Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$
Paul Péringuey, Anne de Roton
TL;DR
The paper addresses the structure of additively left-stable sets in the low-sum regime connected to Freiman's $3k-4$ theorem in ${\mathbb Z}$ and ${\mathbb R}$. The authors develop sharp density bounds for left-stable sets on initial segments using continuous and discrete sumset inequalities, notably $\lambda(A_x)\le d\,h(x/d)$ in the real case and discrete analogs derived from Grynkiewicz's bound. They establish a canonical structural decomposition $A=\min(A)+(A_1\cup I\cup(\mathrm{diam}(A)-A_2))$ with extremal constructions that attain equality, proving the bounds are tight and describing uniqueness properties for extremals at specific points. These results yield precise quantitative constraints on near-left densities and enhance the understanding of the critical configurations described by Freiman's $3k-4$ theorem in both continuous and discrete settings, informing the partitioning of near-extremal sets into left-stable components and arithmetic progressions.
Abstract
We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of integers and when $A$ is a bounded subset of real numbers. More precisely we give a sharp upper bound for the density of $A$ in $[\inf(A),x]$ for $x\le\sup(A)$, and construct sets reaching this density for any given $x$ in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman's $3k-4$ theorem in both cases.