MathDB

Let

A

be a nonempty set of positive integers, and let

N(x)

denote the number of elements of

A

not exceeding

x

. Let

B

denote the set of positive integers

b

that can be written in the form

b=a-a^{\prime}

with

a\in A

and

a^{\prime}\in A

. Let

b_1<b_2<\cdots

be the members of

B

, listed in increasing order. Show that if the sequence

b_{i+1}-b_i

is unbounded, then

\lim_{x\to \infty}\frac{N(x)}{x}=0

B6