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2 Capacities and box-counting dimensions
Throughout this section we will consider images of a non-empty compact set E; since box dimensions are not defined for unbounded sets or for the empty set, and also the box dimensions of a set equal those of its closure, we lose little by doing so. We will assume without always saying so explicitly that E is non-empty.
2.1 Capacity and minimum energy Capacity arguments using potential kernels of the form φ(x) = x −s are widely used in Hausdorff dimension arguments, see for example [10, 11, 14, 16].| | For box-counting dimensions, another class of kernels turns out to be useful. Let s > 0 and r > 0. Throughout this paper we use the potential kernels r s φs(x) = min 1, (x Rn) (2.1) r x ∈ n | | o which were introduced in [4, 6]. Let E Rn be non-empty and compact and let (E) denote the set of Borel probability measures⊂ supported by E. We define the energyM of µ (E) with respect to the kernel φs by ∈M r φs(x y)dµ(x)dµ(y), Z Z r −
2 and the potential of µ at x Rn by ∈ φs(x y)dµ(y). Z r −
s The capacity Cr (E) of E is the reciprocal of the minimum energy achieved by probability measures on E, that is
1 s s = inf φr(x y)dµ(x)dµ(y); (2.2) Cr (E) µ∈M(E) Z Z −
s s note that since φr is continuous and E is compact, 0 < Cr (E) < . (For a non-closed bounded set the capacity is taken to equal that of its closure.) ∞ The following energy-minimising property is standard in potential theory, but it is key for our needs, so we give the proof which is particularly simple for continuous kernels.
Lemma 2.1. Let E Rn be compact and s > 0 and r > 0. Then the infimum in (2.2) ⊂ is attained by a measure µ (E). Moreover 0 ∈M
s 1 φr(x y)dµ0(y) s (2.3) Z − ≥ Cr (E) for all x E, with equality for µ -almost all x E. ∈ 0 ∈ s s Proof. Let µk (E) be such that φr(x y)dµk(x)dµk(y) γ := 1/Cr (E). Then ∈ M − → s µk has a subsequence that is weaklyR convergent R to some µ0 (E); since φ (x y) is ∈ M r − continuous the infimum is attained. Now suppose that φs(w y)dµ (y) γ ǫ for some w E and ǫ > 0. Let δ be r − 0 ≤ − ∈ w the unit point mass atRw and for 0 <λ< 1 let µλ = λδw + (1 λ)µ0 (E). Then − ∈M
s 2 s s φ (x y)dµλ(x)dµλ(y) = λ φ (w w)+2λ(1 λ) φ (w y)dµ0(y) Z Z r − r − − Z r − 2 s + (1 λ) φ (x y)dµ0(x)dµ0(y) − Z Z r − λ2 +2λ(1 λ)(γ ǫ)+(1 λ)2γ ≤ − − − = γ 2λǫ + O(λ2), − which, on taking λ small, contradicts that µ0 minimises the energy integral. Thus in- equality (2.3) is satisfied for all x E, and equality for µ -almost all x is immediate from ∈ 0 (2.2).
2.2 Capacities and box-counting numbers For a non-empty compact E Rn, let N (E) be the minimum number of sets of diameter ⊂ r r that can cover E. Recall that the lower and upper box-counting dimensions of E are defined by
log Nr(E) log Nr(E) dimBE = lim and dimBE = lim , (2.4) log r r→0 log r r→0 − −
3 with the box-counting dimension given by the common value if the limit exists, see for s example [2]. The aim of this section is to prove Corollary 2.4, that the capacity Cr (E) and the covering number N (E) are comparable provided that s n. Note that this is not r ≥ necessarily the case if 0 s < n and we will see in Subsection 2.3 that it is this disparity that leads to formulae for≤ the box dimensions of projections. The next two lemmas obtain lower and upper bounds for Nr(E). Lemma 2.2. Let E Rn be compact and let r > 0. Suppose that E supports a measure µ (E) such that⊂ for some γ > 0 ∈M (µ µ) (x, y): x y r γ. (2.5) × | − | ≤ ≤ Then c N (E) n , (2.6) r ≥ γ where the constant cn depends only on n. In particular (2.6) holds if, for some s> 0,
φs(x y)dµ(x)dµ(y) γ. (2.7) Z Z r − ≤ Proof. Let (E) be the set of half-open coordinate mesh cubes of diameter r (i.e. of C ′ side length r/√n) that intersect E, and suppose that there are Nr(E) such cubes. By Cauchy’s inequality,
2 1 = µ(E)2 = µ(C) CX∈C(E) N ′ (E) µ(C)2 ≤ r CX∈C(E) = N ′ (E) (µ µ) (w, z) C C r × ∈ × CX∈C(E) N ′ (E)(µ µ) (w, z): w z r ≤ r × | − | ≤ N ′ (E) γ ≤ r (2√n)n N (E) γ, ≤ r noting that a set of diameter r can intersect at most (2√n)n of the cubes of (E). C Since 1 (x) φs(x), inequality (2.7) implies (2.5), to complete the proof. B(0,r) ≤ r Lemma 2.3. Let E Rn be non-empty and compact and let s > 0 and r > 0. Suppose ⊂ that E supports a measure µ (E) such that for some γ > 0 ∈M φs(x y)dµ(y) γ for all x E. (2.8) Z r − ≥ ∈ Then c log (diamE/r)+1 n,n⌈ 2 ⌉ if s = n γ Nr(E) c , (2.9) ≤ n,s if s > n γ where cn,s depends only on n and s.
4 Proof. Write M = diamE. For all x E, ∈
⌈log2(M/r)−1⌉ s −ks φr(x y)dµ(y) µ(B(x, r))+ 2 dµ(y) Z − ≤ Z k+1 k Xk=0 B(x,2 r)\B(x,2 r)
⌈log2(M/r)−1⌉ µ(B(x, r))+ 2−ksµ(B(x, 2k+1r)) ≤ Xk=0
⌈log2(M/r)⌉ 2s 2−ksµ(B(x, 2kr)). ≤ Xk=0
′ Let B(xi,r), i = 1,...,Nr(E), be a maximal collection of disjoint balls of radii r with x E, where in this proof N ′ (E) denotes this maximum number. From (2.8), for each i, i ∈ r
⌈log2(M/r)⌉ s s −ks k γ φ (xi y)dµ(y) 2 2 µ(B(xi, 2 r)). ≤ Z r − ≤ Xk=0
Summing over the xi,
′ ⌈log2(M/r)⌉ Nr(E) N ′ (E)γ 2s(1−k) µ(B(x , 2kr)), r ≤ i Xk=0 Xi=1 so, for some k with 0 k log (M/r) , ≤ ≤ ⌈ 2 ⌉ N ′ (E) r ′ s(1−k) k Nr(E)γ log2(M/r)+1 if s = n 2 µ(B(xi, 2 r)) ⌈ ⌉ , (2.10) ≥ N ′ (E)γ2k(n−s)(1 2n−s) if s > n Xi=1 r − the case of s > n coming from comparison with a geometric series. For all x E ∈ a volume comparison using the disjointedness of the balls B(xi,r) shows that at most k n (k+1)n k (k+1)n (2 + 1) 2 of the xi lie in B(x, 2 r). Consequently x belongs to at most 2 ≤ k of the B(xi, 2 r). Thus
′ Nr(E) µ(B(x , 2kr)) 2(k+1)nµ(E) = 2n+s2−s(1−k)2k(n−s) 2n+s2−s(1−k), (2.11) i ≤ ≤ Xi=1 using that s n. Inequality (2.9) now follows from (2.10), (2.11) and the fact that N (E) a N≥′ (E) where a is the minimum number of balls in Rn of diameter 1 that can r ≤ n r n cover a ball of radius 1.
Corollary 2.4. Let E Rn be non-empty and compact and let r> 0. Then ⊂ s s cn,n log2(diamE/r)+1 Cr (E) if s = n cnCr (E) Nr(E) ⌈ s ⌉ . (2.12) ≤ ≤ cn,s Cr (E) if s > n
Proof. By Lemma 2.1 we may find µ (E) satisfying (2.2) and (2.3), so the conclusion follows immediately from Lemmas 2.2∈M and 2.3.
5 2.3 Dimension profiles and box-counting dimensions of images For s> 0 we define the lower and upper s-box dimension profiles of E Rn by ⊆ s s s log Cr (E) s log Cr (E) dimBE = lim and dimBE = lim . (2.13) log r r→0 log r r→0 − − When s n equality of the box dimensions and the dimension profiles is immediate from Corollary≥ 2.4 . Corollary 2.5. Let E Rn. If s n then ⊂ ≥ s s dimBE = dimBE and dimBE = dimBE. Proof. This follows from (2.12) and the definitions of box dimensions (2.4) and of dimen- sion profiles (2.13). The following theorem enables us to obtain upper bounds for the box dimensions of images of sets under Lipschitz or H¨older functions in terms of dimension profiles. Theorem 2.6. Let E Rn be compact and let f : E Rm be an α-H¨older map satisfying ⊂ → f(x) f(y) c x y α (x, y E), (2.14) | − | ≤ | − | ∈ where c> 0 and 0 <α 1. Then ≤ 1 1 mα dim f(E) dimmαE and dim f(E) dim E. B ≤ α B B ≤ α B Proof. From (2.1) and (2.14), for r> 0 and s> 0, r s r s φs(f(x) f(y)) = min 1, min 1, r − f(x) f(y) ≥ c x y α n | − | o n | − | o 1/α sα −s r sα = min 1,c c φ 1/α (x y) x y ≥ 0 r − n | − | o for x, y E, where c = min 1,c−s . ∈ 0 { } For each r we may, by Lemma 2.1, find a measure µ (E) such that for all x E ∈M ∈ 1 mα −1 m −1 m mα φr1/α (x y)dµ(y) c0 φr (f(x) f(y))dµ(y) c0 φr (f(x) w)d(fµ)(w), Cr1/α (E) ≤ Z − ≤ Z − ≤ Z − where fµ (f(E)) is the image of the measure µ under f defined by g(w)d(fµ)(w)= ∈M g(f(x))dµ(x) for continuous g. Then for each z = f(x) f(E), R ∈ R m c0 φr (z w)d(fµ)(w) mα . Z − ≥ Cr1/α (E) By Lemma 2.3
−1 mα N (f(E)) c log (diamf(E)/r)+1 c C 1/α (E), r ≤ m,m⌈ 2 ⌉ 0 r so −1 mα log N (f(E)) log c log (diamf(E)/r +1 c log C 1/α (E) r m,m⌈ 2 ⌉ 0 + r log r ≤ log r α log r1/α − − − and the conclusion follows on taking lower and upper limits as r 0. ց 6 The next theorem enables us to obtain almost sure lower bounds for box dimensions of images of sets. It is convenient to express the condition in probabilistic terms, though for our principal application to projections fω will simply be orthogonal projection from Rn onto the m-dimensional subspace ω G(n, m). ∈ In Theorem 2.7, (ω, , P) is a probability space and (ω, x) fω(x) is a σ( )- measurable function whereF denotes the Borel sets in Rn. 7→ F × B B Theorem 2.7. Let E Rn and let s,γ > 0. Let f : E Rm, ω Ω be a family of ⊂ { ω → ∈ } continuous mappings such that for some c> 0 s P( f (x) f (y) r) cφ γ (x y) (x, y E, r> 0). (2.15) | ω − ω | ≤ ≤ r − ∈ Then, for P-almost all ω Ω, ∈ s dim f (E) γ dims E and dim f (E) γ dim E. (2.16) B ω ≥ B B ω ≥ B Proof. First note that for all µ (E), ∈M E (f µ f µ) (w, z): w z r ω × ω | − | ≤ = E (µ µ) (x, y): f x f y r × | ω − ω | ≤ = P fωx fωy r dµ(x)dµ(y) Z Z | − | ≤ s c φ γ (x y)dµ(x)dµ(y) (2.17) ≤ Z Z r − s ′ s −γt′ using (2.15). If dimBE > t >t> 0 then C γ (E) ri for a sequence ri 0, where ri −i ≥ ց we may assume that 0 Theorem 2.8. Let E Rn be compact. Then for all V G(n, m) ⊂ ∈ m dim π E dimmE and dim π E dim E, (2.18) B V ≤ B B V ≤ B and for γ -almost all V G(n, m) n,m ∈ m m dimBπV E = dimB E and dimBπV E = dimB E. (2.19) m n Proof. Identifying V with R in the natural way, πV : R V is Lipschitz so the inequalities (2.18) follow from Theorem 2.6 taking α = 1. → Now note that values φm(x) are comparable to the proportion of the subspaces V r ∈ G(n, m) for which the r-neighbourhoods of the orthogonal subspaces to V contain x, specifically, for all 1 m < n there are numbers a > 0 such that ≤ n,m φm(x) γ V : π x r a φm(x) (x Rn). (2.20) r ≤ n,m | V | ≤ ≤ n,m r ∈ This standard geometrical estimate can be obtained in many ways, see for example [14, Lemma 3.11]. One approach is to normalise to the case where x = 1 and then estimate the (normalised) (n 1)-dimensional spherical area of S y| :| dist(y,V ⊥) r , that is the intersection of− the unit sphere S in Rn with the ‘tube’∩{ or ‘slab’ of points≤ } within distance r of some (n m)-dimensional subspace V ⊥ of Rn. In particular − γ V : π x π y r a φm(x y) (x, y Rn). (2.21) n,m | V − V | ≤ ≤ n,m r − ∈ Taking Ω = G(n, m) and P = γn,m, this is (2.15) with s = m and γ = 1. Thus (2.19) follows from Theorem 2.7. Whilst equality holds in (2.19) for γ -almost all V G(n, m), dimension profiles n,m ∈ can provide further information on the size of the set of V for which the box dimensions of the projections πV E are exceptionally small; this is analogous to the bounds on the dimensions of exceptional projections that have been obtained for Hausdorff dimensions [11, 13, 14, 16]. Note that G(n, m) is a manifold of dimension m(n m) and thus dim G(n, m) = m(n m), where dim denotes Hausdorff dimension. In− particular the H − H dimension bound for the exceptional set of V given by the following theorem decreases as the deficit m s 0 increases. The proof is similar to that of Theorem 2.8 except − ≥ integration is with respect to a measure ν supported on the exceptional set of V , with (2.20) replaced by an estimate involving ν rather than γn,m. Theorem 2.9. Let E Rn be compact and let 0 s m. Then ⊂ ≤ ≤ s dim V G(n, m) such that dim π E < dim E m(n m) (m s), (2.22) H{ ∈ B V B } ≤ − − − and dim V G(n, m) such that dim π E < dims E m(n m) (m s). (2.23) H{ ∈ B V B } ≤ − − − 8 Proof. Let s K = V G(n, m) such that dim π E < dim E . { ∈ B V B } Suppose for a contradiction that dim K > m(n m) (m s). By Frostman’s Lemma, H − − − see [13, 16], there is a Borel probability measure ν supported on a compact subset of K m(n−m)−(m−s) and c> 0 such that ν(BG(V, ρ)) cρ for all V G(n, m) and ρ> 0, where B (V, ρ) G(n, m) is the ball centre≤ V and radius ρ with respect∈ to some natural locally G ⊂ m(n m)-dimensional metric on the Grassmanian manifold G(n, m). This ensures that the subspaces− in K cannot be too densely concentrated, and analogously to (2.20) and (2.21) we obtain ν V : π x π y r aφs(x y) (x Rn) | V − V | ≤ ≤ r − ∈ for some a > 0, see [13] or [16, Inequality (5.12)] for more details. Taking Ω = G(n, m) P s and = ν with γ = 1 in Theorem 2.7 gives that dimBπV E dimBE for ν-almost all V G(n, m), contradicting the definition of k. The proof for≥ lower box dimensions is ∈ similar. 2.5 Images under stochastic processes Almost immediately after their introduction, Xiao [17, 20] used dimension profiles to determine the packing dimensions of the images of sets under fractional Brownian motions. In our framework, the box dimension analogues follow easily from Theorems 2.6 and 2.7. We first state a result that applies to general random functions. Proposition 2.10. Let X : Rn Rm be a random function, let E Rn be compact and let 0 <α 1. Suppose that → ⊂ (a) for≤ all 0 <ǫ<α there is a random constant M > 0 such that X(x) X(y) M x y α−ǫ (x, y E), (2.24) | − | ≤ | − | ∈ almost surely, and (b) for all ǫ> 0 there is a constant c> 0 r1−ǫ m P X(x) X(y) r c (x, y E, r> 0). (2.25) | − | ≤ ≤ x y α+ǫ ∈ | − | Then, almost surely, 1 1 mα dim X(E) = dimmαE and dim X(E) = dim E. B α B B α B Proof. Note that condition (2.25) implies that (1−ǫ)/(α+ǫ) (α+ǫ)m r (α+ǫ)m P X(x) X(y) r min 1,c c φ − (x y). | − | ≤ ≤ x y ≤ 0 r(1 ǫ)/(α+ǫ) − n | − | o The conclusion is immediate using Theorem 2.6 and Theorem 2.7 taking ǫ arbitrarily small. 9 Index-α fractional Brownian motion (0 <α< 1) is the Gaussian random function X : Rn Rm that with probability 1 is continuous with X(0) = 0 and such that the increments→ X(x) X(y) are multivariate normal with mean 0 and variance x y α, − n | − | see, for example, [2, 9]. In particular, X = (X1,...,Xm) where the Xi : R R are independent index-α fractional Brownian motions with distributions given by → 1 1 t2 P Xi(x) Xi(y) A = exp − dt (2.26) − ∈ √2π x y α Z 2 x y 2α | − | t∈A | − | for each Borel set A R. ⊂ Corollary 2.11. Let X : Rn Rm be index-α fractional Brownian motion (0 <α< 1) → and let E Rn be compact. Then, almost surely, ⊂ 1 1 mα dim X(E) = dimmαE and dim X(E) = dim E. B α B B α B Proof. Index-α fractional Brownian motion satisfies an (α ǫ)-H¨older condition for all − 0 <ǫ<α, so (2.24) is satisfied. Furthermore, for each ǫ> 0 P X(x) X(y) r P ( X (x) X (y) r for all 1 i m | − | ≤ ≤ | i − i | ≤ ≤ ≤ 1 1 t2 m exp − dt ≤ √2π x y α Z 2 x y 2α | − | |t|≤r | − | r1−ǫ m c ≤ x y α(1+ǫ) | − | −ǫ using that exp( u) cǫu for u> 0, giving (2.25). The conclusion follows by Proposition 2.10. − ≤ 2.6 Inequalities We exhibit some inequalities satisfied by the dimension profiles; these were obtained in [5, Section 6] but their derivation is more direct using our capacity approach. s Proposition 2.12. Let E Rn and set either d(s) = dims E or d(s) = dim E. Then ⊂ B B for 0 10 For (2.29) let 0 s/t r−d/(1+(1/s−1/t)d) φs(x y)dµ(y) R−d φt (x y)dµ(y)+ R−d φt (x y)dµ(y) . Z r − ≤ Z R − Z R − If Ct (E) R−d for some R then by Lemma 2.1 there is an energy-minimising measure R ≥ µ (E) such that the right-hand side of this inequality, and thus the left-hand side, ∈ M s 1 −d/((1+(1/s−1/t)d) is at most 2 for µ-almost all x, so Cr (E) 2 r for the corresponding r. Letting R 0, t follows that ≥ ց d(t) d(s) , ≥ 1+(1/s 1/t)d(t) − where d( ) is either the lower or upper dimension profile, which rearranges to (2.29). · Examples show that the inequalities in Lemma 2.12 give a complete characterisation of the dimension profiles that can be attained, see [5, Section 6]. Note also that it follows easily from (2.27) and (2.29) that d : R+ R+ is continuous and, indeed, locally Lipschitz with constant 1. → 3 Packing dimensions In this final section we indicate how the results on box-counting dimensions carry over to the packing dimensions of projections and images of sets. Packing measures and dimensions were introduced by Taylor and Tricot [18, 19] as a type of dual to Hausdorff measures and dimensions, see [2, 14] for more recent expositions. Whilst, analogously to Hausdorff dimensions, packing dimensions can be defined by first setting up packing measures, an equivalent definition in terms of upper box dimensions of countable coverings of a set is often more convenient in practice. For E Rn we may define the packing dimension of E by ⊂ ∞ dimPE = inf sup dimBEi : E Ei ; (3.1) 1≤i<∞ ⊂ n i[=1 o 11 since the box dimension of a set equals that of its closure, we can assume that the sets Ei in (3.1) are all compact. It is natural to make an analogous definition of the packing dimension profile of E Rn ⊂ for s> 0 by ∞ s s dimPE = inf sup dimBEi : E Ei with each Ei compact . (3.2) 1≤i<∞ ⊂ n i[=1 o By virtue of this definition, properties of packing dimension can be deduced from corresponding properties of upper box dimension. For example, we get an immediate analogue of Corollary 2.5. Corollary 3.1. Let E Rn. If s n then ⊂ ≥ s dimPE = dimPE. Similarly, packing dimensions of H¨older images behave in the same way as box dimen- sions in Theorem 2.6. Corollary 3.2. Let E Rn and let f : E Rm be an α-H¨older map satisfying ⊂ → f(x) f(y) c x y α (x, y E), | − | ≤ | − | ∈ where c> 0 and 0 <α 1. Then ≤ 1 dim f(E) dimmαE. P ≤ α P mα Proof. If t> dim E we may cover E by a countable collection of sets Ei, which we may P mα take to be compact, such that dimB Ei < t. By Theorem 2.6 1 mα t dim f(E ) dim E , B i ≤ α B i ≤ α for all i, so the conclusion follows from (3.1). For packing dimension bounds in the opposite direction we need the following property. Rn s Proposition 3.3. Let E be a Borel set such that dimPE > t. Then there exists ⊂ s a non-empty compact F E such that dimP(F U) > t for every open set U such that F U = . ⊂ ∩ ∩ 6 ∅ Note on proof. In the special case where E is compact there is a short proof is based on [1, Lemma 2.8.1]. Let be a countable basis of open sets that intersect E. Let B F = E V : dims (E V ) t . \ ∈ B P ∩ ≤ [ s s Then F is compact and, since dimP is countably stable, dimPF > t and furthermore s dimP(E F ) t. \ ≤ s Suppose for a contradiction that U is an open set such that F U = and dimP(F U) t. As is a basis of open sets we may find V U with V ∩ such6 ∅ that F V =∩ and≤ dims (FB V ) t. Then ⊂ ∈ B ∩ 6 ∅ P ∩ ≤ dims (E V ) max dims (E F ), dims (F V ) t, P ∩ ≤ { P \ P ∩ } ≤ 12 which contradicts that V . ∈ B s ′ For a general Borel set E with dimPE > t we need to find a compact subset E E s ′ ⊂ with dimPE > t which then has a suitable subset as above. Whilst this is intuitively natural, I am not aware of a simple direct proof from the definition (3.2) of packing dimension profiles in terms of box dimension profiles. The proof in [7, Theorem 22] uses a packing-type measure which relates to the s-packing dimension profile in the same way that packing measure relates to packing dimension. This measure on E is infinite, and by a ‘subset of finite measure’ argument E has a compact subset E′ of positive finite measure. ✷ Assuming Proposition 3.3 the packing dimension analogue of Theorem 2.7 follows easily. n m Corollary 3.4. Let E R be a Borel set and let s,γ > 0. Let fω : E R , ω Ω be a family of continuous⊂ mappings such that for some c> 0 { → ∈ } s P( f (x) f (y) r) cφ γ (x y) (x, y E, r> 0). | ω − ω | ≤ ≤ r − ∈ Then, for P-almost all ω Ω, ∈ dim f (E) γ dims E. P ω ≥ P s Proof. Let t< dimPE. By Proposition 3.3 we may find a non-empty compact F E such s s ⊂ that for every open U that intersects F , dimP(F U) > t, so in particular dimB(F U) > t. Rn ∩ ∞ ∩ As is seperable, there is a countable basis Ui i=1 of open sets that intersect F . By Theorem 2.7, for P-almost all ω Ω, { } ∈ s dim f (F U ) γ dim (F U ) > γt (3.3) B ω ∩ i ≥ B ∩ i for each i, and thus for all i simultaneously. For such an ω, let K ∞ be a cover of the compact set f (F ) by a countable { j}j=1 ω collection of compact sets. By Baire’s category theorem there is a k and an open V such that = fω(F ) V fω(F ) Kk. There is some Ui such that fω(F Ui) fω(F ) V , so in∅ particular 6 ∩ ⊂ ∩ ∩ ⊂ ∩ dim (f (F ) K ) dim (f (F ) V ) dim f (F U ) γt B ω ∩ k ≥ B ω ∩ ≥ B ω ∩ i ≥ by (3.3). Since there is such a Kk for every cover of fω(F ) by a countable collection of compact sets K ∞ , we conclude that dim f (E) dim f (F ) γt by (3.1). { j}j=1 P ω ≥ P ω ≥ Corollaries 3.2 and 3.4 can be applied in exactly the same way as Theorems 2.6 and 2.7 to obtain, for example, packing dimension properties of projections and random images. We just state the basic projection result. Theorem 3.5. Let E Rn be Borel. 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