Sum of Elements in Finite Sidon Sets

[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].

A Non-abelian, Non-Sidon, Completely Bounded Set

The purpose of this note is to construct an example of a discrete non-abelian group G and a subset E of G, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.

Sobre sucesiones de Sidon

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.

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

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].

Monochromatic Hilbert Cubes and Arithmetic Progressions

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 $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = \Omega( |A|^3) .$$

The Relationship Between ϵ-Kronecker Sets and Sidon Sets

A 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.

$k$-Fold Sidon Sets

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 $1 \le i \le j\le k$.