Inverse theorems for small restricted sumsets in prime cyclic groups

In a prime cyclic group, which pairs of sets attain the lower bound |A⊕B|=|A|+|B|−2 for restricted sums a+b with a≠b? A proposed inverse theorem classifies them as arithmetic progressions when |A|+|B|≤p−1.

Why it matters and what counts as progress

Why it matters: An inverse theorem describes extremal structure, so an interior-range exception changes the classification rather than only a boundary case.

What counts as progress: A correct equality classification or explicit sets satisfying the non-boundary hypotheses but not the claimed arithmetic-progression conclusion.

How the problem developed

1996 · Open question [1]
Noga Alon, Michael Nathanson and Imre Ruzsa Posed the inverse problem of characterizing equality cases in the restricted-sumset lower bound for prime cyclic groups.
9 Sept 2026 · Theorem [1]
Jiantao Li and Xinyi Liang Released a version of their restricted-sumset inverse theorem and conjecture; the public v2 record states a refined theorem under |A|+|B|≤p−1.
16 Oct 2026 · Counterexample claim [2]
AItoBit project maintainers Registered a Z/13 example with |A|+|B|=11 and restricted sumset size 9, contradicting the stated arithmetic-progression conclusion in Li–Liang version 2.

AI contribution

Counterexamples to the restricted-sumset inverse theorem in arXiv:2609.10148v2

alexander

In ℤ/13ℤ, A={0,1,3,4,5,6,10} and B={0,1,4,6} have |A|+|B|=11≤12. Their restricted sumset is {1,3,4,5,6,7,9,10,11}, of size 9=|A|+|B|−2, but neither A nor B is an arithmetic progression. This contradicts the forward conclusion of Li–Liang Theorem A/3.1 in arXiv:2609.10148v2.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 decide +kernel and independent exact finite enumeration)

Human understandingFull