On sets with small doubling

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1、Onsetswithsmalldoubling∗ShkredovI.D.Annotation.LetGbeanarbitraryAbeliangroupandletAbeafinitesubsetofG.Ahassmalladditivedoublingif

2、A+A

3、≤K

4、A

5、forsomeK>0.ThesesetswerestudiedinpapersofG.A.Freiman,Y.Bilu,I.Ruzsa,M.C.–Chang,B.GreenandT.Tao.Inthearticleweprovethatifwehavesomeminorres

6、trictionsonKthenforanysetwithsmalldoublingthereexistsasetΛ,Λ≪εKlog

7、A

8、suchthat

9、A∩Λ

10、≫

11、A

12、/K1/2+ε,whereε>0.IncontrasttothepreviousresultsourtheoremisnontrivialforlargeK.ForexampleonecantakeKequals

13、A

14、η,whereη>0.Weuseanelementarymethodinourproof.1.Introduction.LetGbeanarbitraryAbel

15、iangroupwithadditivegroupoperation+.SupposethatA,BaretwofinitesubsetsofGanddefinetheirsumsetA+Btobethesetofallpairwisesumsa+bwitha∈A,b∈B.Letlogstandforthelogarithmtobase2.SupposethatAisasetsuchthat

16、A+A

17、≤K

18、A

19、,whereK≥1issmall(forexampleK=loglog

20、A

21、orK=2).Thesesetsarecalledsetswith

22、smalldoubling.Thepropertiesofsuchsetswerestudiedinpapers[4,5,6,7,9,10,13,14,15,16].G.A.Freiman(see[4])provedthefollowingwonderfulresultonthestructureofthesesets.RecallthatasetQ⊆Giscalledad–dimensionalarithmeticprogressionifQ={n0+n1λ1+···+ndλd:0≤λi

23、h.NT]11Mar2007wheremi,ni∈Zandmi≥0.LetG=Z.Theorem1.1(Freiman)LetK≥1bearealnumber,andA⊆Zbeafiniteset.Letalso

24、A+A

25、≤K

26、A

27、.Thenthereexistnumbersd=d(K)andC=C(K)dependonKonlyandd–dimensionalarithmeticprogressionQsuchthat

28、Q

29、≤C

30、A

31、andA⊆Q.Thefunctionsd=d(K)andC=C(K)werestudiedin[6,7].Inpa

32、per[7]M.–C.Changprovedthatd=O(K2log2K)andC=exp(O(K2log2K))(asusualweuseX=O(Y)orX≪YtodenoteanestimateoftheformX≤MYforsomeabsoluteconstantM).nLetnbeapositiveinteger.SetswithsmalldoublingingroupsG=(Z/2Z)wereconsid-neredin[9,17,13,14,16].Forexampleweformulateatheoremfrom[9].Notet

33、hat(Z/2Z)isavectorspace.∗ThisworkwassupportedbyRFFIgrantno.06-01-00383,President’sofRussianFederationgrantN1726.2006.1andINTAS(grantno.03–51–5-70).1nTheorem1.2LetK≥1bearealnumber.LetA⊆(Z/2Z)beasetsuchthat

34、A+A

35、≤2K4K

36、A

37、.ThenAiscontainedinasubspaceHwith

38、H

39、≤K2

40、A

41、.Thereareanothers

42、tructuralresultsonsetswithsmalldoubling.LetAbeasetwithsmalldoublinga