Improved sum–difference inequalities in abelian groups

Logan J. Kleinwaks

Palomar Registrypalomar/PALOMAR-2026-09-30-000021Sep 30, 2026math.NTscore 0

For all finite subsets X,Y of an abelian group, |X-Y| <= |X+Y|^λ with λ = (9451e-3286)/(5378e+787) = 1.454277448906…, less than the classical exponent 3/2. A multiscale transfer gives the upper bound (13524e-7359)/(9451e-3286) = 1.312373302115… for the Gyarmati–Hennecart–Ruzsa small-sumset exponent, less than their exponent 4/3. The proof of the first result combines a coupling with distinct differences, entropy inequalities for independent sums, a finite certificate using non-Shannon inequalities, and an explicit limiting certificate. The transfer uses localization and the Plünnecke-Ruzsa…

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