Year: 2026
Type: preprint
Abstract: We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson--Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of \[ H=A+B,\qquad H\cup\{0\}=A-A,\qquad H=A\mathbin{\widehat{+}} A,\qquad H\cup\{0\}=A\mathbin{\widehat{+}} A, \] where $H$ is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions $H=A+B$, building on the approach introduced by Hanson--Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev--Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on $H\cup\{0\}=A-A$. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.
Source: arXiv (Cornell University)
Authors: Cochrane, Albert, Yip, Chi Hoi, Yoo, Semin
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Institute for Basic Science
Award: G1938303756
OpenAlex ID: W7172379028