Additive decompositions of multiplicative subgroups via differential identities

Abstract

S\'ark\"ozy conjectured that the nonzero quadratic residues modulo a sufficiently large prime have no nontrivial additive decomposition. Hanson and Petridis proved the conjecture for almost all primes, and Kalmynin completed the proof. Kalmynin also developed a general framework for additive decompositions of multiplicative subgroups. More recently, Rudnev and Tyrrell used this framework to classify all additive decompositions of proper multiplicative subgroups of prime fields, showing that the only nontrivial example is the subgroup of order $4$. We give a new self-contained proof of this classification that streamlines the arguments of Kalmynin and of Rudnev and Tyrrell. At the heart of the proof are two new global differential identities. They give an independent proof of Kalmynin's theorem that the two summands have equal size and ultimately reduce the classification to direct coefficient comparisons, avoiding the residue calculations and subsequent arithmetic analysis in the earlier arguments.