Ackelsberg 2026 inverse theorem sumsets sets positive density

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Ethan Ackelsberg, Florian K. Richter, An inverse theorem for sumsets of sets of positive density in the integers. arXiv:2604.12864 (2026).

The paper proves an inverse theorem (Theorem 1.4) for Kneser's sumset inequality in the integers: if A, B are subsets of N whose densities exist, with d(A) > 0, d(A)+d(B) < 1, B meeting every residue class, and d_N(A+B) = d(A)+d(B) along some scale sequence N, then for a subsemigroup H = hN a translate of A lies in H and a translate of B splits into a part B_0 in H and a part B_1 off H, and either they are, up to zero density, lifts of parallel Bohr intervals given by an irrational rotation n -> n*theta, or a degenerate case (2) holds in which B_0 has zero density along N and A and B are invariant, up to zero density along N, under all shifts in H. This is the integer analog of the known compact-group inverse theorem (Theorem 1.2 of Kneser and Griesmer), and the proof combines intermediate-scale U^1 and U^2 seminorms, arithmetic regularity and Gowers-norm machinery, almost-periodicity, and an ergodic-theoretic endgame using Host-Kra seminorms and Furstenberg systems. Sections 15.1-15.2 construct explicit examples in case (2), including a pair whose sumset has lower density d(A)+d(B) < 1 while the upper density of A+B equals 1, so the density of A+B need not exist. For problem 335 (the Erdos-Graham question of characterizing all A, B with d(A+B) = d(A)+d(B), stated as Problem 1.5), the paper gives a full answer under the extra hypothesis that B meets every residue class, and refutes the Erdos-Graham speculation that all such pairs come from Bohr-interval-like constructions: case (2) and a simple divisibility example (Example 1.6, a random subset of the even numbers with d(A) = 1/4, d(A+A) = 1/2) are counterexamples.

Source: https://arxiv.org/abs/2604.12864. The arXiv record (https://arxiv.org/abs/2604.12864, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #335

Results to transcribe.

• Theorem 1.4: Inverse theorem for sumsets in the integers: if d(A) > 0, d(A)+d(B) < 1, B meets every residue class and d_N(A+B) = d(A)+d(B), then A and B decompose over a subsemigroup H = hN and are either lifts of parallel Bohr intervals for an irrational rotation, or satisfy the degenerate shift-invariance condition (2).
• Problem 1.5: Restatement of Erdos-Graham Problem #335: characterize all A, B in N with positive densities and d(A+B) = d(A)+d(B).
• Example 1.6: A random subset of the even numbers has d(A) = 1/4 and d(A+A) = 1/2 almost surely, giving a divisibility-based counterexample to the Erdos-Graham guess that equality forces Bohr-interval structure.
• Proposition 15.1: For every alpha in (0,1) there are a set A with d(A) = alpha and a set B meeting every residue class with d(A+B) = d(A); the paper offers these as explicit examples of case (2) of Theorem 1.4.
• Proposition 15.2: For sequences N, M with N_{s-1}/M_s -> 0 and M_s/N_s -> 0, alpha < 1/h and beta = 1 - 1/h, and sets A* in hN, B* with densities alpha and beta, there are A in hN and B with d(A) = alpha, d(B) = beta, A+h and A agreeing and B and N \ hN agreeing up to zero density along N, A+B of lower density alpha+beta (attained along N), and A, B agreeing with A*, B* up to zero density along M. With suitable random A*, B* this yields a pair with lower density of A+B equal to d(A)+d(B) < 1 but upper density 1, so the asymptotic density of A+B can fail to exist in case (2).