Castelnuovo-Mumford regularity of projective monomial curves via sumsets

Let $A=\{a_0,\dots ,a_{n-1}\}$ be a finite set of $n\ge 4$ 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 $A$, $${\mathcal{C}}_A=\{(v^d:u^{a_1}{v}^{d-a_1}:\cdots :u^{a_{n-2}}{v}^{d-a_{n-2}}:u^d)\mid (u:v)\in {\mathbb{P}}_k^1\}\subset {\mathbb{P}}_k^{n-1}\mathrm{.}$$ 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.