The multifractal spectra of projected measures in Euclidean spaces

Chaos, Solitons and Fractals 11 (2000) 901±921

www.elsevier.nl/locate/chaos

The multifractal spectra of projected measures in Euclidean spaces Toby C. O'Neil 1 Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, Edinburgh, Scotland EH9 3JZ, UK Accepted 29 September 1998

Abstract We use generalised Hausdor€ and packing dimensions to investigate the relationship between the multifractal spectra of orthogonal projections of a measure in Euclidean space and the measure's original multifractal spectrum. Ó 2000 Elsevier Science Ltd. All rights reserved. MSC: 28A75

1. Introduction In this paper we investigate the relationship between the multifractal spectrum of a measure in Euclidean space and the multifractal spectrum of its projection onto a lower dimensional linear subspace. In particular we obtain inequalities relating the generalised Hausdor€ and packing dimensions (introduced by Olsen [17±19], see also [2]) of the original measure to those of its projection. The ®rst investigations in this direction were by Marstrand [14] who showed that for a Borel set, E, in the plane dim…PV E† ˆ minfdim…E†; 1g for almost every line, V , through the origin (here dim …
† denotes the usual Hausdor€ dimension and PV denotes orthogonal projection onto V ). His results were later extended to higher dimensions by Kaufman [13] and Mattila [15]. Their methods involved characterisations of the Hausdor€ dimension of a set in terms of measures supported on the set and led naturally to de®ning the Hausdor€ dimension of a Borel measure l by dim…l† ˆ inffdim…E† : l…E† > 0g: One may then deduce from this de®nition and Marstrand's projection theorem the behaviour of the Hausdor€ dimension of a measure under projection. Recent work of Falconer and Mattila [8], and Falconer and Howroyd [7] has extended these results to packing dimensions of both sets and measures. The Hausdor€ dimension of a measure is only a crude indicator of the measure's size and it is natural to ask whether ®ner approaches are possible. For example, we are often interested in sets of

1

Research supported by EPSRC grant GR/J64429 and completed whilst the author was based at the University of St Andrews. E-mail address: [email protected] (T.C. O'Neil).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 8 ) 0 0 2 5 6 - 2

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T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

points where a measure, l, scales in a particular way and these sets typically have zero l measure: one method to gain information about such sets is to consider the multifractal spectrum of l, see [1,3,17,10]. In this paper we concentrate on investigating the behaviour of the ®ne multifractal spectrum under projections, that is, we investigate the relationship between fl …a† and flV …a† where  fm …a† ˆ dim

log mB…x; r† x : lim ˆa r&0 log r

 :

…1†

This di€ers from [9] where we investigate the coarse multifractal spectrum, that is, we are interested in limits of moment sums taken over boxes of equal size (for other results about the behaviour of the coarse multifractal spectrum under projections see [12]). Our approach is to consider the behaviour of (slightly modi®ed) generalised Hausdor€ and packing dimensions under projections. These dimension functions were introduced by Olsen in [17] and provide a natural mathematical framework within which to study multifractals since, for a wide class of measures, they are related to (1) via the Legendre transform, see [17±20] for some examples. Our results may be summarised as follows (we give full de®nitions in Section 2 and just note here that bql …E† and Bql …E† denote the generalised Hausdor€ and packing dimensions of a set E with respect to the measure l, respectively): 1. For q 6 1, all V 2 G…n; m† and E  Spt l; BqlV …PV E† 6 Bql …E†; q q 2. for q P 1, all V 2 G…n; m† and E  Spt l …E†; m…1 ÿ q†g 6 blV …PV E†; R R l; maxfb ÿs 3. for l such that there is s > m with jx ÿ yj dl…y† dl…x† < 1, we have that for cn;m -a.e. V 2 G…n; m† and q P 0, when m < s < 2m; m…1 ÿ q† 6 bqlV …Spt lV † 6 BqlV …Spt lV † 6 max fm…1 ÿ q†; ÿsq=2g and when s P 2m; bqlV …Spt lV †† ˆ BqlV …Spt lV † ˆ m…1 ÿ q†; 4. for l with s-Ahlfors regular support for some s < m, (see (46)) we ®nd that for all compact sets E  Spt l; cn;m -a.e.V 2 G…n; m† and 0 6 q 6 1; bqlV …PV E† P bql …E†:

2. Generalised Hausdor€ and packing measures Throughout the paper we shall assume that l is a ®nite Borel probability measure with support a compact subset of Rn (n-dimensional Euclidean space). For technical convenience we shall also assume that Spt l  B…0; 1=2†; all the results we prove concerning the behaviour of dimensions hold without this assumption. By G…n; m† we denote the set of m-dimensional linear subspaces, V , of Rn and we let cn;m denote the invariant Haar measure on G…n; m† such that cn;m …G…n; m†† ˆ 1. For V 2 G…n; m† we de®ne the projection map, PV : Rn ! V  Rn , to be the usual orthogonal projection onto V and we de®ne the projection of a measure, lV , by lV …A† ˆ l…PVÿ1 …A††; for A  Rn . Observe that, as l has compact support, Spt lV ˆ PV …Spt l† for all V 2 G…n; m†. We de®ne the local Hausdor€ dimension of a measure l at a point x, denoted b…l; x†, by b…l; x† ˆ lim inf r&0

(

log l…B…x; r†† log r

…2† ) ÿt

ˆ sup t 2 R : lim sup r l…B…x; r†† ˆ 0 : r&0

…3†

From this we de®ne the Hausdor€ dimension of a measure l by dim…l† ˆ sup ft 2 R : b…l; x† P t for l-a:e: xg

…4†

T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

903

and recall that equivalent characterisations of dim…l† include dim…l† ˆ inf f dim…E† : l…E† > 0g  Z Z  dl…x† dl…y† < 1 ; ˆ sup s : jx ÿ yjs

…5† …6†

see [6,16] for further details. (There are equivalent versions of (4) and (5) for the packing dimension of a measure but we shall not use them here.) We continue by introducing multifractal Hausdor€ and packing measures which di€er slightly from those introduced by Olsen in [17] (although in the event that l satis®es a global doubling condition the measures are equivalent). Our main reason for modifying Olsen's de®nition is to allow us to prove results without having to assume that the measures we are investigating satisfy such a doubling condition. We need to avoid this assumption if at all possible because the doubling condition is not necessarily preserved under projections. For q; t 2 R we de®ne the …q; t; l†-Hausdor€ measure of a set E  Spt l by nX o q rt lB…x; 3r† : fB…x; r†g is a d-cover of E ; …7† Hq;t l;d …E† ˆ inf q;t Hq;t l …E† ˆ sup Hl;d …E†:

…8†

d>0

q;t For general E  Rn we then de®ne Hq;t l …E† ˆ Hl …E \ Spt l†. Observe that the centres of the balls in the admissible covers need not be in the set E ± this di€ers from the approach of Olsen and allows us to apply the general theory of Rogers [22] more easily. For q 6 0 it is straightforward to verify that this measure is equivalent to Olsen's multifractal Hausdor€ measures and when l satis®es a global doubling condition, the same is also true for q > 0. We observe that Hq;t l -measure is a Method II measure [22] and that we would obtain the same measures if we worked with covers by open balls instead. Consequently Theorem 23 of [22] gives:

Lemma 2.1. For all q; t 2 R, …q; t; l†-Hausdorff measure is a regular Gd -regular metric measure, all Borel sets q;t are Hq;t l -measurable and each Hl -measurable set of finite measure contains an Fr -set of the same measure. Now let us recall the Besicovitch covering theorem. Theorem 2.2 (Besicovitch covering theorem). There exists a constant f ˆ f…n†, depending only on n such that: If C is a collection of nondegenerate closed balls in Rn with sup f diam B : B 2 Cg < 1 and if C is the set of centres of balls in C, then there exist C1 ; . . . ; Cf  C such that each Ci is a countable collection of disjoint balls in C and C

f [ [

B:

iˆ1 B2Ci

Proof. This is the Besicovitch covering theorem as stated in [4,1.5.2].



We can use this result to de®ne real-valued functions on sets, which are comparable to the generalised Hausdor€ measures but which use covers of the sort in Theorem 2.2: These functions will reduce the number of covers we need to consider in order to prove results about the generalised Hausdor€ dimension. For E  Spt l de®ne ( ) X q rit lB…xi ; 3ri † : fB…xi; ri †gi2N 2 Cd …E† BHq;t l;d …E† ˆ inf i

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and q;t BHq;t l …E† ˆ sup BHl;d …E†; d>0

where



Cd …E† ˆ

fBij gi2N;

f…n†

jˆ1;...;f…n†

: E  [ [Bij ; Bij ˆ B…xij ; rij † jˆ1 i

 and rij 6 d for all i; j; fBij gi2N disjoint for all j :

Proposition 2.3. For all E  Spt l  Rn and all q; t 2 R q;t q;t n Hq;t l …E† 6 BHl …E† 6 9 f…n†Hl …E†:

…9†

Proof. The left-hand inequality is obvious as there are fewer admissible covers in the de®nition of BHq;t l …
†. We now derive the right-hand inequality of (9). Fix d > 0 and suppose that fB…xi ; ri †g is a d-cover of E. Now consider [ f B…x; ri =3† : x 2 B…xi ; ri †  Eg i

b ˆ B…x; ri †  B…xi ; 3ri †. We now use the Beand notice that if B ˆ B…x; ri =3† is in this collection then B sicovitch covering theorem to ®nd countable index sets I1 ; . . . ; If…n† such that E

[

‰B…xi ; ri † \ EŠ 

i

f…n† [

[

B…x; rx †

jˆ1 x2Ij

where for each j, fB…x; rx † : x 2 Ij g is a disjoint family. Moreover, for any x 2 [Ij , rx ˆ ri =3 for some i such that x 2 B…xi ; ri †. An elementary volume estimate allows us to deduce that for each j and all i cardfx 2 Ij : B…x; rx †  B…xi ; 3ri †g 6 9n : Hence f…n† X X jˆ1 x2Ij

q

rxt lB…x; 3rx † 6

f…n† X 1 X jˆ1

6 9n

X

q

iˆ1 x2Ij :B…x;rx †B…xi ;3ri †; rx ˆri =3

f…n† X 1 X jˆ1

6 9n f…n†

iˆ1 1 X iˆ1

3ÿt rit lB…xi ; 3ri †

rxt lB…x; 3rx †

q

q

3ÿt rit lB…xi ; 3ri † :

Thus, as the d-cover was arbitrary we conclude that q;t n BHq;t l;d …E† 6 9 f…n†Hl …E†

which gives (9) on letting d & 0.



q;t Since BHq;t l …
† is comparable to Hl …
† it determines the same generalised dimension function. This shall prove useful later. We now introduce the …q; t; l†-packing measure of a set E  Spt l by nX o q rt lB…x; r=3† : fB…x; r†g is a centred d-packing of E ; Pq;t l;d …E† ˆ sup

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905

q;t Pq;t l;0 …E† ˆ inf Pl;d …E†

…10†

d>0

and

( Pq;t l …E†

ˆ inf

X i

) Pq;t l;0 …Ei †

: E  [Ei :

…11†

i

q;t We extend this de®nition to general E  Rn by de®ning Pq;t l …E† ˆ Pl …E \ Spt l†. For convenience, given B ˆ B…x; r† we let B ˆ B…x; r=3†. Again, this de®nition of packing measure di€ers slightly from that given in Olsen [17]. As with normal Hausdor€ and packing measures we can de®ne Hausdor€ and packing dimension functions for E  Spt l by setting, for q 2 R, q;t bql …E† ˆ supft : Hq;t l …E† ˆ 1g ˆ infft : Hl …E† ˆ 0g;

…12†

q;t Bql …E† ˆ supft : Pq;t l …E† ˆ 1g ˆ infft : Pl …E† ˆ 0g:

…13†

We also de®ne the pre-packing dimension function, Bql;0 …E†, given by q;t Bql;0 …E† ˆ supft : Pq;t l;0 …E† ˆ 1g ˆ infft : Pl;0 …E† ˆ 0g

…14†

(this is a generalisation of the more usual box dimension.) We note that for all q 2 R bql …;† ˆ Bql …;† ˆ Bql;0 …;† ˆ ÿ1: For q ˆ 0 it is easy to see that for E \ Spt l 6ˆ ;, b0l …E† ˆ b…E \ Spt l† and B0l …E† ˆ B…E \ Spt l† where b…
† and B…
† denote the usual Hausdor€ and packing dimensions. We have the following useful characterisation of the generalised packing and pre-packing dimensions. Proposition 2.4. For q 2 R; l a finite, compactly supported, Borel measure on Rd and E  Rd we have Bql;0 …E†

q log Nl;d …E† ˆ lim sup ÿ log d d&0

where q Nl;d …E†

( ˆ sup

X i

…15† ) q

lB…xi ; d=3† : fB…xi ; d†gi is a packing of E \ Spt l

q …E† ˆ 0:† Consequently, (if E \ Spt l ˆ ; we set Nl;d ( ) q [ log Nl;d …Ei † q Bl …E† ˆ inf sup lim sup :E Ei : ÿ log d i d&0 i

Proof. The proof of this is straightforward and mimics that in [5, Proposition 3.8].

…16†



We may now verify that the properties which make Olsen's multifractal measures useful also hold for these, slightly di€erent, measures. We ®rst observe that the generalised packing dimension is larger than the generalised Hausdor€ dimension. Proposition 2.5. For all q; t 2 R there is a constant c depending also on the dimension of the ambient space such that for E  Rn , q;t Hq;t l …E† 6 cPl …E†:

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In particular for all q 2 R, bql …E† 6 Bql …E†: Proof. The proof is a straightforward application of the Besicovitch covering theorem and we omit it here.  The main reason for introducing these measures is encapsulated in the following proposition where estimates are given for the size of sets of points of a given local dimension. Proposition 2.6. If 0 6 aq ‡ Bql …Spt l† then for q P 0 b…fx 2 Spt l : b…l; x† 6 ag† 6 aq ‡ Bql …Spt l†; B…fx 2 Spt l : B…l; x† 6 ag† 6 aq ‡ Bql …Spt l† and for q 6 0 b…fx 2 Spt l : B…l; x† P ag† 6 aq ‡ Bql …Spt l† and B…fx 2 Spt l : b…l; x† P ag† 6 aq ‡ Bql …Spt l†: In addition, if 0 6 aq ‡ bql … Spt l† then for q P 0 b…fx 2 Spt l : B…l; x† 6 ag† 6 aq ‡ bql …Spt l† and for q 6 0 b…fx 2 Spt l : b…l; x† P ag† 6 aq ‡ bql …Spt l†: Proof. The proofs for all these inequalities are very similar to those given for [Propositions 2.5 and 2.6 in [18] and use the usual covering arguments for estimating upper bounds on dimension.  The next proposition describes some of the properties of the functions bql …
† and Bql …
†. Proposition 2.7. Let l be a compact finite Borel measure on Rn and fix E  Spt l. We have that: 1. both bql …E† and Bql …E† are decreasing functions of q, 2. Bql …E† is convex function of q, 3. bql …E† is almost convex in the sense that for p; q 2 R and a 2 ‰0; 1Š blap‡…1ÿa†q …E† 6 aBpl …E† ‡ …1 ÿ a†bql …E†:

…17†

Proof. That both Bql …E† and bql …E† are decreasing functions of q is obvious. The proof that Bpl …E† is a convex function of q follows that given in [18, Section 4.4] and we omit it here. We verify (17). We ®rst show that for a set we have …F † 6 aBpl;0 …F † ‡ …1 ÿ a†bql …F †: bap‡…1ÿa†q l

…18†

Recall, from (15), that Bpl;0 …F † ˆ lim sup d&0

p log Nl;d …F † : ÿ log d

Fix  > 0 and let t ˆ Bpl;0 …F † and s ˆ bql …F †. We shall assume that p P 0, the proof for p < 0 requires only minor modi®cations. Fix 1 > D > 0 such that for d < D

T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

907

p Nl;d …F † 6 dÿtÿ=2 :

Fix 0 < d < D=9 and observe that Proposition 2.3 allows us to ®nd a F ; fB…x; rx † : x 2 Ii ; i ˆ 1; . . . ; f…n†g such that for each i, fB…x; rx † : x 2 Ii g is disjoint and f…n† X X iˆ1

x2Ii

d-cover

of

rxs‡ lB…x; 3rx †q 6 1:

Hence, as a 2 ‰0; 1Š we may use H olders inequality to estimate ap‡…1ÿa†q;at‡…1ÿa†s‡

BHl;d

…F † 6

f…n† X X iˆ1 x2Ii

ˆ

ap‡…1ÿa†q

rxat‡…1ÿa†s‡ lB…x; 3rx †

f…n† X X ÿ x2Ii

iˆ1

6

f…n† X X iˆ1

6 ˆ

x2Ii

f…n† X X

rxs‡ lB…x; 3rx †q

!a

f…n† X X

rxt‡ lB…x; 3rx †

x2Ii

1 X


1ÿa !1ÿa

q rxs‡ lB…x; 3rx †

p

!a

X

kˆ0 x2Ii :2ÿkÿ1 6 rx 6 2ÿk

f…n† X 1 X

x2Ii

iˆ1

!a

f…n† X

iˆ1


a ÿ

p rxt‡ lB…x; 3rx †

iˆ1

iˆ1

6c

rxt‡ lB…x; 3rx †p

p rxt‡ lB…x; 3rx †

X

!a 2

ÿk…t‡†

ÿk p

lB…x; 3
2 †

;

kˆ0 x2Ii :2ÿkÿ1 6 rx 6 2ÿk

where c ˆ maxf2ÿ…t‡† ; 2t‡ g. Now observe that for any k 2 N and all i, as fB…x; 2ÿkÿ1 † : x 2 Ii ; 2ÿkÿ1 < rx 6 2ÿk g is a disjoint family, we have that for all x 2 Ii with 2ÿkÿ1 < rx 6 2ÿk , cardfB…y; 3
2ÿk † : y 2 Iy ; B…y; 3
2ÿk † \ B…x; 3
2ÿk † 6ˆ ;g 6 14n and thus for any and k 2 N and i ˆ 1; . . . ; f…n† X p p 2ÿk…t‡† lB…x; 3
2ÿk † 6 …14n ‡ 1†Nl;9
2 ÿk …F † x2Ii :2ÿkÿ1
6 …14n ‡ 1†…9
2ÿk †

ÿtÿ=2 ÿk…t‡†

2

:

Hence ap‡…1ÿa†q;at‡…1ÿa†s‡=2

BHl;d

…F † 6 9ÿtÿ=2 f…n†…14n ‡ 1†c

1 X

2ÿk=2

kˆ0

6 9ÿtÿ=2 f…n†…14n ‡ 1†c=…1 ÿ 2ÿ=2 †: Thus ap‡…1ÿa†q;at‡…1ÿa†s‡=2

BHl;0

…F † < 1

and so, as  > 0 was arbitrary, we deduce (18). We ®nish the proof by ®xing E  Rn and  > 0 and choosing a countable cover, fEi g, of E such that for all i Bpl;0 …Ei † 6 Bpl …E† ‡ :

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We may then use (18) and the monotonicity of the dimension functions to deduce that blap‡…1ÿa†q …E† ˆ bap‡…1ÿa†q …E \ Spt l† 6 sup blap‡…1ÿa†q …Ei \ E \ Spt l† i

  6 sup aBpl;0 …Ei \ E \ Spt l† ‡ …1 ÿ a†bql …Ei \ E \ Spt l† i

6 a…Bpl …E† ‡ † ‡ …1 ÿ a†bql …E† which gives (17) on taking a sequence of  tending to 0.  For an example of a measure for which bql …Spt l† is not convex, see [17, Section 3.1]. This result immediately allows us to deduce some bounds on the graphs of Bql …
† and bql …
†. Corollary 2.8. We have the following for F  Spt l… Rn †: 1. For q 6 0; bql …F † P b0l …F †…1 ÿ q† ˆ dim…F †…1 ÿ q†

…19†

and Bql …F † P B0l …F †…1 ÿ q† ˆ Dim…F †…1 ÿ q†;

…20†

2. for 0 6 q 6 1; Bql …F † 6 B0l …F †…1 ÿ q† 6 n…1 ÿ q†

…21†

and bql …F † 6 b0l …F †…1 ÿ q† 6 n…1 ÿ q†;

…22†

3. and for q P 1 Bql …F † P bql …F † P B0l …F †…1 ÿ q† P n…1 ÿ q†:

…23†

Proof. Since Bql …F † is a convex function of q, all inequalities involving bounds on it are obvious. To verify (19) for bql …F †, set a ˆ ÿq=…1 ÿ q† and p ˆ 1 in (17). For (22), set p ˆ 1 and a ˆ q in (17). For (21) it suces to observe that 0 6 B0l …F † 6 n and B1l …F † ˆ 0 and use the convexity of Bql …F †. For (23), ®x q P 1, let p ˆ 0, set a ˆ …q ÿ 1†=q and use (17) together with (21).  These basic results give us enough tools to begin investigating the projection properties of these generalised Hausdor€ and packing measures. 3. Projection estimates for general measures In this short section we derive global bounds on the generalised dimension of a projection of a measure in terms of its original generalised dimensions. The bounds in this section apply to all ®nite, compactly supported Borel measures. We begin by investigating the generalised packing dimension for q 6 1. Lemma 3.1. Let l be a finite, compactly supported measure on Rn and E  Spt l. For q 6 1; s 2 R and all V 2 G…n; m† q;s Pq;s lV …PV E† 6 cPl …E†;

…24†

T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

909

where c ˆ c…q; s; n† < 1 depends only on q; s and n, and may be taken to be 1 for q < 0. In particular, BqlV …PV E† 6 Bql …E†:

…25†

Proof. When q < 0 it suce to observe that if fB…xi; ri † : i 2 Ng is a centred d-packing of PV E then fB…yi; ri † : i 2 Ng is a centred d-packing of E (where yi 2 E is such that xi ˆ PV …yi †† and, clearly, q

q

lV B…xi; ri † 6 lB…yi; ri † : This easily gives (24). For 0 6 q 6 1: suppose that F  Rn ; s 2 R and V 2 G…n; m†. Fix d > 0 and let fBi ˆ B…ui ; ri †g be a dpacking of PV F . For each i consider the collection of balls in Rn , fB…x; ri =9† : x 2 F \ PVÿ1 ‰V \ B…ui ; ri =3†Šg: By the Besicovitch covering Theorem 2.2 we may ®nd a constant f ˆ f…n† and index sets, Ji1 ; Ji2 ; . . . ; Jif such that f [ [ B…x; ri =9† F \ PVÿ1 ‰V \ B…ui; ri =3†Š  jˆ1 x2Jj

and fB…x; ri =9† : x 2 Jij g is a disjoint family for each j. Also observe that for ®xed i and j, we may make a simple volume estimate to further subdivide Jij into at most 7n disjoint subfamilies, Jij1 ; Jij2 ; . . . ; Jij7n such that for each j and k, fB…x; ri =3† : x 2 [i Jijk g is a d=3-packing of F . Hence, as 0 6 q 6 1, " #q f X 7n X X X X q s s r lV Bi 6 r lB…x; ri =9† i

i

i

6

i

jˆ1

kˆ1 x2Jijk

f X X XX 7n

i

ˆ 3s

jˆ1

kˆ1 x2Jijk

ris lB…x; ri =9†

f X X X  s X ri 7n

jˆ1

kˆ1

i

x2Jijk

3

q

lB…x; ri =9†

q

6 3s 7n fPq;s l;d=3 …F †: Thus q;s s n Pq;s lV ;d …PV F † 6 3 7 fPl;d=3 …F †

which, on taking the limit as d & 0, gives q;s s n Pq;s lV ;0 …PV F † 6 3 7 fPl;0 …F †:

Thus, given a set E and a cover of it by sets Ei …i ˆ 1; 2; . . .† then PV E  [PV Ei and we ®nd that X q;s X q;s Pq;s PlV ;0 …PV Ei † 6 3s f Pl;0 …Ei †: lV …PV E† 6 i

i

Since the family of covering sets, fEi g was arbitrary we conclude that q;s s n Pq;s lV …PV E† 6 3 7 fPl …E†

which is (24), as required.  We now derive a dual result for generalised Hausdor€ dimension.

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Lemma 3.2. Let l be a finite, compactly supported Borel measure on Rn and suppose that E is a subset of Spt l. Then for q 6 0 and all V 2 G…n; m† bqlV …PV E† 6 bql …E†

…26†

and for q P 1 and all V 2 G…n; m† bqlV …PV E† P maxfm…1 ÿ q†; bql …E†g:

…27†

Proof. The result for q 6 0 follows in the same way as the corresponding result for generalised packing dimension (24). When q ˆ 1, all quantities evaluate to zero so we may assume that q > 1. Fix V 2 G…n; m† and d > 0 and suppose that fBi ˆ B…ui ; ri †g is a d-cover of PV …E†. (We may assume that all ui 2 V .) For each i, we may use the Besicovitch covering theorem to ®nd a constant f ˆ f…n†, depending only on n, and a family of balls fBij ˆ B…xij ; rij † : j 2 Ng with rij ˆ ri =4 which is a d-cover of PVÿ1 …Bi † \ E such that for all y 2 Rn X 1Bij …y† 6 f j

and b i \ V †  [B b ij PVÿ1 … B j

d ˆ B…x; 3r†). Moreover as the B b ij all have the same radius, an elementary volume estimate (where B…xr† shows that for all i X 1B^ij 6 9n : j

Thus lV …B^i † ˆ l…PVÿ1 …B^i \ V †† P 9ÿn fÿ1

X

l…B^ij †:

j

Hence for q > 1 we may deduce that X i

ris …lV

b i †q P 9ÿnq fÿq B

X i

P 9ÿnq f

X ÿq i;j

P 4s 9ÿnq fÿq

X

ris

!q b ij lB

j

b qij ris l B

X i;j

b qij : rijs l B

Consequently, as fBi g was any d-cover of PV …E†, we conclude that s ÿnq ÿq Hqs f Hq;s l;d …E† 6 4 9 lV ;d …PV E†

which implies the lemma since d was arbitrary.



Since the methods used to obtain these results take no account of the geometry of the measure, we have no reason to expect that the bounds they give are sharp and, indeed, it is not hard to construct examples where these bounds fail to be sharp for a set of V of positive measure, see [8,12].

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911

4. Bounds for large measures When l is a measure of (Hausdor€) dimension strictly greater than m (recall, from (4)±(6), the de®nition and equivalent formulations) one would expect that its projection onto an m-dimensional subspace should be, essentially, m-Lebesgue measure. In this section we show that this expectation is nearly correct. We begin by investigating the multifractal spectrum of a measure which is absolutely continuous with respect to Lebesgue measure. Lemma 4.1. Let x be a compactly supported Radon measure on Rd and suppose that x is absolutely continuous with respect to Lebesgue measure on Rd . Then for a Borel set E  Spt l which has positive x measure and for q P 0 Bqx …E† P d…1 ÿ q†: Proof. Fix q P 0 and suppose that f P 0 is such that x ˆ f Ld then, as x…E† > 0, we can ®nd a Borel set A  E of positive Lebesgue measure and k > 0 such that for all u 2 A, f …x† P k. Fix d > 0 and suppose that fBi ˆ B…xi ; ri †g is a d-cover of E. For s 6 d…1 ÿ q† we ®nd X i

q …xB…xi ; 3ri †† ris

P

X i

P kq

ris

!q

Z B…xi ;3ri †

f dL

d

X ÿ
q ris Ld …A \ B…xi ; 3ri †† i

s

P 3ÿs a…m† kq

X

Ld …A \ B…xi ; ri ††

q‡s=d

i

P 3ÿs a…d†s kq Ld …A†; where a…d† denotes the Lebsgue measure of the unit ball in Rd . Hence bqx …E† P s and the inequality follows.  RR If a measure l has ®nite m-energy (that is, jy ÿ xjÿm dl…y† dl…x† < 1) then it is well known (see [16, 9.7]) that for almost every V 2 G…n; m†, lV is absolutely continuous with respect to Hm bV . In the next few results we describe how if l has ®nite s-energy for some s > m then we may deduce a little more about the usual structure of the projections of l. We introduce the following compact notation for use in the following propositions: for a measure x on Rd we say, for p 2 ‰1; 1Š, that x 2 Lp …Rd † if there is a function f 2 Lp …Rd † such that f is the Radon± Nikodym derivative of x with respect to Ld for x-a:e: x. We interpret x 2 Lp …V † in the obvious way when V is a d-dimensional linear subspace. If x 2 Lp …Rd † we let kxkp denote the p-norm of its Radon±Nikodym derivative (with respect to Ld ). Proposition 4.2. Fix p > 1 and suppose that x 2 Lp …Rm † is a compactly supported Radon measure. Then for qP0 Bqx …Spt x† 6 maxfm…1 ÿ q†; ÿmq…p ÿ 1†=pg:

…28†

Proof. For 0 6 q 6 1 the result automatically holds by (21). Suppose that q P p, s > ÿmq…p ÿ 1†=p and that 0 < d < 1=3 is ®xed. Let fBi ˆ B…xi ; ri †g be a centred d-packing of E  Spt x and let Bi ˆ B…xi ; ri =3†. Then we ®nd, on using H older's inequality that

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X i

ris x…Bi † ˆ q

q X Z ris f dLm Bi

i

6 a…m†ÿs=m

X

Lm …Bi †s=m

Z Bi

i

ˆ a…m†

ÿs=m

X

Lm …Bi †

f p dLm

…s=m†‡…q…pÿ1†=p†

q=p  Z

Z B^i

i

6 a…m†

ÿs=m

XZ B^i

i

6 a…m†

ÿs=m

XZ B^i

i

6 a…m†

ÿs=m

Z

p

Bi

f p dLm

1 dLm

f p dLm

q…pÿ1†=p

q=p

q=p !q=p

p

f dL

f dL

m

m

q=p < 1:

q;s q Hence Pq;s x;0 …E† < 1 and thus, in particular, Px …Spt l† < 1 which implies that Bx …Spt x† 6 s and the result follows for q P p. The convexity of Bqx …E† as a function of q (Proposition 2.7) now enables us to deduce the result for 1 < q 6 p. 

We now describe conditions which ensure that a measure has projections which are in Lp for some p > 1. Proposition 4.3. Suppose that m 6 s < n and l is a finite Borel measure on Rn for which the s-energy of l is finite: Z Z 1 dl…x† dl…y† < 1: jx ÿ yjs Then 1. for cn;m -a:e: V 2 G…n; m†, lV 2 L2 …V †. 2. If m < s 6 2m then for 1 6 p < …2m=2m ÿ s† and cn;m -a:e: V 2 G…n; m†, lV 2 Lp …V †. 3. If 2m < s < n then for cn;m -a:e: V 2 G…n; m† the Radon±Nikodym of lV with respect to Hm bV is bounded and essentially continuous. Proof. This is Proposition 3.11 of [9].



Notice that for s ˆ dim…l† it is not necessarily the case that the s-energy of l is ®nite and thus the preceeding proposition does not necessarily tell us anything about the behaviour of l when dim…l† ˆ m. These results now enable us to describe the behaviour of large measures under projection. Corollary 4.4. Suppose that l is a compactly supported Radon measure on Rn and 0 < m 6 s < n are such that Is …l† < 1. Then if 2m < s < n, for cn;m -a:e: V 2 G…n; m† and q P 0 bqlV …Spt lV † ˆ Bql V …Spt lV † ˆ m…1 ÿ q†

…29†

and if m 6 s 6 2m then for cn;m -a:e: V 2 G…n; m† and q P 0 m…1 ÿ q† 6 bqlV …Spt lV † 6 BqlV …Spt lV † 6 maxfm…1 ÿ q†; ÿsq=2g: Proof. Follows directly from the preceeding results.



…30†

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913

That the upper bound in Corollary 4.4 may be sharp can be seen by considering appropriate Cantor `targets' (these targets can be viewed as a natural generalisation of the example considered by Hunt and Kaloshin [12]). That is, ®x 1 >  > 0 and let m…† denote the -dimensional Cantor measure on the unit interval with bqm…† …Spt m…†† ˆ …1 ÿ q† and consider the Cantor target measure , l…†, generated by rotating m…† around the origin, this is the natural measure on f…r; h† : r 2 Spt m…†g (See Example 7.7 of [5].) By observing that there exist bi-Lipschitz maps between quadrants of these targets and the cross product of m…† with Lebesgue measure restricted to the unit interval, it is straightforward to calculate, using results in [18,19], that bql…† …Spt l…†† ˆ …1 ÿ q†…1 ‡ †: Moreover, for all V 2 G…2; 1†, bq‰l…†ŠV …Spt ‰l…†ŠV † ˆ maxf1 ÿ q; ÿ…1 ‡ †q=2g: By taking a suitable countable sum of rescaled and shifted versions of such measures one may obtain the upper bound in Corollary 4.4. One may even, by considering restrictions of these targets to sectors subtended by an angle less than p, show the existence of measures, x, which have bqxV …Spt xV † ˆ maxf1 ÿ q; ÿq=2g for a set of V of positive measure and bqxV …Spt xV † ˆ 1 ÿ q: for a di€erent set of V of positive measure. We have said nothing about the case for q negative. By considering the measure, l, in the plane which has Radon±Nikodym derivative, f , with respect to Lebesgue measure given by f …x† ˆ minf0; …1 ÿ jxj†aÿ1 g where a > 1, we ®nd that for any projection V we have bqlV …Spt lV † ˆ maxf1 ÿ q; ÿaqg: It appears that in order to investigate what happens when q < 0, one should look at the Lp -class of 1=f (when restricted to those points where f is non-zero.) Again by restricting l to particular sectors of the unit disc we may arrange for di€erent behaviour under projections for two di€erent sets of directions of positive measure. Any general theory for the behaviour of measures under projections will have to take these examples into account and, in particular, we cannot expect to ®nd a single parameter which will encapsulate how measures behave under projections for almost every direction. 5. Bounds for measures with small support In this section we use a characterisation of the generalised Hausdor€ dimension in terms of auxiliary measures to investigate the projections of measures with small support. We introduce a local density which typi®es the behaviour of a measure locally under projection and use this to investigate projections for 0 6 q 6 1. We ®rst introduce some useful notation: For a Borel set E  Rn we let M…E† denote the collection of ®nite Borel measures with compact support contained in E. For q; t 2 R, a ®nite compactly supported Borel measure l on Rn and a set E  Spt l with l…E† > 0, we de®ne Mq;t l …E† ˆ fm 2 M…E† : For m-a:e: x;

q

for 0 < r 6 1; mB…x; r† 6 rt lB…x; 3r† g:

…31†

Theorem 5.1 (Frostman lemma). For a compact set E  Spt l with l…E† > 0 we have, for q 2 R bql …E† ˆ supft 2 R : 9m 2 Mq;t l …E† with m 6ˆ 0g:

…32†

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Proof. We provide a brief sketch of the proof in the case when q P 0. We begin by de®ning a net measure on dyadic cubes which determines the same dimension function as our generalised Hausdor€ measure: viz, for a cube Q we set h…Q† ˆ inffh…x; r† : x 2 Q; B…x; r†  Qg; where q

h…x; r† ˆ inffqt lB…x; 3q† : r 6 q 6 1g: (That this does determine the same dimension function is straightforward to check.) The veri®cation of the theorem for this net measure now follows routinely from the theory described by Rogers [22]. For q < 0, one may use the methods described in [11]. As we shall not be using this theorem when q < 0 we omit further details.  This result leads us to de®ne for a measure m, bql …m† ˆ supft 2 R : For m-a:e: x; bql …m; x† P tg;

…33†

where we have de®ned the local …q; l†-Hausdor€ dimension, bql …m; x†, of a measure m at a point x by . bql …m; x† ˆ lim inf r&0 (

log m…B…x; r†† ÿ q log l…B…x; 3r†† log r

ˆ sup t 2 R : lim sup r&0

…34† )

m…B…x; r†† q ˆ 0 : rt ‰l…B…x; 3r††Š

…35†

This allows us to reformulate (32) as bql …E† ˆ supfbql …m† : m 2 M…E†; m 6ˆ 0g:

…36†

It is now straightforward to ®nd our ®rst quanti®cation of the behaviour of bql …
† under projection. Proposition 5.2. For compact sets E  Spt l and q 2 R it is the case that for all V 2 G…n; m† n o bqlV …PV …E†† ˆ sup bqlV …mV † : m 2 M…E†; m 6ˆ 0 : Proof. This result follows immediately from the preceding theorem together with the observation that a ®nite Borel measure x on PV …E† may be pulled back to give a ®nite Borel measure on E (see [16 Theorem 1.18]).  The main advantage of this result is that it means we need to only consider test measures supported on l in the original space to determine the behaviour of a projected measure. It is now possible to characterise bql …E† in terms of appropriately formed energy integrals. In the usual theory of projections, energy integrals prove a useful tool, however in our more general situation the resulting energy integrals appear to be unamenable to analysis and we avoid using them in the sequel. For completeness though, we state the characterisation. Corollary 5.3. For a Borel set E  Spt l and q 2 R   Z Z dm…y† dm…x† < 1 ; bql …E† ˆ sup t 2 R : 9m 6ˆ 0 2 M…E† s:t: hq;t l …x; jy ÿ xj†

…37†

q t where hq;t l …x; r† ˆ r lB…x; 3r† .

In order toR investigate further the behaviour of measures under projections we need to study the properties of xV B…xV ; r† dc…V † for a general measure x. The approach we use here was ®rst used by Falconer and Mattila [8] and further developed in [7].

T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

915

Fix 0 < m 6 n and suppose that x is a Borel probability measure with Spt x  B…0; 1†  Rn . We begin by investigating the behaviour of the xV -measure of a ball in V for V 2 G…n; m† and relate this to local properties of the measure x. To this end we shall investigate the behaviour of the function fx;m …x; r† given by Z xV …B…PV …x†; r†† dcn;m …V †: fx;m …x; r† ˆ G…n;m†

Lemma 5.4. There are constants 0 < cn;m 6 1 6 Cn;m < 1 such that for x 2 Spt l and r > 0 Z ÿm fx;m …x; r† 6 Cn;m minf1; rm jy ÿ xj g dx…y† and

Z cn;m

ÿm

minf1; rm jy ÿ xj

g dx…y† 6 fx;m …x; r†:

…38†

…39†

Consequently, R log minf1; rm jy ÿ xjm g dx…y† ˆ 1: lim r&0 log fx;m …x; r† Also

Z

minf1; rm jy ÿ xj

ÿm

g dx…y† ˆ mrm

Z

1 r

hÿmÿ1 x…B…x; h†† dh:

…40†

Proof. See [16] for (38), the other inequality (39) is obvious. Eq. (40) follows from Fubini's theorem.  This leads us to introduce a kernel function K m : [n Rn n f0g ! …0; 1† by setting K m …z† ˆ minf1; jzj On de®ning m

K  x…x; r† ˆ

Z

ÿm

g:

K m ……y ÿ x†=r† dx…y†

…41†

…42†

we see, from (38) and (39), that in order to understand how the x-measure of balls behaves under typical projections we need only understand the behaviour of K m  x…x; r†. We observe, from (40) that K m  x…x; 2r† 6 2m K m  x…x; r†:

…43†

Some other simple observations one can immediately make are given in the following inequalities: for all V 2 G…n; m†, x…B…x; r†† 6 K m  x…x; r† 6 K m  xV …xV ; r†

…44†

x…B…x; r†† 6 xV …B…xV ; r†† 6 K m  xV …xV ; r†:

…45†

and

Lemma 5.5. Suppose that m is a finite Borel measure on Rn with compact support and 0 < m 6 n, then R log K m  mV …xV ; r† dcn;m …V † ˆ 1: lim r&0 log K m  m…x; r†

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Proof. We use (38) and Fubini's theorem to deduce that   Z Z 1 h K m  mV …xV ; r† dcn;m …V † 6 Cn;m m2 rm mB…x; h† dh: hÿmÿ1 log r r However (45) immediately implies that Z Z m K  mV …xV ; r† dcn;m …V † P mV B…xV ; r†; dcn;m …V † which together with (39) gives Z K m  mV …xV ; r† dcn;m …V † P cn;m K m  m…x; r†: It is now straightforward, using the characterisation given in (40), to verify the lemma.



In order to proceed in our investigation of the behaviour of l under projection we need to introduce an assumption on the structure of its support: we need to be able to assume the existence of a uniform measure with support containing the support of l. More precisely: We de®ne a set E  Rn to be s-Ahlfors regular (with measure x and constant c) if there is a Radon measure x and 0 < c < 1 such that Spt x  E and for all x 2 Spt x and 0 < r 6 1 cÿ1 rs 6 xB…x; r† 6 crs :

…46†

We observe that Rn is itself n-Ahlfors regular and an m-plane V 2 G…n; m† is m-Ahlfors regular. It is easy to see that an s-Ahlfors regular set has packing dimension less than or equal to s. The reason for introducing this notion is that it allows us to derive growth estimates on measures supported on Ahlfors regular sets. We begin by deriving some elementary estimates on measures supported on such sets. Lemma 5.6. Suppose that a Borel set E  Rn is s-Ahlfors regular for some 0 6 s 6 n with measure x and constant c. Then for all finite Borel measures l with compact support contained in E we have for k > 1; 0 < r < 1=…2k† and M > 1, lfx : lB…x; kr† P MlB…x; r†g 6 c2 22s M ÿ1 ks l…Rn †: Proof. This lemma is a generalisation of Lemma 2.1 in [8] (which in turn is based on a result in [21]). In [8,21], they use the fact that Rn is n-Ahlfors regular (with measure Ln ). We omit further details.  Following the method in [8] it is now straightforward to show that no measure l can have too many points where the measure of a ball grows too quickly. Proposition 5.7. Suppose that a Borel set E  Rn is an s-Ahlfors regular set and l is a finite Borel measure with compact support contained in E. Then there is a constant c > 0 such that for 0 < a < 1 and  > 0, for l-a:e: x there is r0 > 0 such that for 0 < r < r0  s…1‡† h lB…x; h† 6 c lB…x; r† r for ra 6 h 6 1. Proof. The proof is exactly the same as that given in [8, Lemma 2.2], using Lemma 5.6 above in place of their Lemma 2.1.  The main use of this result R is that it allows us to estimate, for m supported on an s-Ahlfors regular set (where s < m), the value of mV B…xV ; r† dc…V † from above by mB…x; r†.

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Lemma 5.8. Fix 0 < m 6 n. Suppose that m is a compactly supported, finite Borel measure on Rn with support contained in an s-Ahlfors regular set for some 0 < s 6 m. Then for all n > 0 and m-a:e: x there are r0 > 0 and c > 0 such that for 0 < r 6 r0 K m  m…x; r† 6 crÿn mB…x; r† and

Z G…n;m†

…47†

K m  mV …xV ; r† dcn;m …V † 6 crÿn mB…x; r†:

…48†

Proof. Both these results rely on the bound on the growth of m given in Proposition 5.7. We only prove (48) here; the proof of (47) is similar. Fix 1 > n > 0 and choose 0 < a < 1 such that s…1 ÿ a†…1 ‡ n† < n. First observe that as Spt m is contained in an s-Ahlfors set then B…m† 6 s and thus for m-a:e: x there is 0 < r0 6 1 such that for 0 < r 6 r0 we have mB…xr† P rm‡n : Further from Proposition 5.7, for m-a:e: x there is 0 < r1 < r0 such that for 0 < r 6 r1 and ra 6 h 6 1  s…1‡n† h mB…x; r†: mB…x; h† 6 c r We can now ®nd 0 < r2 6 r1 such that for 0 < r < r2 , r…1ÿa†…mÿs…1‡n†† j log rj 6 rÿn : 1=a

Thus we ®nd that for m-a:e: x and 0 < r < R  r2 Z K m  mV …xV ; r† dcn;m …V † G…n;m†

ˆ mrm

Z

hÿmÿ1

2 m

6 Cn;m m r

"

Z

1 r

2

that Fubini's theorem gives

Z mV B…xV ; r† dcn;m …V † dh hÿmÿ1 log…h=r†mB…x; r† dh a

6 Cn;m m mB…x; r †

Z r

ra

 ÿmÿ1     h h h log d r r r Z

‡

R ra

#  ÿmÿ1       h h h r m m…Rn † mB…x; h† d ‡ log : r r r R m

Thus, as     Z ra  ÿmÿ1 h h h log d 6 mÿ2 ; r r r r and

Z

 ÿmÿ1     h h h mB…x; h† d log r r r a r Z R=r 6 cmB…x; r† H ÿ‰mÿs…1‡n†Š log H dH R

raÿ1

6

   cmB…x; r† 1 r…1ÿa†…mÿs…1‡n†† ÿ …1 ÿ a† log r m ÿ s…1 ‡ n† m ÿ s…1 ‡ n†

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we deduce that Z K m  mV …xV ; r† dcn;m …V † 6 c0 rÿn mB…x; r† G… n;m†

for some constant c0 , as required.



Corollary 5.9. Suppose that x is a compactly supported, finite Borel measure on Rd , then for x-a:e: x and for any sequence ri & 0 lim inf i!1

log xB…x; ri † log K d  x…x; ri † ˆ lim inf i!1 log ri log ri

…49†

log xB…x; ri † log K d  x…x; ri † ˆ lim sup : log ri log ri i!1

…50†

and lim sup i!1

Proof. Both equations follow directly from (44) and (47) (with m ˆ d).



Proposition 5.10. Fix 0 < m < n and suppose that 0 < s 6 m is such that Spt l is s-Ahlfors regular. For all finite Borel measures m with support contained in Spt l we find that for q P 0 and almost every V 2 G…n; m† bqlV …mV † P bql …m†:

…51†

Proof. Since bqlV …mV † is de®ned to be supft 2 R : For mV -a:e: xV ; bqlV …mV ; xV † P tg it suces to show that for almost every V 2 G…n; m† and mV -a:e: xV bqlV …mV ; xV † P bql …m; x†: Fix t < bql …m; x† to ensure that lim sup r&0

mB…x; r† q ˆ 0: rt lB…x; 3r†

Now observe that from (48), for all n > 0 and m-a:e: x there is c > 0 and r0 > 0 such that for 0 < r 6 r0 Z K m  mV …xV ; r† dcn;m …V † 6 crÿn mB…x; r†: G…n;m†

Hence for 0 < r 6 r0 , as q P 0, Z Km  mV …xV ; r† mB…x; r† dcn;m …V † 6 tÿn q: tÿ2n l B…x ; 3r†q lB…x; 3r† r r V G…n;m† V Thus, on choosing 0 < r1 < r0 such that for 0 < r 6 r1 mB…x; r†

q rt lB…x; 3r†

6 1;

we can estimate, from Markov's inequality, that for any r 6 r1 and C > 0,  cn;m V 2 G…n; m† : K m  mV …xV ; r† P Crtÿ2n lV B…xV ; 3r†q 6 crn =C:

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919

On recalling (43), we deduce that for all k 2 N  q V : 9r 2 …2ÿkÿ1 ; 2ÿk Š; K m  mV …xV ; r† > Crtÿ2n lV B…xV ; 3r†   CM  V : K m  mV …xV ; 2ÿkÿ1 † > ÿm 2ÿ…k‡1†…tÿ2n† lV B…xV ; 3
2ÿkÿ1 †q 2  Gn;k ; say (where M ˆ minf1; 2tÿ2n g). Hence we may use the Borel±Cantelli lemma to deduce that for almost every V 2 G…n; m† lim sup r&0

K m  mV …xV ; r† q ˆ 0: rtÿ2n lB…xV ; 3r†

Thus, on choosing a sequence ni & 0, we conclude that for m-a:e: x and almost every V 2 G…n; m† bqlV …mV ; xV † P bql …m; x†; as required.



Corollary 5.11. Fix 0 < m < n. For l a compactly supported finite Borel measure on Rn with the support of l an s-Ahlfors regular set for some 0 6 s 6 m we have for all compact sets E  Spt l; q P 0 and almost every V 2 G…n; m† bqlV …E† P bql …E†:

…52†

Proof. This follows directly from Proposition 5.2 and (51).  In fact, from Lemma 3.2 we know this result for all V for q P 1. It is straightforward to verify that any compact set which is self-similar and satis®es strong separation is automatically s-Ahlfors regular for s equal to its packing dimension (just choose x to be s-dimensional Hausdor€ measure restricted to the set). Hence for any measure whose support is contained in such a set with s 6 m we may use (52) to deduce a lower bound for its generalised Hausdor€ dimension function for projections for 0 6 q 6 1. In particular, for self-similar measures satisfying strong separation we obtain the following result. Corollary 5.12. Fix 0 < m < n and suppose that l is a self-similar measure on Rn with support of dimension s 6 m. Then for cn;m -a:e: V and 0 6 q 6 1, bqlV …Spt lV † ˆ bqlV …Spt lV † ˆ bql …Spt l†: Proof. We just use Corollary 5.11 together with (25).



(In fact this result will hold for any quasi self-similar measure, see [20] for de®nitions and examples.) We are left with the problem of ascertaining which sets are s-Ahlfors regular for some s. It is clear that if a set is s-Ahlfors regular then s must be at least equal to the set's packing dimension and it seems unlikely that all sets are necessarily s-Ahlfors regular for s equal to their packing dimension. However I do not know of a counterexample. 6. Open problems One can investigate the behaviour of bql …m† for a test measure m a little further. Lemma 6.1. Suppose that m 2 M…Spt l† and q P 0. Then for almost every V 2 G…n; m† bqlV …mV † 6 bq;m l;m …m†;

…53†

920

T.C. O'Neil / Chaos, Solitons and Fractals 11 (2000) 901±921

where q;m bq;m l;m …m† ˆ supft 2 R : bl;m …m; x† P t for m-a:e: xg

and

(

bq;m l;m …m; x†

) K m  m…x; r† ˆ sup t 2 R : lim sup t m q ˆ 0 : r K  l…x; 3r† r&0

Proof. It is clear, from (45), that for any V 2 G…n; m†. mV B…x; r† K m  mV  …xV ; r† 6 q q : rt lV B…x; 3r† rt K m  l…x; 3r† The lemma now follows an application of Markov's inequality and the Borel±Cantelli lemma.



At this point we would like to use Lemma 6.1 together with Proposition 5.2 to conclude that for almost every V , bqlV …PV E† 6 supfbq;m l;m …m† : m 2 M…E†; m 6ˆ 0g:

…54†

Unfortunately this involves an illegal exchange of quanti®ers and is not true in general: this is clearly illustrated when l is 1-dimensional Hausdor€ measure restricted to the unit circle. It is easy to show that for all m 2 M…Spt l† we have bq;m l;m 6 1 ÿ q (with equality when m ˆ l) but, recall from earlier, that for all V 2 G…2; 1†, bqlV …Spt lV † ˆ maxf1 ÿ q; ÿq=2g. This apparent contradiction for q P 2 is resolved by recognising that, although for each test measure, m, the set of exceptional directions has zero measure, the measure which realises the supremum for a given direction, V , may have that V as an exceptional direction and thus (53) does not apply for this particular measure and direction. In fact, for this example the supremum for a given direction is realised when m is a Dirac measure on a point of Spt l which has tangent perpendicular to V and, in this case, the exceptional direction for m is, indeed, V . One possible way to overcome this problem is to introduce a parameter which measures the deviation of lV B…xV ; r† from K m  l…x; r† (for r > 0) and attempt to analyse its behaviour. It seems that such a parameter would depend heavily on the geometry of the support of l and would probably result in a classi®cation of measures similar to that in the classical theory of recti®ability. If the measure l is sucently small (that is, its support has small packing dimension) then one would hope that the set of directions where (54) fails is also small ± this may prove a fruitful line of investigation. References [1] D. Bessis, G. Paladin, G. Turchetti, S. Vaienti, Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, Journal of Statistical Physics 51 (1988) 109±134. [2] G. Brown, G. Michon, J. Peyriere, On the multifractal analysis of measures, Journal of Statistical Physics 66 (1992) 775±790. [3] R. Cawley, R.D. Mauldin, Multifractal decomposition of Moran fractals, Advances in Mathematics 92 (1992) 196±236. [4] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. [5] K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990. [6] K.J. Falconer, Techniques in Fractal Geometry, Wiley, New York, 1997. [7] K.J. Falconer, J.D. Howroyd, Packing dimensions of projections and dimension pro®les, Mathematical Proceedings of the Cambridge Philosophical Society 121 (1997) 269±286. [8] K.J. Falconer, P. Mattila, The packing dimension of projections and sections of measures, Mathematical Proceedings of the Cambridge Philosophical Society 119 (1996) 695±713. [9] K.J. Falconer, T.C. O'Neil. Convolutions and the geometry of multifractal measures, to appear in Mathematische Nachrichten. [10] T.C. Halsey, M.H. Jensen, L.P. Kadano€, I. Procaccia, B.J. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Physics Reviews A 33 (1986) 1141±1151. [11] J.D. Howroyd, On dimension and on the existence of sets of ®nite positive Hausdor€ measure, Proceedings of the London Mathematical Society (3) 70 (1995) 604±881. [12] B.R. Hunt, V.Y. Kaloshin, Which dimensions of fractal measures are preserved by typical projections? Preprint. [13] R. Kaufman, On Hausdor€ dimension of projections, Mathematika 15 (1968) 153±155.

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