Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

1.6 MB·English

by  Sabrina Kombrink

LATTICE-TYPE SELF-SIMILAR SETS WITH PLURIPHASE GENERATORS
FAIL TO BE MINKOWSKI MEASURABLE
SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
5
1
0 Abstract. A long-standing conjecture of Lapidus claims that under certain conditions,
2 self-similar fractal sets fail to be Minkowski measurable if and only if they are of lat-
ticetype. ThetheoremwasestablishedforfractalsubsetsofRbyFalconer,Lapidusand
n v.Frankenhuijsen,andtheforwarddirectionwasshownforfractalsubsetsofRd,d ≥2,
a byGatzouras.Sincethen,muchefforthasbeenmadetoprovetheconverse.Inthispaper,
J
weproveapartialconversebymeansofrenewaltheory. Ourproofallowsustorecover

5 severalpreviousresultsinthisregard,butismuchshorterandextendstoamoregeneral
1 setting;severaltechnicalconditionsappearinginpreviousversionsofthisresulthavenow
beenremoved.

]
R
P
.
h 1. Introduction
t
a We address the Minkowski measurability (see Def. 2.1) for self-similar sets in Rd. In
m
particular,weattempttocharacterizethispropertyintermsofthelatticepropertiesofthe
[ underlying iterated function system (IFS). Let S = {S ,...,S } denote an IFS in which
1 N
1 eachSi isacontractivesimilarityactingonRd,calledaself-similarsysteminthesequel.
v Further,letF ⊆Rd denotetheself-similarsetwhichistheuniquenon-emptycompactset
4 satisfying F = (cid:83)N S F; see[Hut81]. IfthescalingratioofS isdenotedbyr,thenthe
76 IFSissaidtobelai=t1ticei ifthereisanr > 0suchthateachri canibewrittenasrkiiforsome
integerk ∈N(seeDef.2.13),otherwise,theIFSissaidtobenonlattice.
3 i
0 For d = 1, it was proven (see [Fal95, LvF12, KK12]) under very general conditions
. that F is Minkowski measurable if and only if the IFS is nonlattice. These conditions
1
include that the Minkowski dimension D of F lies strictly between 0 and 1, and that the
0
5 IFS satisfies the open set condition (OSC), see Def. 2.2. It was conjectured in [Lap93,
1 Conj. 3] that this equivalence statement should remain true for d ≥ 2, when d − 1 <
v: D < d. In [Gat00], Gatzouras was able to prove and strengthen (under the OSC) one
i direction of this conjecture, namely, that for arbitrary d ∈ N and D ∈ (0,d), the self-
X
similar attractor F ⊂ Rd is Minkowski measurable when the IFS is nonlattice. It is an
r
openproblemtoprovetheconverse,andthishasbeenaveryactivearea;see,forexample,
a
[LP10, LPW11, LPW13, RW13, Kom11, DKO¨+13]. Our results in this paper give some
further progress towards establishing the converse, i.e., showing that the attractor of a
latticeself-similarsystemisnotMinkowskimeasurable. Atthesametimewedemonstrate
thatitisessentialtoexcludesetsofintegerMinkowskidimension Dfromtheconjecture
butthatitisplausibletoextendLapidus’conjecturetothesettingofnon-integerD∈(0,d);
seeRem.1.2.
2010MathematicsSubjectClassification. Primary: 11M41,28A12,28A75,28A80,52A39,52C07,Sec-
ondary:11M36,28A78,28D20,42A16,42A75,52A20,52A38.
Keywordsandphrases. Self-similarset,latticeandnonlatticecase,Minkowskidimension,Minkowskimea-
surability,Minkowskicontent.
VersionofJanuary16,2015.
1
2 SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
Weworkinasettingwhichincludes–tothebestofourknowledge–alltheprevious
casesinwhichtheMinkowskimeasurabilityoflatticeself-similarsetshasbeenaddressed
(see the detailed discussion at the very end of the introduction) and which extends the
class of sets covered in several directions. For instance, we do not require the set F to
possess a compatible feasible open set O satisfying the OSC, that is, one which satisfies
bdO ⊂ F. Thisallowsinparticulartotreatself-similarsetsofanyMinkowskidimension
DandremovestheassumptionD>d−1,whichispresentinallpreviousworkknownto
theauthors. Insteadofcompatibility,wewillassumethroughoutthatthefeasibleopenset
Oweworkwithsatisfiesthefollowingadditionalconditions:
• (StrongOSC)O∩F (cid:44)∅;
• (Projectioncondition)S O⊆π−1(S F)fori=1,...,N.
i F i
Here π denotes the metric projection onto F (see Def. 2.7) and A denotes the closure
F
of A ⊂ Rd. It follows from results in [BHR06] that one can always find a feasible open
set satisfying both the strong OSC (SOSC) and the projection condition whenever OSC
is satisfied (the “central open set”; see Rem. 2.8). Therefore, these two conditions alone
do not restrict at all the class of self-similar sets considered; they should be seen as a
convenientchoiceoffeasibleopensetwemakeinordertosimplifytheproblem. Theonly
further (rather restrictive) assumption we require is the following. We suppose that the
ε-parallelsetF ofF (see(2.1))iswell-behavedintheset
ε
(cid:91)N
Γ=Γ(O):=O\ S O. (1.1)
i
i=1
Moreprecisely,itisrequiredthattheparallelvolumeλ (F ∩Γ)ofFrestrictedtothesetΓ
d ε
(whereλ isLebesguemeasure)ispiecewisepolynomialinthevariableε. Inthiscasewe
d
calltheset F pluriphasewithrespecttoΓ(andwecall F monophasewithrespecttoΓif
thisparallelvolumeisapolynomial),seeDef.2.9andthediscussionafterwardsfordetails.
Thepluriphaseconditionisasimplifyingassumptiononthegeometryof F whichwould
ideallyberemovedinfuturework. Note: thedefinitionsofpluriphaseandmonophaseare
extended here to the present more general setting. In case of a compatible feasible set it
reduces to the pluriphase/monophase assumption made in earlier work on this topic, e.g.
in[DKO¨+13,Kom11,LPW13].). SeealsoRem.2.11.
Themainresultsofthispaperaresummarizedinthefollowingstatement.
Theorem 1.1. Let F ⊂ Rd be a self-similar set which is the attractor of a lattice self-
similarsystemS = {S ,...,S }, N ≥ 2satisfyingtheOSC.Let D := dim F denoteits
1 N M
Minkowskidimension.
(i) IfD=dimaffF (wherethelatteristhedimensionoftheaffinehullofF),thenF
isMinkowskimeasurable. Inparticular,thisistrueforD=d.
(ii) SupposeD<disnotanintegerandthereexistsastrongfeasiblesetOsatisfying
the projection condition such that F is pluriphase with respect to the set Γ(O).
ThenF isnotMinkowskimeasurable.
(iii) SupposeD<disanintegerandthereexistsastrongfeasiblesetOsatisfyingthe
projectionconditionsuchthatF ispluriphasewithrespecttothesetΓ(O). Then
FisMinkowskimeasurableifandonlyifcertainalgebraicrelationsinvolvingthe
dataofthepluriphaserepresentationaresatisfied. Inparticular,theserelations
areneversatisfiedinthecasewhenF ismonophasewithrespecttoΓ(O).
Part(ii)isreformulatedandprovedinThm3.4;amorepreciseformulationofpart(iii),
thecasewhen Disaninteger,isgiveninThm3.6. Westressthatinthesituationofpart
LATTICENONMEASURABILITYINTHEPLURIPHASECASE 3
(iii)bothcasesarepossible: latticesetsinRd ofintegerMinkowskidimensionD < dcan
beMinkowskimeasurableornot;seeRem.1.2.
Asforpart(i),wecangiveashortproofimmediately.
ProofofThm.1.1(i). As a function of ε, the tubular volume λ (F ) is continuous and
d ε
strictlyincreasingon(0,∞)foranycompactsetF ⊆Rd,andthusM (F):=lim λ (F )=
d ε(cid:38)0 d ε
λ (F); seeDef.2.1forthefulldefinitionofMinkowskicontentM . Inotherwords, the
d d
limit must exist and it is not difficult to see that it coincides with λ (F). Now, if F is a
d
self-similar set with dim F = d satisfying OSC, then it is well known that F has in-
M
terior points (see e.g. [Sch94, PW12]) and therefore M (F) = λ (F) > 0 . Thus F
d d
is Minkowski measurable as claimed. Now, if F ⊆ Rd is a self-similar set such that
dim F = dimaffF, then Minkowski measurability follows from working in the affine
M
hull and observing that Minkowski measurability is independent of the dimension of the
ambientspace;cf.[Res13,Kne55]andthereferencestherein. (cid:3)
Remark1.2. Theaboveproofindicatesthatifdim F = dimaffF,then F isMinkowski
M
measurableregardlessofwhetheritislatticeornonlattice. Thesignificanceofthispoint
isasfollows: onemaybenaturallyledtosupposethattheoriginalconjectureofLapidus
maybeextendedtoincludesetswithMinkowskidimensionD∈(0,d)(insteadofrequiring
D ∈ (d−1,d), but Thm. 1.1(i) shows there are (somewhat trivial) counterexamples with
dim F =dimaffF. ForthecasewhenD<disaninteger(asinThm.1.1(iii)),aclassof
M
examplesofMinkowskimeasurablelatticesetsisdiscussedinSec.4,Ex.1. Atthispoint,
allknownexamplesofthistypecanberepresentedastheembeddinginRdofaself-similar
setinRD. Itmaywellbethatthisistheonlywaysuchathingispossible.
The proofs of the other parts of Thm. 1.1 are obtained using the elementary tools of
probabilistic renewal theory. These are combined with recent results in [Win14], where
the projection condition was observed to be essential for deriving renewal equations in
termsofthegeneratorofanassociatedtiling. Basedontherenewaltheorem,weobtainin
Thm.3.1andCor.3.2acharacterizationofMinkowskimeasurabilityintermsofaperiodic
function p(see(3.2))whichcanbeexpressedcompletelyintermsoftheparallelvolume
λ (F ∩Γ)ofFrestrictedtoΓ.Thisresultmaybeofindependentinterestforfutureworkas
d ε
itdoesnotrequirethepluriphaseconditionandthusappliestoall(nontrivial)self-similar
sets. Weusethisresulttoproveourmainresults,thenon-Minkowskimeasurabilityinthe
casewhenF ispluriphasew.r.t.thesetΓ,byexploitingandrefininganideain[Kom11].
Beforemovingontotheresults,weexplaininmoredetailtheimprovementsobtained
here compared to previous results from [DKO¨+13, Kom11, LPW13]. The assumptions
of [Kom11] and [LPW13] only differ slightly, and combining [Kom11, Thm. 2.38] with
[LPW13, Thm. 5.4] the resulting nonmeasurability result applies to the case when the
followingrequirementsaremet:
(a) Theopensetcondition(seeDef.2.2)holdsforafeasibleopensetOwithbdO⊆
F,whichinparticularimpliesd−1< D<d.
(b) TheD-dimensionalouterMinkowskicontentofOisfinite(see[LPW13,Def.5.2]).
(c) ThegeneratorG =O\(cid:83)N S O(seeDef.2.5)hasonlyfinitelymanyconnected
i=1 i
components.
(d) EachconnectedcomponentofthegeneratorGismonophase(seeDef.2.9).
In our main result, Thm. 1.1, we remove (a)–(c), and replace (d) with the more general
conditionthat F ispluriphasew.r.t.Γ. Thesignificanceofremoving(a)isthattheresults
of the present paper (in particular, Thm. 1.1) cover sets of any Minkowski dimensions
D ≤ d. Other notable improvements of the present article are the comparatively shorter
4 SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
andsimplerproofsbasedonprobabilisticrenewaltheory. In[LPW13],themorepowerful
(butalsomorecomplicated)apparatusoffractalspraysandcomplexdimensionswasused;
theapproachtakenin[Kom11]usesrenewaltheoryinsymbolicdynamics(motivatedby
[Lal89]) yielding results for the more general class of self-conformal sets. When the re-
newaltheoremforsymbolicdynamicsisrestrictedtotheself-similarsetting,itboilsdown
to the probabilistic renewal theorem (as used in [Lal88]) which we apply directly here.
This direct application makes the proofs significantly shorter and simpler. It should be
notedthatundertheadditionalassumptionthatOcoincideswiththeinterioroftheconvex
hullofF,aresultsimilarto[LPW13,Thm.5.4]/[Kom11,Thm.2.38]wasindependently
provenin[DKO¨+13]usingMellintransforms. See[DKO¨U¨13a,DKO¨U¨13b,DKO¨U¨14]for
furtherinterestingandrelatedresults.
The structure of the article is as follows. In Sec. 2 we lay the foundations for stating
and proving our main results in Sec. 3. The final section, Sec. 4, is devoted to examples
demonstratingourfindings.
2. Preliminaries
Wepresenttheterminologyrequiredtostateandproveourmaintheorems.
2.1. Minkowski measurability. Let A be a compact subset in Euclidean space Rd and
ε≥0. Theε-parallelsetofA(orε-tubularneighborhoodofA)is
A (cid:66){x∈Rd :d(x,A)≤ε}, (2.1)
ε
whered(x,A)(cid:66)inf{(cid:107)x−a(cid:107):a∈ A}istheEuclideandistanceofxtothesetA.
AtubeformulaforAisanexplicitformulaforλ (A ),asafunctionofε,whereλ de-
d ε d
notesthed-dimensionalLebesguemeasure;see[LP10,LPW11,LPW13]foradiscussion
of fractal tube formulas. The volume λ (A ) is referred to as the ε-parallel volume of A
d ε
andwecallλ (A ∩B)theε-parallelvolumeofAinsideBforanyBorelsetB⊆Rd.
d ε
Definition 2.1. Let A be a compact subset of Euclidean space Rd. For 0 ≤ α ≤ d, we
denoteby
M (A)(cid:66) lim εα−dλ (A ) (2.2)
α d ε
ε→0+
theα-dimensionalMinkowskicontentofAwheneverthislimitexists(asavaluein[0,∞]).
IfM (A)existsandsatisfies0 < M (A) < ∞,thenAiscalledMinkowskimeasurable
α α
(of dimension α), and dim A (cid:66) α is the Minkowski dimension of A. If the limit in
M
(2.2) does not exist, one may consider the logarithmic Cesa`ro average known as the (α-
dimensional) average Minkowski content (which always exists in the case of self-similar
setsA,see[Gat00]). Itisdefinedby
1 (cid:90) 1 dε
M (A)(cid:66) lim εα−dλ (A ) .
α δ→0+ |lnδ| δ d ε ε
wheneverthislimitexists.
2.2. Self-similartilingsandtheirgenerators. LetS = {S ,...,S }, N ≥ 2beaniter-
1 N
atedfunctionsystem(IFS),whereeachS isasimilaritymappingofRd withscalingratio
i
r,where0<r <1. ThenwecallS aself-similarsystem. ForA⊆Rd,wewrite
i i
(cid:91)N
SA(cid:66) S (A). (2.3)
i
i=1
LATTICENONMEASURABILITYINTHEPLURIPHASECASE 5
Theself-similarsetFgeneratedbytheIFSS istheuniquecompactandnonemptysolution
ofthefixed-pointequationF =SF;cf.[Hut81],alsocalledtheattractorofS.
We study the parallel volume of the attractor by studying the parallel volume inside a
certaintilingofitscomplement,whichisconstructedviatheIFSasdescribedbelow. The
tilingconstructionwasintroducedin[Pea07]anddevelopedin[PW12], wheretilingsby
opensetswerestudied;seealso[Pea06,LP10,LPW11,LPW13].Inthispaperweconsider
self-similar tilings in a generalized sense, with the tiles not necessarily being open (see
Def.2.5). Theconstructionofaself-similartilingrequirestheIFStosatisfytheopenset
conditionandanontrivialitycondition,asdescribedinthefollowingtwodefinitions.
Definition 2.2. A self-similar system S = {S ,...,S } satisfies the open set condition
1 N
(OSC)ifandonlyifthereisanonemptyopensetO⊆Rd suchthat
S (O)⊆O, i=1,...,N and
i (2.4)
S (O)∩S (O)=∅, i(cid:44) j.
i j
Inthiscase,Oiscalledafeasibleopenset for{S ,...,S }; see[Hut81,Fal03,BHR06].
1 N
IfadditionallyO∩F (cid:44)∅,thenOiscalledastrongfeasibleopenset.
Itwasshownin[Sch94]thatifaself-similarsystemsatisfiesOSC,thenitpossessesa
strongfeasibleopenset.
Definition 2.3. A self-similar set F, which is the attractor of a self-similar system S =
{S ,...,S } satisfying OSC, is said to be nontrivial if there exists a feasible open set O
1 N
suchthat
O(cid:42)SO, (2.5)
whereSOdenotestheclosureofSO;otherwise,F iscalledtrivial.
This condition is needed to ensure that the set Γ = O\SO in Def. 2.5 has nonempty
interior. ItturnsoutthatnontrivialityisindependentoftheparticularchoiceofthesetO. It
isshownin[PW12]thatF istrivialifandonlyifithasnonemptyinterior,whichamounts
tothefollowingcharacterizationofnontriviality:
Proposition2.4([PW12,Cor.5.4]). LetF ⊆Rdbeaself-similarsetwhichistheattractor
ofaself-similarsystemsatisfyingOSC.ThenFisnontrivialifandonlyifFhasMinkowski
dimensionstrictlylessthand.
Unless explicitely stated otherwise, all self-similar sets considered here are assumed
to be nontrivial, and the discussion of a self-similar tiling T implicitly assumes that the
correspondingattractorF isnontrivialandthatthecorrespondingsystemsatisfiesOSC.
Denotethesetofallfinitewordsformedbythealphabet{1,...,N}by
(cid:91)∞
W (cid:66) {1,...,N}k. (2.6)
k=0
Foranywordw = w w ...w ∈ W,letr (cid:66) r ·...·r andS (cid:66) S ◦···◦S . In
1 2 n w w1 wn w w1 wn
particular,ifw∈Wistheemptyword,thenr =1andS =Id.
w w
Definition2.5. LetObeafeasibleopensetfor{S ,...,S }. Theself-similartilingT(O)
1 N
associatedwiththeIFS{S ,...,S }isthecollectionofopensets
1 N
T(O)(cid:66){Sw(G)...w∈W}, (2.7)
where the open set G (cid:66) O\SO is called the generator of the tiling. We call the tiling
T(cid:101)(O)(cid:66){Sw(Γ)...w∈W}generatedby
Γ=Γ(O)(cid:66)O\SO, (2.8)
6 SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
O G S(G) S2(G) S3(G) (O)
O G S(G) S2(G) S3(G) S4(G) (O)
O G S(G) S2(G) S3(G) (O)
Figure 1. Fromtoptobottom: aKochcurvetiling, aSierpinskigaskettiling, anda
Sierpinskicarpettiling. Ineachoftheseexamples,thesetOistheinterioroftheconvex
hullofF,andthesetFismonophasew.r.t. Γ(seeDef.2.9). TheKochcurvetilingdoes
notsatisfythecompatibilitycriterionbdO⊆Fbuttheothertwoexamplesdo.
aself-similartilinginageneralizedsense,withthetilesnotnecessarilybeingopen.
Remark2.6. Self-similartilingsgeneratedbyGwereintroducedin[Pea06,Pea07,PW12]
and further studied in [LP08, LP10, LPW11, LPW13, DDKU¨10, DKO¨U¨14, DKO¨+13,
KR12]. Thenomenclaturestemsfromthefact(provedin[PW12,Thm.5.7])thatT(O)is
anopentilingofOinthesensethat
(cid:91)
O= S (G), (2.9)
w
w∈W
wherethetilesS (G)arepairwisedisjointopensets.
w
We will find it more useful to work in terms of the set Γ instead ofG for most of the
sequel. ObservethatG ⊆ ΓandthatΓ\G ⊂ (cid:83) bdS O. IfF isassumedtobenontrivial,
i i
thenthesetΓhasnonemptyinteriorandwelet
g(cid:66)sup{d(x,F)... x∈Γ}=sup{d(x,F)... x∈G} (2.10)
denote the maximal distance of a point in Γ to F. The reason for the use of Γ is that
Lebesguemeasureisnotstablewithrespecttotheclosureoperation:onemayhaveλ (U)<
d
λ (U)foranopensetU ⊆Rd. Weremarkthat
d
(cid:91)n
O=Γ∪SO=Γ∪SΓ∪S2O=···= SkΓ∪Sn+1O, (2.11)
k=0
wherealltheunionsaredisjoint,andhence(2.9)implies
(cid:91)
O= S Γ. (2.12)
w
w∈W
Therefore,T(cid:101)(O)fromDef.2.5givesatilingofO,wherethetilesS Γarepairewisedisjoint
w
butnotnecessarilyopen,justifyingthetermself-similartilinginageneralizedsense.Also,
(2.11)allowsforthefollowingnicedecompositionoftheε-parallelvolumeoftheattractor
whichisusedintheproofofThm.3.1:
(cid:88)N
λ (F )= λ (F ∩S O)+λ (F ∩Γ)+λ (F \O). (2.13)
d ε d ε i d ε d ε
i=1
LATTICENONMEASURABILITYINTHEPLURIPHASECASE 7
ThisrepresentationisparticularlyusefulforsetsOsatisfyingtheprojectioncondition.
Definition 2.7. For a compact set A ⊆ Rd, we let π denote the metric projection onto
A
A. It is defined on the set of points x ∈ Rd which have a unique nearest neighbour y in
Abyπ (x) = y. LetObeafeasibleopensetoftheself-similarsystem{S ,...,S }with
A 1 N
attractorF. ThenOissaidtosatisfytheprojectionconditionif
S O⊆π−1(S F) fori=1,...,N. (2.14)
i F i
Iftheprojectionconditionissatisfiedthen
F ∩S O=(S F) ∩S O (2.15)
ε i i ε i
foreachε>0andi=1,...,N (see[Win14,Lem.3.19]).
Remark2.8. Thecentralopensetisaparticularchoiceoffeasibleopensetthatexistsfor
anyIFSsatisfyingtheOSC;itisdefinedandstudiedin[BHR06]. Becauseofitsdefinition
intermsofmetricprojections(see[BHR06]),itiseasytoseethatthecentralopensetwill
always satisfy the projection condition. It is also clear that F is contained in the central
open set, so the strong condition is also automatically satisfied. Therefore, it is always
possibletofindastrongfeasibleopensetwhichsatisfiestheprojectioncondition,aslong
astheOSCholds. Aproofofthesefactsisgivenin[Win14,Prop.3.17].
Definition 2.9. For a given IFS and a fixed feasible open set O, we call the attractor F
pluriphasewithrespecttothesetΓ = Γ(O)(asdefinedin(2.8))ifandonlyifthereexists
afinitepartitionoftheinterval(0,∞)withpartitionpoints0 =: a < a < ··· < a <
0 1 M−1
a :=gsuchthat,forε>0,
M
(cid:88)M (cid:88)d
λ (F ∩Γ)= 1 (ε) κ εd−k+1 (ε)λ (Γ), (2.16)
d ε (am−1,am] m,k (g,∞) d
m=1 k=0
forsomeconstantsκ ∈R,where1denotesacharacteristicfunctionandgisasin(2.10).
m,k
We assume that the representation in (2.16) is given with M minimal, so that for each
m = 1,...,M,thereexistsak ∈ {0,...,d}withκ (cid:44) κ . Imposingminimalityof M,
m,k m−1,k
wecallF monophasewithrespecttoΓifandonlyifM =1intheaboverepresentation.
Remark2.10. Atthetimeofwriting,thereisnoknowncharacterizationofthepluriphase
ormonophaseconditionsintermsoftheself-similarsystem{S }N . However,itisknown
i i=1
from [KR12] that a convex polytope in Rd is monophase (with Steiner-like function of
class Cd−1) iff it admits an inscribed d-dimensional Euclidean ball (i.e., a d-ball tangent
toeachfacet). ThisincludesregularpolygonsinR2 andregularpolyhedrainRd,aswell
as all triangles and higher-dimensional simplices. Furthermore, it was recently shown in
[KR12]that(undermildconditions),anyconvexpolyhedroninRd (d ≥ 1)ispluriphase,
therebyresolvingintheaffirmativeaconjecturemadein[LP08,LP10,LPW11]. Werefer
to[KR12]forfurtherrelevantinterestingresults.
Remark2.11. In[LP08,LP10]thenotionsmonophase,pluriphaseandthesymbol“g”were
introduced for the generator of a self-similar tiling satisfying the compatibility condition
bdO ⊆ F. The reader should be aware that these terms in the present paper only match
previous usage in the literature for the case when bdO ⊆ F. In this context, the upper
endpointgoftherelevantintervalinDef.2.9wasdefinedastheinradius(cid:101)gofG,i.e.,the
maximal radius of an open metric ball contained in the set G. Moreover, the generator
G (as a set) was called monophase if λd(G−ε) is polynomial in ε for ε ∈ (0,(cid:101)g), where
G−ε := {x ∈ G ... d(x,Gc) ≤ ε} and pluriphase if λd(G−ε) is piecewise polynomial in ε
8 SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
SO
O 3
SO SO
1 2
F
G G F Ç G
-e e
Figure2. ASierpinskigaskettilingalternativetotheonefromFig.1.Here,Oisnotthe
interioroftheconvexhullofF,butratherthecentralopensetdiscussedin[BHR06].The
setFispluriphase(butnotmonophase)w.r.t. Γ,whilethesetΓ(asaset)ismonophase.
Atright,thesets√Γ−ε(cid:66)(bdΓ)ε∩ΓandFε∩Γareshownforseveralvaluesofε.Forthis
example,g=(2 3)−1and(cid:101)g=1/8;seeRem.2.11. Thisexamplewillbefurtherstudied
inSec.4,Ex.2.
forε ∈ (0,(cid:101)g). However, Fig.2showsanexamplewhereg (cid:44)(cid:101)gandwhereG (asaset)is
monophasebutF isnotmonophasew.r.t.Γ. WereturntothisexampleinSec.4,Ex.2. It
isclearthattheinradiusandthenotionsmono-andpluriphaseforsetsarenottheproper
conceptforthesituationwherebdO (cid:42) F. Itisalsoclearfromthisobservationthat(2.10)
andDef.2.9arethenaturalextensionstothepresent(moregeneral)setting.
Remark2.12. Someexamplesofself-similartilingsassociatedtofamiliarfractalsetsare
shown in Fig. 1. In each case, there is a connected monophase generator. In Fig. 2 an
alternativetilingassociatedwiththeSierpinskigasketisprovided. Here, thegeneratoris
notconnected.
2.3. Lattice,nonlatticeandtherenewaltheorem.
Definition 2.13. Consider a family of similarity mappings S = {S ,...,S }, and let r
1 N i
denotethescalingratioofS . Thefamilyissaidtobeoflatticetypeiffthereisanr > 0
i
such that each scaling ratio ri can be written as ri = rki for some integer ki, and to be of
nonlatticetypeotherwise. Thereisasmallestnumberr >0forwhichtheaforementioned
representation can be found. We always use this minimal r, and we say that S is lattice
withbaser.
Anextendeddiscussionoftheimplicationsofthelattice/nonlatticedichotomymaybe
foundin[LvF12,Thm.3.6]. Thelattice/nonlatticedichotomyalsoappearsinprobabilistic
renewal theory, where the usual nomenclature is “arithmetic/nonarithmetic”. For more
details,see[Fel68,§XIII],[Fal97,§7]or[Win08,§4].
For use in the sequel, we include here a version of the renewal theorem formulated
foradiscreteprobabilitydistribution(cid:80)N pδ ,whereδ isapointmass(Diracmeasure)
i=1 i yi y
concentratedaty∈R. Thistheoremwillbeappliedtothedistribution
(cid:88)N
rDδ , (2.17)
i yi
i=1
whereDisthesimilaritydimensionofF. Thesimilaritydimensionistheuniquepositive
realnumberαthatsatisfiestheMoranequationrα+rα+···+rα =1,i.e.,theuniqueD>0
1 2 N
thatmakes(2.17)intoadiscreteprobabilitydistribution.
Theorem2.14(RenewalTheorem(see[Fal97,Cor.7.3]or[Win08,§4]). Letp ,...,p ∈
1 N
(0,1) satisfy (cid:80)N p = 1, and let y ,...,y > 0. Let z : R → R be a function with a
i=1 i 1 N
discretesetofdiscontinuitieswhichsatisfies
|z(t)|≤c e−c2|t|, forallt∈R, (2.18)
1
LATTICENONMEASURABILITYINTHEPLURIPHASECASE 9
for some constants 0 < c ,c < ∞. Also, let Z : R → R be the unique solution of the
1 2
renewalequation
(cid:88)N
Z(t)=z(t)+ pZ(t−y) (2.19)
i i
i=1
whichsatisfieslim Z(t)=0. Thenthefollowingholds:
t→−∞
(i) If{y ,...,y }⊆h·Zandh>0ismaximalassuch,then
1 N
h(cid:88)
Z(t)∼ z(t−(cid:96)h), ast→∞. (2.20)
η
(cid:96)∈Z
(ii) Iftheredoesnotexisth>0suchthat{y ,...,y }⊆h·Z,then
1 N
1(cid:90) ∞
limZ(t)= z(τ) dτ. (2.21)
t→∞ η −∞
Here,η(cid:66)(cid:80)N y p. Moreover,inbothcases,wehave
i=1 i i
1 (cid:90) T 1(cid:90) ∞
lim Z(t) dt= z(t) dt. (2.22)
T→∞T 0 η −∞
In(2.20),thenotationg∼ f,ast→∞,meansthatgisasymptoticto f ast→∞inthe
sensethatforanyδ>0,thereisanumbers= s(δ)forwhich
(1−δ)f(t)≤g(t)≤(1+δ)f(t), forallt≥ s. (2.23)
Remark 2.15. Iftheself-similarsystem{S ,...,S }islattice,thenthereexistr > 0and
1 N
ki ∈ Nsuchthatri = rki,whereri denotesthescalingratioofSi fori = 1,...,N. Inthis
case{−lnr ,...,−lnr } = {−k lnr,...,−k lnr} ⊆ −lnr·Zandthus,weareincase(i)
1 N 1 N
of the renewal theorem. On the other hand, if {S ,...,S } is nonlattice, then we are in
1 N
case(ii)oftherenewaltheorem.
Remark2.16(Abriefdictionary). Therenewaltheoremaboveisgivenintermsoftheaddi-
tivevariablet∈Rbutwillbeappliedinthecontextofthemultiplicativevariableε∈(0,g].
Forthereader’sconvenience,weofferthefollowingtranslationofsymbolscorresponding
tothechangeofvariablesε=e−t:
e−t t→∞ −lng≤t<∞ e−h t−(cid:96)h 1(t−lng) p
h i
ε ε→0 0<ε≤g r −ln(r−(cid:96)ε) −log (g−1ε) rD
r i
3. Statementandproofofthemainresults
InordertoproveThm.1.1,wefirstproveatheoremwhichprovidesinformationonthe
asymptotic behavior of the parallel volume of the self-similar set F ⊆ Rd under weaker
conditions. This intermediate result is of independent interest as it does not require the
pluriphaseconditiontobesatisfied,andthusmayprovideanavenueforeventuallyremov-
ingthishypothesis. Inanalogyto(2.23)wesaythat f ∼ gasε → 0iff f ◦h ∼ g◦has
t →∞,whereh(t):=exp(−t). If f,g:(0,∞)→(0,∞)thisisequivalenttoassumingthat
lim f(ε)/g(ε)=1.
ε→0
Theorem 3.1. Let F ⊂ Rd be the attractor of a self-similar system S = {S ,...,S }
1 N
satisfyingtheOSCandletr denotethecontractionratioofS fori=1,...,N. AssumeF
i i
10 SABRINAKOMBRINK,ERINP.J.PEARSE,ANDSTEFFENWINTER
isnontrivial(i.e.D := dim (F) < d). LetObeanarbitrarystrongfeasiblesetsatisfying
M
theprojectioncondition,Γ:=O\SOandgasin(2.10). IfS islatticewithbaserthen
lnr
εD−dλ (F )∼ p(ε), asε→0, (3.1)
d ε (cid:80)N rDlnr
i=1 i i
where p:(0,g]→Risdefinedby
(cid:88)
p(ε)(cid:66)εD−d r(cid:96)(D−d)λ (F ∩Γ), forε>0. (3.2)
d r(cid:96)ε
(cid:96)∈Z
Moreover,forε∈(rg,g], phasthealternativerepresentation
 
p(ε)=εD−drDλ−dd(Γ−)1 +(cid:88)∞ r(cid:96)(D−d)λd(Fr(cid:96)ε∩Γ). (3.3)
(cid:96)=0
If one can show that the periodic function p is non-constant, then Thm. 3.1 implies
that limsup εD−dλ (F ) > liminf εD−dλ (F ) and hence that F is not Minkowski
ε→0 d ε ε→0 d ε
measurable.Ontheotherhand,ifthefunctionpisconstant,then(3.1)impliesimmediately
that F isMinkowskimeasurable. Notethatinbothcases pisastrictlypositivefunction.
Thisisforinstanceobviousfrom(3.3)sincethefirsttermisstrictlypositiveandallterms
inthesecondsummationarenon-negative. Wesavetheseimportantobservationsforlater
use:
Corollary3.2. UnderthehypothesisofThm.3.1,theself-similarsetFisMinkowskimea-
surableifandonlyifthefunctionp(givenby(3.2)or(3.3))isconstant,thatis,p(ε)=Cfor
someconstantC > 0andallε > 0. Moreover,inthiscasethe D-dimensionalMinkowski
contentofF isgivenby
lnr
M (F)= ·C.
D (cid:80)N rDlnr
i=1 i i
NotethatThm.3.1andCor.3.2applytoallnontrivialself-similarsetssatisfyingOSC.
There is no monophase or pluriphase condition present and the projection condition on
its own does not impose any restrictions. There exists always a strong feasible set O for
whichitissatisfied,seeRem.2.8. Onlytrivialself-similarsetsareexcluded. Inthiscase,
the statement of Thm. 3.1 does not make sense, since Γ = ∅ and thus p ≡ 0. But such
setsarealwaysMinkowskimeasurableandthereisnoneedforastatementlikethis. (Note
thatforatrivialself-similarsetF ⊂Rd,thed-dimensinoalMinkowskicontentisgivenby
M (F)=λ (F).)
d d
ProofofThm.3.1. WedecomposetheparallelvolumeofF through
λ (F )=λ (F \O)+λ (F ∩O). (3.4)
d ε d ε d ε
Forthefirstsummandontherighthandsideof (3.4), wenotethatSO ⊆ OsothatOc ⊆
SOc ⊆(SOc) . By[Win08,Cor.5.6.3]weknowthatthereexistc,γ>0suchthat
ε
λ (F ∩(SOc) )≤cεd−D+γ, forε∈(0,1). (3.5)
d ε ε
Note that it is this estimate which requires the hypothesis that O is a strong feasible set.
Equation(3.5)impliesλ (F \O)≤cεd−D+γ,whence
d ε
limεD−dλ (F \O)=0. (3.6)
d ε
ε→0