I was reading Henry Mann's proof for the Schnirelmann-Landau Conjecture from 1942 which can be found in JSTOR here
Today, the Schnirelmann-Landau Conjecture is known as Mann's Theorem:
$$d(C) \ge \min(d(A)+d(B),1)$$
Here is how Mann defines his terms:
Let $A(B,C)$ be sets of positive integers. Form $A^0,B^0$ by adjoining $0$ to $A$ and $B$ respectively. Let $A(n)$ be the number of positive integers in $A$ that are $\le n$. The greatest lower bound of the quotients $A(n)/n$ is called the density of $A$. Let $C^0$ consist of all integers of the form $a + b(a \in A^0,b \in B^0).$
Here's how defines $n_r$:
Let $n_1 < n_2 < \cdots$ be the numbers of $\overline{C}$ (i.e. numbers not in $C$); then $C(n_r) = n_r - r$.
When it comes to the first step in the proof, Mann divides up $[1,n_r]$ into three sets:
We divide up the numbers $\le n_r$ into three sets: numbers of $B$, numbers of the form $n_r - a(a\in{A^0})$, numbers of $L_r$ where $L_r$ denotes the set of all positive numbers $\le n_r$, that are neither in $B$ nor of the form $n_r - a(a \in{A^0})$. Denote by $l_r$ the number of integers in $L_r$. These three sets are disjoint.
How can we be sure that the numbers of $B$ and the numbers of the form $n_r - a(a\in{A^0})$ are disjoint?
For example, if $n_r=5, \{1,3\} \subseteq B, \{2\} \subseteq A$, then:
$B_{n_r} = \{1, 3\}, A_{n_r-a} = \{3\}, L_{r} = \{4\}$ which are not disjoint.
I am clearly missing one of Mann's assumptions. What assumption does Mann make that guarantees that the three sets are disjoint?
Edit: Thanks to Ted Shifrin, I've realized my misunderstanding! My example above is wrong because of sumsets. If $n_r = 5$, then $n_r - a \notin B$ since by assumption $n_r \notin C$ and $C = A + B$ where $+$ is the sumset.