On bounded basis of integers

Abstract

Let Z be the set of integers and N the set of positive integers. For a nonempty set A of integers and any integer n, denote rA(n) by the number of representations of n of the form n=a+a′, where a⩽a′ and a,a′∈A, and dA(n) by the number of (a,a′) with a,a′∈A such that n=a−a′. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, and S(n)=−S(|n|) for negative integers n. In 2004, Nešetřil and Serra (2004) [6] proved that there is a set A of integers satisfying rA(n)=1 for all integers n and |S(x)⋃S(y)|⩽4|S(x+y)| for x,y∈A. In this paper, we obtain a stronger result by adding the restriction that dA(n)=1 for all positive integers n. Furthermore, we also prove that, (i) there is a set A1 of integers satisfying rA1(n)=1 for all n∈Z, the set consisting of positive integers n with dA1(n)=0 has density one, and |S(x)⋃S(y)|⩽4|S(x+y)| for x,y∈A1; (ii) there is a set A2 of integers satisfying dA2(n)=1 for all n∈N, the set consisting of integers n with rA2(n)=0 has density one, and |S(x)⋃S(y)|⩽4|S(x+y)| for x,y∈A2.