{"id":"palomar/PALOMAR-2026-09-30-000021","url":"https://github.com/kleinwaks/sum-difference/tree/9bd88c1a14ecedc6ecc6d96eb6403a62d9fc0989","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-30-000021-v1.json","repo":"https://github.com/kleinwaks/sum-difference","tags":["math.NT","math.CO","msc:11B30","msc:11B75","msc:94A17","additive-combinatorics","sumset-inequalities","information-theory"],"score":0,"title":"Improved sum–difference inequalities in abelian groups","trust":"high","domain":"math.NT","source":"palomar","authors":["Logan J. Kleinwaks"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-30","reactions":{},"formalized":true,"formalizes":[{"title":"Improved sum–difference inequalities in abelian groups","identifier":"","relationship":"formalizes"},{"title":"Information inequalities for five random variables","identifier":"https://arxiv.org/abs/2512.23316v2","relationship":"formalizes"},{"title":"Entropy and set cardinality inequalities for partition-determined functions","identifier":"https://arxiv.org/abs/0901.0055","relationship":"formalizes"}]}