Uniform iterated-sumset growth at the first doubling threshold

Definitions and mathematical statement

Let $A$ be a finite nonempty subset of $\mathbb Z$. For a positive integer $n$, define the iterated sumset \[ nA=\{a_1+\cdots+a_n:a_1,\ldots,a_n\in A\}, \] where repeated summands are allowed. Thus $2A=A+A$, not the dilate $\{2a:a\in A\}$. All cardinalities count distinct integers. For a real parameter $k>1$, define the optimal uniform multiplier \[ f(n,k)=\sup\left\{\frac{|nA|}{|A|}: A\subseteq\mathbb Z\text{ finite and nonempty}, |2A|<k|A|\right\}. \] The supremum, rather than an assumed maximum, is appropriate because sets of arbitrarily large cardinality are allowed. It is the smallest constant in $|nA|\le f(n,k)|A|$ valid simultaneously for all sets meeting the strict doubling hypothesis.

The problem asks to determine this function and, especially, its growth as $n\to\infty$ with $k$ fixed. Estimates should be uniform in $A$, including its cardinality, diameter, and additive structure. For an individual fixed set, the behaviour of $|nA|$ can be much more constrained than the supremum over all eligible sets, which may vary with $n$. The strict sign in $|2A|<k|A|$ is part of the source question and matters at threshold values of $k$. For $k>1$ singletons are eligible; the range avoids an empty family at $k\le1$. The case $n=1$ gives the normalization $f(1,k)=1$.

Short English statement

What is the optimal uniform bound on repeated sumset growth when the original set has doubling ratio strictly below a fixed threshold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001128 (AIM-COMBINATORICS-0253), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 3.16. Original workshop question underlying AIM-COMBINATORICS-0253. https://aimath.org/WWN/additivecomb/additivecomb.pdf.