Hypergraph Regularity Lemmas and the Inverse Littlewood-Offord Problem in Additive Combinatorics

This monograph establishes a rigorous structural isomorphism between the decomposition factors of the Gowers-Rödl hypergraph regularity lemma and the generalized arithmetic progression (GAP) structures characterizing the Inverse Littlewood-Offord problem. We analyze the concentration probability of random walks i=1n i vi in abelian groups, where i are Bernoulli random variables. By encoding the additive energy of the set V={v1, …, vn} into a k-uniform hypergraph functional, we demonstrate that the "structured" component of the regularity partition corresponds precisely to the algebraic generators of the GAP containing V. Conversely, we prove that the "pseudorandom" error term in the regularity decomposition implies a sharp upper bound on the small-ball probability, effectively linearizing the inverse theorem of Tao and Vu via higher-order Fourier analysis. We further discuss the implications of this correspondence for the singularity probability of random discrete matrices.