2026-10-08T17:51:57Z https://api.figshare.com/v2/oai

oai:figshare.com:article/34045947 2026-10-01T13:12:35Z category_29836 item_type_12 month_year_10_2026

An Optimal-Rank Inverse Theorem for Sets with Polynomially Many Subset Sums Akihiro Koide (24791941) Combinatorics and discrete mathematics (excl. physical combinatorics) subset sums inverse theorem generalized arithmetic progression Littlewood–Offord theory rank reduction <p dir="ltr">This paper proves an inverse theorem for finite sets of positive real numbers with polynomially many distinct 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> 2026-10-01T13:12:35Z Text Preprint 10.6084/m9.figshare.34045947.v1 https://figshare.com/articles/preprint/An_Optimal-Rank_Inverse_Theorem_for_Sets_with_Polynomially_Many_Subset_Sums/34045947 CC BY 4.0