A two-sided transfer principle for the least doubling of 3-AP-free sets

Definitions and mathematical statement

For a finite set $A\subseteq\mathbb Z$, its sumset is $A+A=\{a+b:a,b\in A\}$; equal summands are permitted and each resulting integer is counted once. Say $A$ is progression-free if there are no three distinct elements $x,y,z\in A$ with $x+z=2y$. Equivalently, $a,a+d,a+2d$ cannot all lie in $A$ for any integer $d\ne0$. For each positive integer $n$, define \[ f(n)=\min\{|A+A|:A\subseteq\mathbb Z, |A|=n, A\text{ is progression-free}\}. \] The family is nonempty, and possible sumset sizes are positive integers, so the minimum is well defined even though the ambient integers are unbounded.

Freiman's problem asks how small $f(n)$ can be, especially its sharp growth as $n\to\infty$. One may equivalently seek the smallest possible doubling ratio $f(n)/n$ under the progression-free condition. A lower bound must hold for every eligible integer set; a matching upper bound requires actual sets of size $n$ with that small a sumset. No upper bound on the diameter or largest element of $A$ is part of the hypothesis. Translating or multiplying every element by the same nonzero integer preserves both the condition and the sumset cardinality. The target therefore concerns the additive structure of the set rather than how it has been placed in an interval.

Short English statement

What is the smallest sumset of an n-element integer set containing no nonconstant three-term arithmetic progression?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001087 (AIM-COMBINATORICS-0212), 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 1.6. Original workshop question underlying AIM-COMBINATORICS-0212. https://aimath.org/WWN/additivecomb/additivecomb.pdf.