  Let $A=\{a_0,\ldots,a_{n-1}\}$ be a finite set of $n\geq 4$ non-negative
relatively prime integers such that $0=a_0<a_1<\cdots<a_{n-1}=d$. The $s$-fold
sumset of $A$ is the set $sA$ of integers that contains all the sums of $s$
elements in $A$. On the other hand, given an infinite field $k$, one can
associate to $A$ the projective monomial curve $\mathcal{C}_A$ parametrized by
$A$, \[ \mathcal{C}_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots
:u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid \
(u:v)\in\mathbb{P}^{1}_k\}\subset\mathbb{P}^{n-1}_k\,. \] The exponents in the
previous parametrization of $\mathcal{C}_A$ define a homogeneous semigroup
$\mathcal{S}\subset\mathbb{N}^2$. We provide several results relating the
Castelnuovo-Mumford regularity of $\mathcal{C}_A$ to the behaviour of the
sumsets of $A$ and to the combinatorics of the semigroup $\mathcal{S}$ that
reveal a new interplay between commutative algebra and additive number theory.
