers with the property that all sums of two elements is different, namely 𝑎+𝑎𝑗′ then the equation {𝑎′} is verified. We will call these sets Sidon sets. The problem is knowing the maximum number of elements that a Sidon set can contain in the interval [1, 𝑁]. We present examples that show the need 𝑖, 𝑎𝑗} = {𝑎𝑖′, 𝑎𝑗 of knowing the size of the interval [1, 𝑁] where the Sidon set will be located to know the size 𝐹(𝑁) of the Sidon set. Ruzsa I. Z. (1998) proved the existence of an infinite Sidon succession such that its size 𝐵 ( 𝑁 ) > 𝑁. In this paper, we rewrite Ruzsa proof in detail, introducing a substantial modification in the proof, by substituting the successions for the succession of the arguments of Gauss integers 𝑎 + 𝑖𝑏 = 𝑝 with 0 < 𝑎 < 𝑏, 𝑎 and 𝑏 integers and 𝑝 prime. {log 𝑝} 𝑝, 𝑟𝑖𝑚𝑜     Keywords: Sets of Sidon, Sums of two elements. 

scholarly journals Weighted p-Sidon sets

1996 ◽  
Vol 61 (1) ◽  
pp. 73-95 ◽  
Author(s):  
K. E. Hare ◽  
D. C. Wilson
Keyword(s):  
Compact Groups ◽  
Sidon Sets ◽  
Sidon Set ◽  
Dual Object

AbstractA weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object Ĝ is shown never to be central (p − 1, p)-Sidon for 1 ≦ p < 2, nor central (1 + ε, 2)-Sidon for ε > 0. Local (p, p)-Sidon sets are shown to be identical to local Sidon sets. Examples are constructed of infinite non-Sidon sets which are central (2p − 1, p)-Sidon, or (p − 1, p)-Sidon, for 1 < p < 2. Full m-fold FTR sets are proved not to be central (a, 2m/(m + 1))-Sidon for any a > 1.

scholarly journals $k$-Fold Sidon Sets

10.37236/3860 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Javier Cilleruelo ◽  
Craig Timmons
Keyword(s):  
The Other ◽  
Sidon Sets ◽  
Sidon Set ◽  
Other Hand ◽  
Positive Integers

Let $k \geq 1$ be an integer.  A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3 + c_4 = 0$.  We prove that for any integer $k \geq 1$, a $k$-fold Sidon set $A \subset [N]$ has at most $(N/k)^{1/2} + O((Nk)^{1/4})$ elements. Indeed we prove that given any $k$ positive integers $c_1<\cdots <c_k$, any set $A\subset [N]$ that contains only trivial solutions to $c_i(x_1-x_2)=c_j(x_3-x_4)$ for each $1 \le i \le j \le k$, has at most $(N/k)^{1/2}+O((c_k^2N/k)^{1/4})$ elements. On the other hand, for any $k \geq 2$ we can exhibit $k$ positive integers $c_1,\dots, c_k$ and a set $A\subset [N]$ with $|A|\ge (\frac 1k+o(1))N^{1/2}$, such that $A$ has only trivial solutions to $c_i(x_1 - x_2) = c_j (x_3 -  x_4)$ for each $1 \le i \le j\le k$.

Reconstruction of the Ancient Numeral System in Basque Language

Philology ◽  
2019 ◽  
Vol 4 (2018) ◽  
pp. 157-172
Author(s):  
FERNANDO GOMEZ-ACEDO ◽  
ENEKO GOMEZ-ACEDO
Keyword(s):  
Cardinal Number ◽  
Counting System ◽  
Cardinal Numbers ◽  
Original Meaning ◽  
Diachronic Development ◽  
Insight Into

Abstract In this work a new insight into the reconstruction of the original forms of the first Basque cardinal numbers is presented and the identified original meaning of the names given to the numbers is shown. The method used is the internal reconstruction, using for the etymologies words that existed and still exist in Basque and other words reconstructed from the proto-Basque. As a result of this work it has been discovered that initially the numbers received their name according to a specific and logic procedure. According to this ancient method of designation, each cardinal number received its name based on the hand sign used to represent it, thus describing the position adopted by the fingers of the hand to represent each number. Finally, the different stages of numerical formation are shown, which demonstrate a long and diachronic development of the whole counting system.

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