Abstract

Let h,k≥2 be integers. We say a set A of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from A. A set of positive integers A is called a Bh[g] set if every positive integer can be represented as the sum of h terms from A in at most g different ways. In this paper we prove the existence of Bh[1] sets which are asymptotic bases of order 2h+1 by using probabilistic methods.

Highlights

  • Let N denote the set of positive integers

  • We say a set A ⊂ N is an asymptotic basis of order k if there exists a positive integer n0 such that Rk,A(n) > 0 for n > n0

  • Sos asked if there exists a Sidon set which is an asymptotic basis of order 3

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Summary

Introduction

Let N denote the set of positive integers. Let h, k ≥ 2 be integers. In [1] and [9] this result was improved on by proving the existence of a Sidon set which is an asymptotic basis of order 4. If k = h, the generalization of the well known conjecture of Erdos and Turan asserts that there does not exist a Bh[g] set which is an asymptotic basis of order h. This conjecture is still unsolved even for h = 2. We prove the case k = 2h + 1 by using probabilistic methods. We give a short survey of the probabilistic method we will use

Probabilistic tools
Proof of Theorem 1
Full Text
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