A sum product estimates in finite fields and applications英文文献资料

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1、GAFA,Geom.funct.anal.Vol.14(2004)27–57
cBirkh¨auserVerlag,Basel20041016-443X/04/010027-31DOI10.1007/s00039-004-0451-1GAFAGeometricAndFunctionalAnalysisASUM-PRODUCTESTIMATEINFINITEFIELDS,ANDAPPLICATIONSJ.Bourgain,N.KatzandT.TaoAbstract.LetAbeasubsetofafinitefieldF

2、:=Z/qZforsomeprimeq.If

3、F

4、δ<

5、A

6、<

7、F

8、1−δforsomeδ>0,thenweprovetheestimate

9、A+A

10、+

11、A·A

12、≥c(δ)

13、A

14、1+εforsomeε=ε(δ)>0.Thisisafinitefieldanalogueofaresultof[ErS].WethenusethisestimatetoproveaSze-mer´edi–Trottertypetheoreminfinitefields,andobtainanewestimatefortheErd¨osdistanc

15、eprobleminfinitefields,aswellasthethree-dimensionalKakeyaprobleminfinitefields.1IntroductionLetAbeanon-emptysubsetofafinitefieldF.WeconsiderthesumsetA+A:={a+b:a,b∈A}andtheproductsetA·A:={a·b:a,b∈A}.Let

16、A

17、denotethecardinalityofA.Clearlywehavethebounds

18、A+A

19、,

20、A·A

21、≥

22、A

23、.T

24、heseboundsareclearlysharpwhenAisasubfieldofF;howeverwhenAisnotasubfield(oranaffinetransformationofasubfield)thenweexpectsomeimprovement.Inparticular,whenFisthecyclicfieldF:=Z/qZforsomeprimeq,thenFhasnopropersubfields,andoneexpectssomegainwhen1

25、A

26、

27、F

28、.Thefirstmainresul

29、tofthispaperistoshowthatthisisindeedthecase.Theorem1.1(Sum-productestimate).LetF:=Z/qZforsomeprimeq,andletAbeasubsetofFsuchthat

30、F

31、δ<

32、A

33、<

34、F

35、1−δforsomeδ>0.Thenonehasaboundoftheform
max

36、A+A

37、,

38、A·A

39、≥c(δ)

40、A

41、1+ε(1)forsomeε=ε(δ)>0.28J.BOURGAIN,N.KATZANDT.TAOGAFAWenote

42、thatoneneedsboth

43、A+A

44、and

45、A·A

46、ontheleft-handsidetoobtainanestimateofthistype;fortheadditiveterm

47、A+A

48、thiscanbeseenbyconsideringanarithmeticprogressionsuchasA:={1,...,N},andforthemultiplicativeterm

49、A·A

50、thiscanbeseenbyconsideringageomet-ricprogression.Thustheabovee

51、stimatecanbeviewedasastatementthatasetcannotbehavelikeanarithmeticprogressionandageometricprogres-sionsimultaneously.ThissuggestsusingFreiman’stheorem[Fr]toobtaintheestimate(1),butthebestknownquantitativeboundsforFreiman’stheorem[C]areonlyabletogainalogarithmic

52、factorin

53、A

54、overthetrivialbound,asopposedtothepolynomialgainof

55、A

56、εinourresult.Wedonotknowwhattheoptimalvalueofεshouldbe.IfthefinitefieldFwerereplacedwiththeintegersZ,thenitisan