I am trying to understand the proof by Gelfond & Linnik that:
$$d(A+B) \ge d(A) + d(B) - d(A)d(B)$$
Here's what I understand:
Let $A$, $B$ be infinite sequences of integers starting with $0$ where $0 < a_1 < a_2 < \dots$ and $0 < b_1 < b_2 < \dots$
Let $C$ be an infinite sequence of integers such that $C = A + B$
Let $A(n)$, $B(n)$, $C(n)$ be the number of integers in the sequence less than or equal to $n$ not including $0$.
Let $a_k$, $a_{k+1}$ be two adjacent integers in the sequence $A$. If there is any numbers in between, then they are:
$$a_k + 1, a_k + 2, \dots, a_k + l = a_{k+1}-1$$
So, it follows that the count of numbers in between $a_k$ and $a_{k+1}$ that are found in $C$ is $B(l)$.
From this observation, Gelfond & Linnik conclude:
$$C(n) \ge A(n) + \sum_{(l)}B(l)$$
where the "sum is taken over the intervals between the numbers of the form $a_k$ and $a_{k+1}$."
I am confused about what the above means?
I am also confused about the observations that follow:
Gelfond & Linnik observe that: $B(l) \ge d(B)*l$ where $d(B)$ is the density of $B$ defined as:
$$d(B) = \inf_{n} \frac{B(n)}{n}$$
I am unclear how $l$ can be taken from each interval $a_k$ and $a_{k+1}$ in one case and then be used later as a single value. What does $l$ equal?
Further below, Gelfond & Linnik note that $\sum{l} = (n - A(n))$
I would appreciate it if someone could explain what $\sum_{(l)} {B(l)}$ means and explain how it is clear that $\sum(l) = (n - A(n))$.