An Optimal-Rank Inverse Theorem for Sets with Polynomially Many Subset Sums

Almost all elements lie in a proper symmetric generalized arithmetic progression of polynomial size, with an optimal rank bound determined by the subset-sum growth exponent. The exceptional proportion tends to zero uniformly over the stated class of sets.

The proof combines inverse Littlewood–Offord containment, a subspace theorem, and integer coordinate elimination. The progression-size exponent and the decay of the exceptional proportion are not optimized.</p><p><br></p><p dir="ltr">The deposit includes the manuscript PDF, LaTeX source, and two exact-arithmetic Python verification scripts with reference data. These check finite coordinate identities, projection collisions, box inclusions, and grid-counting examples; the general theorem is proved in the manuscript.</p>