Small Doubling in the Integers: Near-Progressions, Excess, and Experimental Classification

This note investigates near-extremal configurations for the classical sumset inequality |A + A| ≥ 2|A| − 1 in the integers, focusing on finite sets A ⊂ ℤ with small excess b(A) := |A + A| − (2|A| − 1) ∈ {0, 1, 2}. Using Freiman’s 3k−4 theorem, we establish that such sets are contained in short arithmetic progressions with at most two missing points. We complement this structural result with an exhaustive computational classification, up to affine equivalence, of all such sets of size |A| ≤ 10. The resulting canonical representatives reveal a variety of local defect patterns (endpoint shifts and interior holes) while confirming the theoretical containment bounds. We present detailed tables and examples for n = 10, include the complete enumeration code in an appendix, and formulate concrete stability conjectures suggesting that only finitely many perturbation patterns occur for each fixed excess, even for arbitrarily large n.