On bounded basis of integers

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 r A ( n ) by the number of representations of n of the form n = a + a ′ , where a ⩽ a ′ and a , a ′ ∈ A , and d A ( 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, Nesetřil and Serra (2004) [6] proved that there is a set A of integers satisfying r A ( 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 d A ( n ) = 1 for all positive integers n. Furthermore, we also prove that, (i) there is a set A 1 of integers satisfying r A 1 ( n ) = 1 for all n ∈ Z , the set consisting of positive integers n with d A 1 ( n ) = 0 has density one, and | S ( x ) ⋃ S ( y ) | ⩽ 4 | S ( x + y ) | for x , y ∈ A 1 ; (ii) there is a set A 2 of integers satisfying d A 2 ( n ) = 1 for all n ∈ N , the set consisting of integers n with r A 2 ( n ) = 0 has density one, and | S ( x ) ⋃ S ( y ) | ⩽ 4 | S ( x + y ) | for x , y ∈ A 2 .