I am still trying to get my head around the basic properties of Schnirelmann Density.
If I'm reading PlanetMath.org correctly, it states that if $d(A) + d(B) \ge 1$, then $d(A+B)=1$
Here's the exact wording:
and also if $\sigma{A} +\sigma{B} \ge 1$, then $\sigma(A+B)=1$.
This seems counter intuitive to me. For example, let $A =$ the odd integers: $\left\{1, 3, 5, \dots\right\}$. Let $B = A$.
Then, $d(A)=\frac{1}{2}$ and $d(B) = \frac{1}{2}$ so that $d(A)+d(B)=1$ but $d(A+B) = \frac{1}{2}$.
Here's my understanding of Schnirelmann Density.
$A,B$ are infinite sequences of integers starting with $0$ with in sequential order such as $0, a_1, a_2, \cdots$ where $0 < a_1 < a_2 < \cdots$
Schnirelmann density is defined as: $$d(A) = \inf\limits_{n}\frac{A(n)}{n}$$
where: $$A(n) = \sum\limits_{0<a_i\le{n}}{1}$$
- So, it is clear that: $$0 \le \frac{A(n)}{n} \le 1$$
Is PlanetMath wrong? Am I misunderstanding their statement about Schnirelmann Density?
No, PlanetMath is right. It's just that they mean sumset when they add sets, i.e., $A+B=\mathbb{N}$ not $A+B=A$.
A sumset is the set of all possible sums from elements in $A$ and in $B$. Since $1\in A$, then you can add 1 to any odd integer and get $\mathbb{N}$. https://en.wikipedia.org/wiki/Sumset
Additionally, Wikipedia explicitly says that they mean sumset in their article on the density. https://en.wikipedia.org/wiki/Schnirelmann_density