A remark of Ruzsa's construction of an infinite Sidon set

Discrete Mathematics, 2005

A sequence (a i) of integers is weak Sidon or well-spread if the sums a i + a j , for i < j, are all different. Let f (N) denote the maximum integer n for which there exists a weak Sidon sequence 0 a 1 < • • • < a n N. Using an idea of Lindström [An inequality for B 2-sequences, J. Combin. Theory 6 (1969) 211-212], we offer an alternate proof that f (N) < N 1/2 + O(N 1/4), an inequality due to Ruzsa [Solving a linear equation in a set of integers I, Acta. Arith. 65 (1993) 259-283]. The present proof improves Ruzsa's bound by decreasing the implicit constant, essentially from 4 to √ 3.

arXiv (Cornell University), 2021

For a given positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum value of n such that, in every partition of the set [1, n] into k parts, there exists a part that contains two distinct pairs of numbers with the same sum. In other words, there is a part that is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with h-tuples, such that in every partition of [1, n] into k parts, there exists a part that contains two distinct h-tuples with the same sum. Alternatively, there is a part that is not a B h set. The second generalization considers the scenario where the interval [1, n] is substituted with a nonnecessarily symmetric d-dimensional box of the form d i=1 [1, n i ]. For the general case of h ≥ 3 and non-symmetric boxes, before applying our method to obtain the Ramsey-type result, we needed to establish an upper bound for the corresponding density parameter.

Acta Arithmetica, 2007

Helou (Media, PA) and J. Pihko (Helsinki) 1. Introduction. Given a subset A of N = {0, 1, 2,. . .} and a positive integer h, the h-representation function by A is the function which to each integer n ≥ 0 associates the number r h (A, n) of h-tuples of elements of A whose sum is equal to n. The study of such functions, their properties and their characterizations are the focus of much attention in additive number theory. In particular, they are used to study some important notions such as that of basis. The set A is called an h-basis (resp. an asymptotic h-basis) of N if every integer n ≥ 0 (resp. every sufficiently large integer n) is the sum of h elements of A. The set A is called a Sidon set if all the sums a + b, with a, b ∈ A and a ≤ b, are distinct, i.e. if r 2 (A, n) ≤ 2 for all integers n ≥ 0. An open problem due to P. Erdős, A. Sárközy and V. T. Sós [1] asks if there exists a Sidon set which is an asymptotic 3-basis of N. This problem was mistakenly presented in [3] as asking if a Sidon set can be an asymptotic 2-basis of N. Even though the negative answer to the latter question is an easy consequence of some well-known properties, it is not extant in published explicit form, and it would therefore not be without interest to give a proof using some new ideas. In this paper, we give some new properties of the 2-representation function of an infinite subset A of N, of intrinsic interest. We then apply them to give a simple proof of the fact that a Sidon set cannot be an asymptotic 2-basis of N. We then turn to the real open problem of the existence of a Sidon set which is an asymptotic 3-basis of N, and we provide a partial answer by proving that a Sidon set cannot be a 3-basis of N. We also give an algorithm for finding, if any exists, a Sidon set A such that, from a specific point on, every integer is the sum of three elements of A.

Arkiv för matematik, 1993

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure µ and dual group Γ. Let E be a Sidon subset of Γ with Sidon constant S(E). Let r n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all α > 0:

Pacific Journal of Mathematics, 1981

G be any compact connected group with dual hypergroup G. We establish in this paper a criterion by which the existence of an infinite Sidon set in G can be decided from the structure of G.