Questions tagged [sumset]

Let $A,B$ be sets. Their sumset $A+B$ is $$ A+B = \{a+b : a\in A, b\in B\} $$Sumsets are a fundamental tool in additive number theory. The Schnirelmann density $\sigma$, named after Lev Schnirelmann, is a measure of how dense a subset of integers is in $\mathbb{N}^+$: $$ \sigma(S) = \inf_{n\in\mathbb{N}^+} \frac{|S \cap[n]|}{n} $$ Questions about representations can be rephrased in terms of this density; for instance, if $S=\{k^2\}_{k=0}^{\infty}$, Lagrange's four-squares theorem is equivalent to the statement $$ S+S+S+S=\mathbb{N}\cup \{0\} $$Similarly, Waring's Problem is the affirmative statement that for all $n\in\mathbb{N}^+$, if $S_n=\{k^n\}_{k=0}^{\infty}$, there is an integer $g(n)$ such that $$ \underbrace{S_n+\cdots +S_n}_{g(n)}=\mathbb{N} $$

Sumsets are not as well-behaved with real numbers. For example, if $\mathcal{C}$ is the Cantor set and $m$ is Lebesgue measure, $m(\mathcal{C})=0$ but $m(\mathcal{C}+\mathcal{C})=2$.