Representation functions, Sidon sets and bases

Integers, 2018

A set $\mathcal{A}$ is said to be an additive $h$-basis if each element in $\{0,1,\ldots,hn\}$ can be written as an $h$-sum of elements of $\mathcal{A}$ in {\it at least} one way. We seek multiple representations as $h$-sums, and, in this paper we make a start by restricting ourselves to $h=2$. We say that $\mathcal{A}$ is said to be a truncated $(\alpha,2,g)$ additive basis if each $j\in[\alpha n, (2-\alpha)n]$ can be represented as a $2$-sum of elements of $\mathcal{A}$ in at least $g$ ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated $(\alpha,2,g)$ additive basis with high or low probability.

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.

Journal of Number Theory, 2004

Let A be an infinite subset of natural numbers, nAN and X a positive real number. Let rðnÞ denotes the number of solution of the equation n ¼ a 1 þ a 2 where a 1 pa 2 and a 1 ; a 2 AA: Also let jAðX Þj denotes the number of natural numbers which are less than or equal to X and belong to A: For those A which satisfy the condition that for all sufficiently large natural numbers n we have rðnÞa1; we improve the lower bound of jAðX Þj given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible.

2006

Moulton and Develin have investigated the notion of representing various sets S of positive integers, of size m say, as subset sums of smaller sets. The rank of a given set S is the smallest positive integer rk(S) ≤ m such that S is represented by an integer set of size rk(S). In this note, we primarily consider sets of the form {1, 2 m , 3 m ,. . .} for positive integer m ≥ 2. Given a positive integer k, we ask for the smallest M such that {1, 2 m , 3 m ,. .. , k m } is independent for all m ≥ M , and provide some answers. We then use a result of Sprague to show that any nondecreasing positive integer sequence a = {a 1 , a 2 ,. . .} that grows polynomially, and in particular the set {1, 2 m , 3 m ,. . .} for fixed exponent m, has limiting rank zero.

2011

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).