ModernStochastics:TheoryandApplications2(2015)371–389 DOI:10.15559/15-VMSTA44 6 1 0 2 n a J On packing dimension preservation by distribution 6 functions of random variables with independent ] ˜ R Q-digits P . h t OleksandrSlutskyi a m DepartmentofMathematicalAnalysisandDifferentialEquationsofDragomanov [ NationalPedagogicalUniversityofUkraine,Ukraine 1 v [email protected](O.Slutskyi) 5 3 Received:16October2015,Revised:13December2015,Accepted:13December2015, 1 Publishedonline:23December2015 1 0 Abstract Thearticleisdevotedtofindingconditionsforthepackingdimensionpreservation ˜ . bydistributionfunctionsofrandomvariableswithindependentQ-digits. 1 The notion of “faithfulness of fine packing systems for packing dimension calculation” 0 is introduced, and connections between this notion and packing dimension preservation are 6 1 found. : v Keywords Packingdimensionofaset,Hausdorff–Besicovitchdimensionofaset, ˜ i faithfulnessoffinepackingsystemforpackingdimensioncalculation,Q-expansionofreal X numbers,packing-dimension-preservingtransformations r 2010MSC 28A78,28A80 a 1 Introduction Let (M,ρ) be a metric space. Suppose that the Hausdorff–Besicovitch dimension dim [8]iswelldefinedin(M,ρ).Atransformationf : M → M iscalleddimen- H sion-preservingtransformation[13]orDP-transformationif dim f(E) =dim (E), ∀E ⊂M. H H Let G(M,dim ) be the(cid:0)set of(cid:1)all DP-transformationsdefined on (M,ρ). It is H easytoseethatGformsagroupw.r.t.thecompositionoftransformations.Itiswell knownthatanybi-Lipschitztransformationbelongstothisgroup[8].However,Gis ©2015TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense. www.i-journals.org/vmsta 372 O.Slutskyi essentially wider than the group of all bi-Lipschitz transformations. In 2004, some sufficientconditionsforbelongingofdistributionfunctionsofrandomvariablewith independents-adicdigitstogroupGwasprovedbyG.Torbinetal.[2].Thereexist alotofDP-functionsthatarenotbi-Lipschitz. Sufficientconditionsfordistributionfunctionsofrandomvariableswithindepen- dents-adicdigitstobeDP havebeenfoundbyG.Torbin[13]in2007.Thesecon- ditionsweregeneralizedforQbyG.Torbin[14]andlaterforQ∗-andQ˜-expansions byS.Albeverio,V.Koshmanenko,M.Pratsiovytyi,andG.Torbin[3,4]. Recently, G. Torbin and M. Ibragim proved rather general sufficient conditions fordistributionfunctionsofrandomvariableswithindependentQ˜-digitstobeinDP- class.Thenotionoffinecoveringsystemfaithfulnessfordim calculation[5]plays H animportantroleintheproofoftheseconditions.Thisnotiongivesusthepossibility toconsidercoveringsbysetsfromsomefamilyΦandtobesurethata“dimension” calculated in such a way is equal to dim . Faithfulness of the family of all s-adic H cylinders (if s is fixed) have been proven by Billingsley [6] in 1961. Faithfulness of the family of Q-cylinders have been proven by M. Pratsiovytyi and A. Turbin [16] in 1992, and faithfulness of the family of Q∗-cylinders (under the condition of separation from zero of the corresponding coefficients) have been proven by S. AlbeverioandG.Torbin[1]in2005.Itisnecessarytoremarkthatthelastresultcan beeasilygeneralizedtoQ˜-expansionunderasimilarcondition. In1982,C.Tricot[15]introducedthenotionofpackingdimensiondim .Thisdi- P mensionisinsomesensedualtotheHausdorff–Besicovitchdimension:thedefinition ofdim ofasetF isbasedonε-coveringsofthisfigure,butthedefinitionofdim H P isbasedonε-packings(thecountablesetsofdisjointopenballsB (r ,c ),k ∈ N, k k k withradiir 6 εandcentersc ∈ F).Thepackingdimensionhasall“good”prop- k k erties of a fractal dimension, such as the countable stability. Therefore, proving or disprovingsimilarresultsfordim isimportant.Forexample,weconsiderthegroup P ofpacking-dimension-preservingtransformations(orPDP-transformations). Definition1.1. Thetransformationf issaidtobeaPDP-transformationif ∀E ⊂M, dim f(E) =dim (E). P P (cid:0) (cid:1) Therearealotofproblemswithprovingofmanyconjecturesfordim because P workwithpackingsisessentiallymorecomplicatedthanworkwithcoverings[10]. These problemsare solving bit by bit. For example, M. Das [7] has proven the Billingsleytheoremforpackingdimension;J.Li[9]obtainedsomesufficientcondi- tionsfordistributionfunctionsofrandomvariableswithindependentQ˜-digitstobe inPDP-class.Namely,J.Lihasproventhefollowingtheorem. Theorem1.1. LetF bethedistributionfunctionofarandomvariableξ withinde- ξ pendentQ˜-representation.If inf q = q > 0 and inf p = p > 0, then F i,j ij ∗ i,j ij ∗ ξ preservesthepackingdimensionifandonlyif h +h +···+h 1 2 k limsup =1, b +b +···+b k→∞ 1 2 k whereh =− nj p lnp andb =− nj p lnq . j i=1 ij ij j i=1 ij ij P P PDP-transformations 373 In Remark 4.2 at the and of article [9], we read: “The conditions inf q = i,j ij q > 0 andinf p = p > 0playanimportantrolein theproofofthetheorem. ∗ i,j ij ∗ Openquestion:Whatcanwesayaboutthetopicifweremovetheseconditions?” S.Albeverio,M.Pratsiovytyi,andG.Torbin[3]removedtheconditioninf p = i,j ij p >0inasimilarsituationforDP-transformations. ∗ In case of packingdimension,the approachof [3] is complicatedbecause it re- quiresappropriateresultsaboutthefinepackingsystem faithfulnessforpackingdi- mensioncalculation.Eventhedefinitionofthefinepackingsystemfaithfulnessisa problembecausecentersofallballsinpackingsshouldbeinthesetthedimensionof whichiscalculated. Theaimofthispaperistoproposesomealternativedefinitionofthepackingdi- mension,uncenteredpackingdimensionordimP(unc).Intheproposeddefinition,the condition“thecentersofballsshouldbeinthefigurethedimensionofwhichiscal- culated”inthedefinitionofdim isreplacedby“everyballshouldhaveanonempty P intersectionwiththefigure.”Weprovethat,insomewideclassofmetricspaces(in- cluding Rn), the value of packing dimension with uncentered balls is matching to the value of classical packing dimension. Introduction of the fine packing system faithfulnessnotionis verysimple in the case of proposeddefinition.It allows usto provefaithfulness(undertheconditionofseparationfromzeroofthecoefficients)of a Q˜-cylindersystem and sufficientconditionsforthe distributionfunctionof a ran- domvariablewithindependentQ˜-digitstobeinthePDP-class.Thecorresponding theoremisthemainresultofthepaper. Theorem1.2. Letinf q :=q .Supposethatq >0.Let i,j ij min min q T := k :k ∈N,p < min ; k 2 (cid:26) (cid:27) T :=T ∩{1,2,...,k}; k ln 1 B :=limsup j∈Tk pj . k k→∞ P LetF be the distributionfunctionofa randomvariableξ with independentQ˜- ξ representation.ThenF preservesthepackingdimensionifandonlyif ξ dim µ =1; P ξ (B =0. 2 Packingdimension Letusrecallthe definitionofpackingdimensionin the formgiven,forexample,in [8]. Definition2.1. LetE ⊂ M andε > 0. A finite orcountablefamily{E }of open j ballsiscalledanε-packingofasetEif 374 O.Slutskyi 1. |E |6εforalli; i 2. c ∈E, i∈N,wherec isthecenteroftheballE ; i i i 3. E ∩E =∅foralli,j,i6=j. i j Remark2.1. Theemptysetofballsisapackingofanyset. Definition2.2. LetE ⊂M,α>0,ε>0.Thentheα-dimensionalpackingpremea- sureofaboundedsetE isdefinedby Pα(E):=sup |E |α , ε i (cid:26) i (cid:27) X where the supremum is taken over all at most countable ε-packings{E } of E (if j E =∅forallj,thenPα(E)=0). j ε Definition2.3. Theα-dimensionalpackingquasi-measureofasetE isdefinedby Pα(E):= limPα(E). 0 ε ε→0 Definition2.4. Theα-dimensionalpackingmeasureisdefinedby Pα(E):=inf Pα(E ):E ⊂ E , 0 j j (cid:26) j (cid:27) X [ wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M. j j Definition2.5. Thenonnegativenumber dim (E):=inf α:Pα(E)=0 P iscalledtheuncenteredpackingdimensio(cid:8)nofasetE ⊂M(cid:9). 3 Uncenteredpackingdimension Definition3.1. LetE ⊂ M andε > 0. A finite orcountablefamily{E }of open j ballsiscalledanuncenteredε-packingofasetE if 1. |E |6εforalli; i 2. E ∩E 6=∅; i 3. E ∩E =∅foralli,j,i6=j. i j Remark3.1. Theemptysetofballsisanuncenteredpackingofanyset. Definition 3.2. Let E ⊂ M, α > 0, ε > 0. Then the uncentered α-dimensional packingpremeasureofaboundedsetE isdefinedby Pα (E):=sup |E |α , ε(unc) i (cid:26) i (cid:27) X wherethesupremumistakenoverallatmostcountableuncenteredε-packings{E } i ofE. PDP-transformations 375 Definition3.3. Theuncenteredα-dimensionalpackingquasi-measureofa setE is definedby Pα (E):= limPα (E). 0(unc) ε(unc) ε→0 Definition3.4. Uncenteredα-dimensionalpackingmeasureisdefinedby Pα (E):=inf Pα (E ):E ⊂ E , (unc) 0(unc) j j (cid:26) j (cid:27) X [ wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M. j j Remark3.2. If(M,ρ) = R1 andα = 1,thentheα-dimensionalpackingmeasure anduncenteredα-dimensionalpackingmeasurearetheLebesguemeasure. Definition3.5. Thenonnegativenumber dimP(unc)(E):=inf α:P(αunc)(E)=0 . iscalledtheuncenteredpackingdimension(cid:8)ofasetE ⊂M. (cid:9) Theorem3.1. Let(M,ρ)beametricspace.LetC ∈N.Ifforallr >0andforany openballI with|I|=8r,thereexistatmostN(I)ballsI , i∈{1,..., N(I)}such i thatI ⊂I, i∈{1,...,N(I),|I |=r, i∈{1,...,N(I)},and N(I)≤C.Then i i dimP(unc)(E)=dimP(E). Proof. Step1.LetusprovetheinequalitydimP(unc)(E)>dimP(E). Bythedefinitionsandsupremumpropertywehave Pα (E)>Pα(E). r(unc) r Bythelimitpropertyofinequalitieswehave Pα (E)>Pα(E). 0(unc) 0 Hence, Pα (E)>Pα(E). (unc) LetdimP(unc)(E)=α0.BythedefinitionofdimP(unc)(E)wehave ∀ε>0, Pα0+ε(E)=0. (unc) Therefore, ∀ε>0, Pα0+ε(E)=0, 0 and,consequently, dim (E)6α . P 0 Hence,itfollowsthatdimP(unc)(E)>dimP(E),whichisourclaim. Step2.LetusshowthatdimP(unc)(E)6dimP(E). IfdimP(unc)(E)=0,thenthestatementistrue. 376 O.Slutskyi LetusconsiderthecasedimP(unc)(E)6=0.Fix0<t<s<dimP(unc)(E). Sinces<dimP(unc)(E),wehave Ps (E)=+∞, (unc) Ps (E)=+∞. 0(unc) Therefore, ∀r >0, Ps (E)=+∞. r(unc) From this and from the supremum property, it follows that there is an uncentered packingV :={E }ofthesetE with i |E |s >1. (1) i i X LetusdividethepackingV intoclasses V := E :2−k−1 6|E |<2−k . k i i Letn bethenumberofballsV(cid:8).Wewillshowthat (cid:9) k k ∃k0 :nk0 >2k0t 1−2t−s . Toobtainacontradiction,supposethat (cid:0) (cid:1) n <2kt 1−2t−s forallk. k Then (cid:0) (cid:1) |E |s < 2−ks·n < 2−ks·2kt 1−2t−s = 1−2t−s · 2t−s k =1, i k i k k k X X X (cid:0) (cid:1) (cid:0) (cid:1) X(cid:0) (cid:1) whichcontradictsourassumption(1). Therefore,such k exists. Letus considerV . We denotebyA ,A ,...,A 0 k0 1 2 nk0 theballsinV ,thatis, k0 V ={A ,A ,...,A }. k0 1 2 nk0 Fixr :=2−k0−1.ThentheradiusofanyAiislessthanr.LetTibeapointofAi suchthatT ∈A ∩E.LetV′bethesetofballswiththecentersT andradiusr,that i i i is, V′ = A′ :A′ =B(T ,r) . i i i Fix (cid:8) (cid:9) V∗ = A∗ :A∗ =B(T ,4r) . i i i LetusdividethesetV′intoc(cid:8)lassesK1,K2,...,Kl(cid:9)asfollows. 1. LetustakeaballA′ =A′ andputitinK togetherwithallotherballsA′ ∈ V′suchthatA′ ∩Aj′1 6=∅1. 1 i i j1 2. LetustakeanarbitraryballA′ ∈ V′\K andputitinK togetherwithall otherballsA′ ∈V′\K suchjt2hatA′ ∩A′1 6=∅. 2 i 1 i j2 PDP-transformations 377 3. LetuscontinuethiswayuntilV′\(K ∪K ∪···∪K )6=∅.Sincethenumber 1 2 l ofelementsinasetV′isafinite,wecanfindsuchanumberl. Now suppose that the balls A′ and A′ intersect each other. In other words, i j ρ(T ,T )62r.Therefore,A ⊂A∗. i j j i TheradiusofA isgreaterthanr/2.Bythetheoremcondition,therearenomore j thanC disjointballswithradiusr/2inaballwithradius4r Therefore,therearenomorethanC ballsinanyclassK . i Moreover,inthecasei < m,theballsA′ andA′ donotintersecteachother. ji jm Indeed,supposeotherwise.ThenA′ isinaclassK orinaclasswithnumberless jm i thani. Hence, V′′ = A′ ,A′ ,...,A′ j1 j2 jl is a centered packing of a set E, a(cid:8)nd the t-volume of(cid:9)this packing is less than the t-volumeoftheuncenteredpackingV nomorethanC times.Therefore, k0 A′ji t >nk0 · 2−Ck0t >2k0t 1−2t−s · 2−Ck0t = 1−C2t−s. V′′ X(cid:12) (cid:12) (cid:0) (cid:1) (cid:12) (cid:12) Fromthisitfollowsthat 1−2t−s Pt (E)> . 2−k0 C Bytheinequality2−k0 <rweget 1−2t−s Pt(E)> forallr>0. r C Consequently,asr →0,wegettheinequality 1−2t−s Pt(E)> . 0 C LetusshowthatPt(E)> 1−2t−s.Recallthedefinition C Pt(E)=inf Pt(E ):E ⊂ E , 0 j j (cid:26) j (cid:27) X [ wheretheinfimumistakenoverallatmostcountablecoveringsE ofasetE. j Let {Ej} be an at most countable covering of E. Since dimP(unc)(E) > s, thereisj0 suchthatdimP(unc)(Ej0) > s (bythecountablestabilityofthepacking dimensiondimP(unc)).Inotherwords,wehave Ps (E )=+∞, (unc) j0 Ps (E )=+∞. 0(unc) j0 WeconcludebythepartofthetheoremalreadyprovedforE that 1−2t−s Pt(E )> 0 j0 C 378 O.Slutskyi and 1−2t−s Pt(E )> . 0 j C j X Butthepreviousinequalityistrueforanarbitrarycovering{E }ofasetE andfor j theinfimumforallcoverings.Therefore, 1−2t−s Pt(E)> C and dim (E)>t. P Sincet-dimP(unc)(E)canbeapproximatedby0,wegetdimP(E)>dimP(unc)(E), whichcompletestheproof. Corollary3.1. IfM =Rn,thendimP(unc)(E)=dimP(E). Proof. Let B be a ball with radius 8r, B be a ball with radius r, and λ be the 8r r n-dimensionalLebesguemeasure.Then λ(B )=8n·λ(B ). 8r r Therefore,we canputnomorethanC = 8n disjointballswithradiir in aball withradius8r,whichcompletestheproof. 3.1 Packingdimensionwithrespecttothefamilyofsets LetΦbeafamilyofballsinametricspace(M,ρ). Definition3.6. LetE ⊂M,α>0,ε>0.Thentheα-dimensionalpackingpremea- sureofaboundedsetE withrespecttoΦisdefinedby Pα(E,Φ):=sup |E |α , ε i (cid:26) i (cid:27) X where the supremum is taken over all uncentered ε-packings {E } ⊂ Φ of E (if i {E }=∅,thenPα(E,Φ)=0). i ε Definition3.7. Theα-dimensionalpackingquasi-measureofasetE w.r.t.Φisde- finedby Pα(E,Φ):= limPα(E,Φ). 0 ε ε→0 Definition3.8. Theα-dimensionalpackingmeasurew.r.t.Φisdefinedby Pα(E,Φ):=inf Pα(E ,Φ):E ⊂ E , 0 j j (cid:26) j (cid:27) X [ wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M. j j PDP-transformations 379 Definition3.9. Thenonnegativenumber dim (E,Φ):=inf α:Pα(E,Φ)=0 P iscalledthepackingdimensionofasetE(cid:8)⊂M w.r.t.Φ. (cid:9) Remark3.3. Inthedefinitionofdim (E,Φ),weuseduncenteredpacking.Butwe P willdenotethisdimensionwithoutindex(unc)because: 1. WewillworkinRn.Inthisspace,centeredanduncenteredpackingdimensions areequal; 2. Thecenteredpackingdimensionw.r.t.somefamilyofballsisnotdefined. Theorem3.2. dimP(E,Φ)6dimP(unc)(E). Proof. LetΦ bethefamilyofallopenballsofM.Then 0 Pα (E)=Pα(E,Φ ). r(unc) r 0 SinceΦ⊆Φ ,bythesupremumpropertywehave 0 Pα(E,Φ)6Pα(E,Φ ). r r 0 Bytheinequalityforpackingpremeasuresitfollowsthat dimP(E,Φ)6dimP(unc)(E), whichprovesthetheorem. 4 Faithfulnessoftheopenballsfamiliesforpackingdimensioncalculation Definition4.1. SupposethatsomeopenballsfamilyΦsatisfiesthefollowingcondi- tion:forallE ⊂M,dimP(unc)(E,Φ)=dimP(unc)(E).ThenΦissaidtobefaithful foruncenteredpackingdimensioncalculation. Remark4.1. ThenotionoffaithfulnessisintroducedfortheHausdorff–Besicovitch dimensiondim [11].Itisclearthat H ∀Φ⊂2M,dim (E,Φ)>dim (E). H H Theorem 4.1 (The sufficient condition for the open-ball family to be faithful for packingdimensioncalculation). Supposethat 1. Φisafamilyofintervalsfrom[0;1]; 2. ∃C >0:∀(a;b)⊂[0;1],∃∆(a;b)∈Φsuchthat: (a) a+b ∈∆(a,b); 2 (b) ∆(a,b)⊂(a;b); (c) b−a >C. |∆(a,b)| ThenΦisafaithfulopen-ballfamilyforpackingdimensioncalculation. 380 O.Slutskyi Proof. Let E be anyset, α > 0, and r > 0. Let{E } = {(a ;b )} be a family of i i i disjointintervalssuchthat ai+bi ∈E andb −a <r. 2 i i Thenthefollowinginequalityholds: |E |α 6 ∆(a ,b ) α·Cα. i i i i i X X(cid:12) (cid:12) (cid:12) (cid:12) Takingthesupremum(overallsetsofintervals{E }satisfyingthepreviouscon- i ditions),wehave Pα(E)6 sup ∆(a ,b ) α·Cα. r i i {Ei} (cid:12) (cid:12) Anyset of intervals{∆(a ,b )} satis(cid:12)fies the co(cid:12)nditionsfrom the Pα (E,Φ) i i r(unc) definition.So, sup ∆(a ,b ) α·Cα 6Pα (E,Φ)·Cα. i i r(unc) {Ei} (cid:12) (cid:12) Therefore, (cid:12) (cid:12) Pα(E)6Pα (E,Φ)·Cα. r r(unc) Takingthelimitofbothsides,wehave Pα(E)6Pα (E,Φ)·Cα. 0 0(unc) TakingtheinfimumoverallpossiblecoveringsofthesetE,wehave Pα(E)6Pα (E,Φ)·Cα (unc) and dimP(E)6dimP(unc)(E,Φ). Since[0;1]⊂R1,itfollowsthat dimP(E)=dimP(unc)(E) and dimP(unc)(E)6dimP(unc)(E,Φ). Using dimP(unc)(E)>dimP(unc)(E,Φ) forallΦ, weobtainthatΦisafaithfulopen-ballfamilyforthepackingdimensioncalculation. ˜ 5 SufficientconditionsforQ-expansioncylindricintervalfamilytobefaithful TheQ˜-expansionofrealnumbersisageneralizationofs-expansionandQ-expansion andwasdescribed,forexample,in[4]. Theorem5.1. LetΦbethesystemofcylindricintervalsofsomeQ˜-expansion.Sup- posethat infq =q >0. ij min i,j ThenΦisafaithfulballfamilyforpackingdimensioncalculation.