On the structure of the hh-fold sumsets

Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$, and~$A=\left\{a_{0}, a_{1}, \ldots, a_{k}\right\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\subseteq[0, c_{t}-2]$, $D_{t} \subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \cup\left[c_{t}, h a_{k}-d_{t}\right] \cup\left(h a_{k-1}-D_{t}\right)$$ for all~$h \geq\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.Comment: 8 page

On the structure of tt-representable sumsets

Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$ elements of $A$. We prove that $(hA)^{(t)}$ is ``structured'' for $$h \gtrsim \frac{1}{e} m\ell t^{1/\ell},$$ and prove a similar theorem for $A\subseteq \mathbb{Z}^d$ and $h$ sufficiently large, with some explicit bounds on how large. Moreover, we construct a family of sets $A = A(m,\ell,t)\subseteq \mathbb{Z}_{\geq 0}$ for which $(hA)^{(t)}$ is not structured for $h\ll m\ell t^{1/\ell}$.Comment: 21 page

Castelnuovo-Mumford regularity of projective monomial curves via sumsets

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.Comment: 18 pages, 2 figures, 1 tabl

On the structure of the hh-fold sumsets

Let $A$ be a set of nonnegative integers. Let $(h A)^{(t)}$ be the set of all integers in the sumset $hA$ that have at least $t$ representations as a sum of $h$ elements of $A$. In this paper, we prove that, if $k \ge 2$, and $A=\left\lbrace a_{0}, a_{1}, \dots , a_{k}\right\rbrace$ is a finite set of integers such that $0=a_{0