Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 165–186

HAUSDORFF AND PACKING DIMENSIONS, INTERSECTION MEASURES, AND SIMILARITIES

Maarit Järvenpää

University of Jyväskylä, Department of Mathematics P. O. Box 35, FIN-40351 Jyväskylä, Finland; amj@math.jyu.fi

Abstract. Let μ and ν be Radon measures on R^{n} with compact supports. We study the Hausdorff, dim_{H}, and packing dimension, dim_{p}, properties of the intersection measures μ ∩ f_{♯}ν when f runs through the similarities of R^{n} and f_{♯}ν is the image of ν under f. These measures can be regarded as natural measures on spt μ ∩ f(spt ν), where spt is the support of a measure. Using the relations between Hausdorff dimensions of sets and measures, we show that if dim_{H}(μ × ν) = dim_{H} μ + dim_{H} ν > n and if the t-energy of ν is finite for all 0 < t < dim_{H} ν < n, then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

ess inf{dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} = dim_{H} μ + dim_{H} ν − n.

Here θ_{n} is the unique orthogonally invariant Radon probability measure on the orthogonal group of R^{n}, denoted by 𝒪_{n}, ℒ^{1} is the Lebesgue measure on the open interval (0, ∞), and τ_{z}∘g∘δ_{r}: R^{n} → R^{n} is the similarity τ_{z} ∘g∘δ_{r}(a) = rga+z. By relating packing dimensions of intersection measures to certain integral kernels, we prove that if the s-energy of μ is finite and the t-energy of ν is finite for some 0 < s < n and 0 < t < n with s+t > n, then for θ_{n} ×ℒ^{1} almost all (g, r) ∈ 𝒪_{n} ×(0, ∞) we have

ess inf{dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} = d_{μ,ν},

where d_{μ,ν} is a constant depending only on the measures μ and ν. We also deduce corresponding equalities for the upper Hausdorff and upper packing dimensions.

1. Introduction

Let A and B be Borel sets in R^{n}. The relations between the Hausdorff dimensions, dim_{H}, of A, B, and f(B), when f runs through the similarities of R^{n}, were studied by Mattila in [9]. He showed that if A is ℋ ^{s} measurable and B is ℋ ^{t} measurable such that 0 < ℋ ^{s}(A) < ∞ and 0 < ℋ ^{t}(B) < ∞ for some 0 < s < n and 0 < t < n with s + t ≥ n, then

(1.1)  dim_{H} A ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})B ≥ s + t − n

for ℋ ^{s} ×ℋ ^{t} ×θ_{n} ×ℒ^{1} almost all (x, y, g, r) ∈ A×B ×𝒪_{n} ×(0, ∞). Here ℋ ^{s} is the s-dimensional Hausdorff measure, τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y}: R^{n} → R^{n} is the similarity

1991 Mathematics Subject Classification: Primary 28A80, 28A12.

τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y}(a) = rg(a − y) + x, θ_{n} is the unique orthogonally invariant Radon probability measure on the orthogonal group of R^{n} denoted by 𝒪_{n}, and ℒ^{1} is the Lebesgue measure on the open interval (0, ∞). In general the opposite inequality in (1.1) is false, but it holds under the additional assumption that the set B has positive t-dimensional lower density at all of its points (see [9, Theorem 6.13]). For more information on results related to these questions see also [7].

Let μ and ν be Radon measures on R^{n} with compact supports. The purpose of this paper is to study Hausdorff and packing dimension analogues of (1.1) for intersection measures μ∩f_{♯}ν when f runs through the similarities of R^{n} and f_{♯}ν is the image of ν under f. These intersection measures introduced by Mattila in [9] can be regarded as natural measures on spt μ ∩ f(spt ν), where spt is the support of a measure. Using the relations between the Hausdorff dimensions of sets and measures, we show that if the t-energy of ν is finite for all 0 < t < dim_{H} ν < n, then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ dim_{H} ν + dim_{H} ν − n

for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0. Here (τ_{z} ∘ g ∘ δ_{r})_{♯}ν is the image of ν under the similarity τ_{z} ∘ g ∘ δ_{r}: R^{n} → R^{n}, τ_{z} ∘ g ∘ δ_{r}(a) = rga + z and ℒ^{n} is the Lebesgue measure on R^{n}. If we suppose in addition to the above assumptions that dim_{H}(μ × ν) = dim_{H} μ + dim_{H} ν > n, then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

ess inf{ dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0}

= dim_{H} μ + dim_{H} ν − n. (1.2)

We also prove corresponding results for the upper Hausdorff dimension.

We continue the work by Järvenpää ([4] and [5]) on packing dimension, dim_{p}, properties of intersection measures. In [4] it is shown that if the (n − t)-energy of μ is finite and the t-energy of ν is finite for some 0 < t < n, and if dim_{H} μ + dim_{H} ν > n, then

dim_{p} μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν ≥ max\{\frac{dim_{H} ν dim_{p} μ(dim_{H} μ + dim_{H} ν − n)}{n dim_{H} μ − (n − dim_{H} ν) dim_{p} μ},
\frac{dim_{H} μ dim_{p} ν(dim_{H} μ + dim_{H} ν − n)}{n dim_{H} ν − (n − dim_{H} μ) dim_{p} ν}\}

for μ × ν × θ_{n} × ℒ^{1} almost all (x, y, g, r) ∈ R^{n} × R^{n} × 𝒪_{n} × (0, ∞). In [5] a corresponding result is proved when we take isometries as the transformation group in place of similarities. The methods we are using when considering packing dimensions of intersection measures are influenced by the theory for projections of measures introduced by Falconer and Howroyd in [1] and by the methods in [6] where sections of measures were considered instead of general intersections. We show that the following analogue of (1.1) holds: if the s-energy of μ is finite and the t-energy of ν is finite for some 0 < s < n and 0 < t < n with s + t > n, then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have (1.3) ess inf{dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} = d_{μ,ν}.

Here d_{μ,ν} is a constant obtained by convolving the product measure μ × ν with a certain kernel. Corresponding results for the upper packing dimension are also deduced. If we consider isometries instead of similarities, then (1.2) holds if we assume that dim_{H}(μ × ν) = dim_{H} μ + dim_{H} ν > n and that the t-energy of ν is finite for all \frac{1}{2}(n + 1) < t < n. This can be proved using the methods of Section 3. For isometries the methods of Section 5 cannot be used. Then an integration with respect to r is not involved, which makes things more difficult.

Equalities (1.2) and (1.3) cannot be strengthened to a result saying that dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν or dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν would be almost surely constant. To see this, let μ_{1} and μ_{2} be suitably chosen measures on R^{n} supported by two disjoint balls such that there is A ⊂ 𝒪_{n} × (0, ∞) with positive θ_{n} × ℒ^{1} measure such that for any (g, r) ∈ A either W_{(z,−z)/2} ∩ (spt μ_{1} × (g ∘ δ_{r})(spt ν)) = ∅ or W_{(z,−z)/2} ∩ (spt μ_{2} × (g ∘ δ_{r})(spt ν)) = ∅ for all z ∈ R^{n} (for the notation see Chapter 2). Then by choosing the dimensions (both Hausdorff and packing) of the measures μ_{1} and μ_{2} in a suitable way, we can find for any (g, r) ∈ A sets B_{g,r}^{1} ⊂ R^{n} and B_{g,r}^{2} ⊂ R^{n} with positive ℒ^{n} measures such that the dimension of the measure π_{♯}[(μ_{1} × (g ∘ δ_{r})_{♯}ν + μ_{2} × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}] is big for z ∈ B_{g,r}^{1} and small for z ∈ B_{g,r}^{2} .

Acknowledgements. I am grateful for the hospitality I received when visiting the Mathematical Institute at the University of St. Andrews where this research was done. For financial support I am indebted to the Academy of Finland and to the Vilho, Yrjö, and Kalle Väisälä Fund. I thank Dr. Esa Järvenpää for useful discussions related to this work.

2. Notation and preliminaries

We denote by d(x, A) = inf{|x − a| : a ∈ A} the distance between x ∈ R^{n} and a non-empty set A ⊂ R^{n}. The open interval in R∪{−∞, ∞} with end points a, b ∈ R ∪ {−∞, ∞} is denoted by (a, b). For the corresponding closed interval we use the notation [a, b]. Further, B(x, r) is the closed ball of centre x ∈ R^{n} and radius 0 < r < ∞, and α(n) = ℒ^{n}(B(0, 1)) where ℒ^{n} is the Lebesgue measure on R^{n}. For 0 ≤ s < ∞, the s-dimensional Hausdorff measure is denoted by ℋ ^{s}.

If μ is a measure on a set X, we denote by f_{♯}μ the image of μ under a function f: X → Y , that is,

f_{♯}μ(A) = μ(f^{−1}(A))

for all A ⊂ Y . The restriction of μ to a set B ⊂ X is denoted by μ | B, that is,

μ | B(A) = μ(B ∩ A)

for all A ⊂ X. For 0 < t < n, the t-energy of a Radon measure μ on R^{n} is defined by

I_{t}(μ) = ∫ ∫ |x − y|^{−t} dμx dμy.

Note that if μ is a finite Radon measure on R^{n} and I_{t}(μ) < ∞, then I_{s}(μ) < ∞ for all 0 < s < t. Let μ and ν be measures on a set X. The measure μ is said to be absolutely continuous with respect to ν, if μ(A) = 0 for any A ⊂ X with ν(A) = 0. In this case we write μ ≪ ν.

Let m and n be integers with 0 < m < n. The Grassmann manifold, which consists of all (n−m)-dimensional linear subspaces of R^{n}, is denoted by G_{n,n−m}. For all V ∈ G_{n,n−m}, let V^{⊥} ∈ G_{n,m} be the orthogonal complement of V , and P_{V^{⊥}}: R^{n} → V^{⊥} the orthogonal projection onto V^{⊥}. We use the notation 𝒪_{n} for the orthogonal group of R^{n} consisting of all linear maps g: R^{n} → R^{n} preserving distance, that is, |g(x) − g(y)| = |x − y| for all x, y ∈ R^{n}. The unique invariant Radon probability measure on 𝒪_{n} is denoted by θ_{n}.

A map f: R^{n} → R^{n} is a similarity, if there is 0 < r < ∞ such that |f(x) − f(y)| = r|x − y| for all x, y ∈ R^{n}. In this case for some z ∈ R^{n}, g ∈ 𝒪_{n}, and 0 < r < ∞ we have

f = τ_{z} ∘ g ∘ δ_{r},

where τ_{z}: R^{n} → R^{n} is the translation τ_{z}(x) = x + z, and δ_{r}: R^{n} → R^{n} is the homothety δ_{r}(x) = rx.

For the definition of intersection measures we need the following definition of sliced measures. Let m and n be integers with 0 < m < n. Let μ be a finite Radon measure on R^{n}, V ∈ G_{n,n−m}, and V_{a} = {v + a : v ∈ V } for all a ∈ V^{⊥}. For ℋ ^{m} almost all a ∈ V^{⊥} there is a Radon measure μ_{V,a} on V_{a} such that

(2.1)  ∫ φ dμ_{V,a} = lim_{δ→0}(2δ)^{−m} ∫_{V_{a}(δ)} φ dμ

for all non-negative continuous functions φ on R^{n} with compact support (see [10, Chapter 10]). Here

V_{a}(δ) = {y ∈ R^{n} : d(y, V_{a}) ≤ δ}.

The measure μ_{V,a} is the slice of μ by the plane V_{a}. Obviously,

spt μ_{V,a} ⊂ spt μ ∩ V_{a},

where spt is the support of a measure. Further, if P_{V^{⊥}}_{♯}μ ≪ ℋ ^{m} | V^{⊥} and f is a non-negative Borel function on R^{n} with ∫ f dμ < ∞, then

(2.2)  ∫_{V^{⊥}} ∫ f dμ_{V,a} dℋ ^{m}a = ∫ f dμ

(see [8, Lemma 3.4 (4)]).

Now we are ready to define intersection measures. We use the method from [9], which is the same as used in [3, 4.3.20] in connection with the construction of intersection currents. Let W be the diagonal of R^{n} × R^{n}, that is,

W = {(x, y) ∈ R^{n} × R^{n} : x = y}.

Let μ and ν be Radon measures on R^{n} with compact supports. The intersection measure μ ∩ f_{♯}ν, where f = τ_{z} ∘ g ∘ δ_{r} for some z ∈ R^{n}, g ∈ 𝒪_{n}, and 0 < r < ∞, is constructed by slicing the product measure μ × (g ∘ δ_{r})_{♯}ν by the n-plane

W_{(z,−z)/2} = {(x, y) ∈ R^{n} × R^{n} : x − y = z},

and by projecting this sliced measure to R^{n} by the projection π: R^{n} ×R^{n} → R^{n}, π(x, y) = x. Hence

μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν = 2^{n/2}α(n)^{−1}π_{♯}[(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}]

provided that the sliced measure (μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2} exists. This is the case for ℒ^{n} almost all z ∈ R^{n}. Clearly,

spt μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ⊂ spt μ ∩ (τ_{z} ∘ g ∘ δ_{r}) spt ν.

If φ is a non-negative lower semicontinuous function on R^{n}, then (2.1) gives

(2.3)  \begin{aligned}∫ φ dμ & ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν \\ & ≤ lim_{δ→0} α(n)^{−1}δ^{−n} ∫_{\{(x,y):|S_{g,r}(x,y)−z|≤δ\}} φ(x) d(μ × ν)(x, y). \end{aligned}

Here S_{g,r}: R^{n} × R^{n} → R^{n} is defined by S_{g,r}(x, y) = x − rgy.

Note that if S_{g,r_{♯}}(μ × ν) ≪ ℒ^{n}, then P_{W^{⊥}}_{♯}(μ × (g ∘ δ_{r})_{♯}ν) ≪ ℋ ^{n}|W^{⊥}, and the disintegration formula (2.2) implies that

(2.4)  ∫_{W^{⊥}} ∫ f d(μ × (g ∘ δ_{r})_{♯}ν)_{W,a} dℋ ^{n}a = ∫ f d(μ × (g ∘ δ_{r})_{♯}ν)

provided that f is a non-negative Borel function with ∫ f d(μ × (g ∘ δ_{r})_{♯}ν) < ∞. Further, by (2.4),

ℋ ^{n}({a ∈ W^{⊥} : (μ × (g ∘ δ_{r})_{♯}ν)_{W,a}(R^{n} × R^{n}) > 0}) > 0,

which gives

ℒ^{n}({z ∈ R^{n} : μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0}) > 0.

The following definitions of dimensions will be used throughout this paper. The Hausdorff and packing dimensions of a finite Radon measure μ on R^{n} are defined by

dim_{H} μ = sup{u ≥ 0 : lim sup_{h→0} h^{−u}μ(B(x, h)) = 0 for μ almost all x ∈ R^{n}}

and

dim_{p} μ = sup{u ≥ 0 : lim inf_{h→0} h^{−u}μ(B(x, h)) = 0 for μ almost all x ∈ R^{n}}.

We will also consider the upper Hausdorff and upper packing dimensions defined as follows

dim_{H}^{∗} μ = sup{u ≥ 0 : μ({x ∈ R^{n} : lim sup_{h→0} h^{−u}μ(B(x, h)) = 0}) > 0}

and

dim_{p}^{∗} μ = sup{u ≥ 0 : μ({x ∈ R^{n} : lim inf_{h→0} h^{−u}μ(B(x, h)) = 0}) > 0}.

Equivalently, these dimensions can be determined by using Hausdorff and packing dimensions of sets. In fact,

dim_{H} μ = inf{dim_{H} A : A is a Borel set and μ(A) > 0},
dim_{p} μ = inf{dim_{p} A : A is a Borel set and μ(A) > 0},
dim_{H}^{∗} μ = inf{dim_{H} A : A is a Borel set and μ(R^{n} \ A) = 0},

and

dim_{p}^{∗} μ = inf{dim_{p} A : A is a Borel set and μ(R^{n} \ A) = 0}.

The following lemma gives a relation between finiteness of energies and Hausdorff dimensions of measures. It is an immediate consequence of the relation between Riesz capacities and Hausdorff dimensions of Borel sets.

Lemma 2.5. If μ is a Radon measure on R^{n} with 0 < μ(R^{n}) < ∞ and with I_{t}(μ) < ∞, then dim_{H} μ ≥ t.

3. Hausdorff dimension and intersection measures

Lemma 3.1. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that dim_{H}(μ×ν) > n. If (g, r) ∈ 𝒪_{n} ×(0, ∞) is such that S_{g,r_{♯}}(μ×ν) ≪ ℒ^{n}, then

ess inf{ dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0}
≤ dim_{H}(μ × ν) − n.

Proof. Consider u ≥ 0 such that dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0. Since dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≤ dim_{H}(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2} , we have

u ≤ dim_{H}(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}

for ℋ ^{n} almost all (z, −z)/2 ∈ W^{⊥} with (μ × (g ∘ δ_{r})_{♯})_{W,(z,−z)/2}(R^{n} × R^{n}) > 0. The fact that dim_{H}(μ×(g ∘δ_{r})_{♯}ν) = dim_{H}(μ×ν) > n, gives with [6, Lemma 3.1]

u ≤ dim_{H} μ × (g ∘ δ_{r})_{♯}ν − n = dim_{H}(μ × ν) − n

and the claim follows. Note that [6, Lemma 3.1] holds for W since P_{W^{⊥}}_{♯}(μ × (g ∘ δ_{r})_{♯}ν) ≪ ℋ ^{n} | W^{⊥}.

For the purpose of proving that the opposite inequality holds in Lemma 3.1 under some additional assumptions we need the following result.

Lemma 3.2. Let μ and ν be Radon measures on R^{n} with compact supports and let B ⊂ R^{n} be a Borel set. If (g, r) ∈ 𝒪_{n} ×(0, ∞) is such that S_{g,r_{♯}}(μ×ν) ≪ ℒ^{n}, then

(μ | B) ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν = (μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν) | B

for ℒ^{n} almost all z ∈ R^{n}.

Proof. Let A ⊂ R^{n}. Since P_{W^{⊥}}_{♯}(μ × (g ∘ δ_{r})_{♯}ν) ≪ ℋ ^{n} | W^{⊥}, we have by [6, Lemma 3.2] for ℒ^{n} almost all z ∈ R^{n}

((μ∩(τ_{z} ∘ g ∘ δ_{r})_{♯}ν) | B)(A) = 2^{n/2}α(n)^{−1}π_{♯}[(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}](B ∩ A)
= 2^{n/2}α(n)^{−1}(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}((B × R^{n}) ∩ (A × R^{n}))
= 2^{n/2}α(n)^{−1}((μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2} | (B × R^{n}))(A × R^{n})
= 2^{n/2}α(n)^{−1}((μ × (g ∘ δ_{r})_{♯}ν) | (B × R^{n}))_{W,(z,−z)/2}(A × R^{n})
= 2^{n/2}α(n)^{−1}((μ|B) × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(A × R^{n})
= 2^{n/2}α(n)^{−1}π_{♯}[((μ|B) × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}](A)
= (μ | B) ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A).

Lemma 3.3. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that I_{t}(ν) < ∞ for all 0 < t < dim_{H} ν < n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ dim_{H} μ + dim_{H} ν − n

for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0.

Proof. Let 0 < t < dim_{H} ν < n. It is enough to prove that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ dim_{H} μ + t − n

for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0. The claim follows by taking a sequence (t_{i}) tending to dim_{H} ν from below.

We may assume that dim_{H} μ + t > n. For all (g, r) ∈ 𝒪_{n} × (0, ∞) define

C_{g,r} = {z ∈ R^{n} : μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0}.

Since

{z ∈ C_{g,r} : dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν < dim_{H} μ + t − n}
= ⋃_{i=1}^{∞} ⋃_{j=1}^{∞}{z ∈ C_{g,r} : there is a Borel set A ⊂ R^{n} such that

dim_{H} A < dim_{H} μ + t − n − \frac{1}{i} and μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A) > \frac{1}{j}\},

it suffices to show that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) the set

E_{g,r} = {z ∈ C_{g,r} : there is a Borel set A ⊂ R^{n} such that
dim_{H} A < u + t − n and μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A) > ε}

has ℒ^{n} measure zero for fixed u < dim_{H} μ and ε > 0.

Let u < v < dim_{H} μ. Then lim sup_{h→0} h^{−v}μ(B(x, h)) = 0 for μ almost all x ∈ R^{n}, and so,

(3.4)  lim_{i→∞} μ(R^{n} \ B^{i}) = 0,

where

B^{i} = \{x ∈ R^{n} : μ(B(x, h)) ≤ h^{v} for all 0 < h ≤ \frac{1}{i}\}

is a Borel set. Further, I_{u}(μ | B^{i}) < ∞ for all i. By [9, Theorem 6.6] we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) that S_{g,r_{♯}}((μ | B_{i}) × ν) ≪ ℒ^{n} for all i. This together with (3.4) gives S_{g,r_{♯}}(μ × ν) ≪ ℒ^{n} for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞). By [9, Theorem 6.7] and Lemma 3.2, for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have for ℒ^{n} almost all z ∈ R^{n}

(3.5)  I_{u+t−n}((μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν) | B^{i}) < ∞

for all i. For all i and (g, r) ∈ 𝒪_{n} × (0, ∞) define

D_{g,r}^{i} = {z ∈ C_{g,r} : (μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}((R^{n} \ B^{i}) × R^{n}) > α(n)2^{−n/2}ε}.

Now the disintegration formula (2.4) gives that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞)

μ × (g ∘ δ_{r})_{♯}ν((R^{n} \ B^{i}) × R^{n})
= ∫ (μ × (g ∘ δ_{r})_{♯}ν)_{W,a}((R^{n} \ B^{i}) × R^{n}) d(ℋ ^{n} | W^{⊥})a

= 2^{n/2}α(n)^{−1} ∫ (μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}((R^{n} \ B^{i}) × R^{n}) dℒ^{n}z

≥ εℒ^{n}(D_{g,r}^{i})

for all i, which gives by (3.4)

(3.6)  lim_{i→∞} ℒ^{n}(D_{g,r}^{i}) = 0.

Let (g, r) ∈ 𝒪_{n} × (0, ∞) such that (3.5) and (3.6) hold. We will prove that for ℒ^{n} almost all z ∈ E_{g,r} we have z ∈ D_{g,r}^{i} for all i. Then the claim follows by (3.6). Let z ∈ E_{g,r} such that (3.5) holds. Then there is a Borel set A ⊂ R^{n} such that dim_{H} A < u+t−n and μ∩(τ_{z}∘g∘δ_{r})_{♯}ν(A) > ε. Consider a positive integer i. Now μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A \ B^{i}) > ε. In fact, if μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A \ B^{i}) < ε, then μ∩(τ_{z}∘g∘δ_{r})_{♯}ν(A∩B^{i}) > 0, and by (3.5) and Lemma 2.5, dim_{H}(A∩B^{i}) ≥ u+t−n, which is a contradiction. Hence μ∩(τ_{z} ∘g ∘δ_{r})_{♯}ν(R^{n} \B^{i}) > ε, and so z ∈ D_{g,r}^{i}.

Lemmas 3.1 and 3.3 together imply the following theorem. Note that as shown above under the assumptions of Theorem 3.7 we have S_{g,r_{♯}}(μ × ν) ≪ ℒ^{n} for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞).

Theorem 3.7. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that dim_{H}(μ × ν) = dim_{H} μ + dim_{H} ν > n and I_{t}(ν) < ∞ for all 0 < t < dim_{H} ν < n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

ess inf{ dim_{H} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0}
= dim_{H} μ + dim_{H} ν − n.

This result is analogous to the packing dimension case (see Theorem 5.9) since

dim_{H} μ + dim_{H} ν − n = dim_{H}(μ × ν) − n
= sup{u ≥ 0 : lim sup_{h→0} h^{−u} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0
for μ × ν almost all (x, y) ∈ R^{n} × R^{n}}.

This can be verified in the same way as [6, Remark 3.9].

4. Upper Hausdorff dimension and intersection measures

The following lemma is an analogue of Lemma 3.1 for the upper Hausdorff dimension.

Lemma 4.1. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that dim_{H}(μ×ν) > n. If (g, r) ∈ 𝒪_{n} ×(0, ∞) is such that S_{g,r_{♯}}(μ×ν) ≪ ℒ^{n}, then

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{H}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0} ≤ dim_{H}^{∗}(μ × ν) − n.

Proof. Since dim_{H}(μ × (g ∘ δ_{r})_{♯}ν) = dim_{H}(μ × ν) > n and dim_{H}^{∗}(μ × (g ∘ δ_{r})_{♯}ν) = dim_{H}^{∗}(μ × ν), we have by [6, Lemma 4.2]

sup{u ≥ 0 : ℋ ^{n}({a ∈ W^{⊥} : dim_{H}^{∗}(μ×(g∘δ_{r})_{♯}ν)_{W,a} ≥ u}) > 0} ≤ dim_{H}^{∗}(μ×ν)−n.

Note that [6, Lemma 4.2] holds for W since P_{W^{⊥}}_{♯}(μ × (g ∘ δ_{r})_{♯}ν) ≪ ℋ ^{n} | W^{⊥}. Using the fact that dim_{H}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≤ dim_{H}^{∗}(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2} , this gives the claim.

Lemma 4.2. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that I_{t}(ν) < ∞ for all 0 < t < dim_{H}^{∗} ν < n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{H}^{∗} μ∩(τ_{z}∘g∘δ_{r})_{♯}ν ≥ u}) > 0} ≥ dim_{H}^{∗} μ+dim_{H}^{∗} ν−n.

Proof. Let 0 < t < dim_{H}^{∗} ν < n. It is sufficient to prove that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{H}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0} ≥ dim_{H}^{∗} μ + t − n.

The claim follows by taking a sequence (t_{i}) tending to dim_{H}^{∗} ν from below.

We may assume that dim_{H}^{∗} μ + t > n. We will prove that for any u < dim_{H}^{∗} μ − t − n we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞)

ℒ^{n}(E_{g,r}) > 0

for

E_{g,r} = {z ∈ R^{n} : dim_{H}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}.

The desired result follows then by taking a sequence (u_{i}) tending to dim_{H}^{∗} μ+t−n.

Let u < v < dim_{H}^{∗} μ + t − n. Then for some positive integer i we have μ(C_{i}) > 0, where

C_{i} = \{x ∈ R^{n} : μ(B(x, h)) ≤ h^{v−t+n} for all 0 < h ≤ \frac{1}{i}\}

is a Borel set. Further, μ × (g ∘ δ_{r})_{♯}ν(C_{i} × R^{n}) > 0 for all (g, r) ∈ 𝒪_{n} × (0, ∞). Now I_{u−t+n}(μ | C_{i}) < ∞, whence [9, Theorem 6.6] implies that S_{g,r_{♯}}((μ | C_{i}) × ν) ≪ ℒ^{n} for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞). This together with the disintegration formula (2.4) gives for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) that

(4.3)  ℒ^{n}(D_{g,r}^{i}) > 0,

where

D_{g,r}^{i} = {z ∈ R^{n} : ((μ | C_{i}) × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(C_{i} × R^{n}) > 0}.

By [9, Theorem 6.7], we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞)

(4.4)  I_{u}((μ | C_{i}) ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν) < ∞

for ℒ^{n} almost all z ∈ R^{n}. It is enough to show that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have z ∈ Ẽ_{g,r} for ℒ^{n} almost all z ∈ D_{g,r}^{i}, where

Ẽ_{g,r} = {z ∈ R^{n} : dim_{H}^{∗}(μ | C_{i}) ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}.

Then (4.3) implies the claim. Consider (g, r) ∈ 𝒪_{n} × (0, ∞) such that (4.3) and (4.4) hold. Let z ∈ D_{g,r}^{i} such that (4.4) holds. If z ∉ Ẽ_{g,r}, then there is a Borel set A ⊂ R^{n} such that dim_{H} A < u and (μ | C_{i}) ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n} \ A) = 0. Now A ∩ C_{i} is a Borel set and μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(A ∩ C_{i}) > 0. This gives by Lemma 2.5 and (4.4) that dim_{H}(A ∩ C_{i}) ≥ u, which is a contradiction. Thus z ∈ Ẽ_{g,r}.

Now we obtain:

Theorem 4.5. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that dim_{H} μ + dim_{H} ν > n, dim_{H}^{∗}(μ × ν) = dim_{H}^{∗} μ + dim_{H}^{∗} ν, and I_{t}(ν) < ∞ for all 0 < t < dim_{H}^{∗} ν < n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{H}^{∗} μ∩(τ_{z}∘g∘δ_{r})_{♯}ν ≥ u}) > 0} = dim_{H}^{∗} μ+dim_{H}^{∗} ν−n.

Proof. The claim follows from Lemmas 4.1 and 4.2. Lemma 4.1 is applicable since dim_{H}(μ × ν) ≥ dim_{H} μ + dim_{H} ν > n and since, using the methods of the proof of Lemma 3.3, we see that S_{g,r_{♯}}(μ × ν) ≪ ℒ^{n} for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞).

This result is analogous to the upper packing dimension case (see Theorem 6.3) since we have here

dim_{H}^{∗} μ + dim_{H}^{∗} ν − n = dim_{H}^{∗}(μ × ν) − n
= sup{u ≥ 0 : μ × ν({(x, y) ∈ R^{n} × R^{n} :

lim sup_{h→0} h^{−u} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0}) > 0}.

5. Packing dimension and intersection measures

Let μ and ν be finite Radon measures on R^{n}. As in the projection (see [1]) and section (see [6]) cases, we relate intersection measures to certain integral kernels. We will show that it is the limiting behaviour of the product measure μ × ν against the kernel considered in [6] in connection with sections of measures that determines the packing dimensions of intersection measures almost everywhere.

The quantities

d_{μ,ν} = sup{u ≥ 0 : lim inf_{h→0} h^{−u} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0
for μ × ν almost all (x, y) ∈ R^{n} × R^{n}}

and the upper analogue

d_{μ,ν}^{∗} = sup{u ≥ 0 : μ × ν({(x, y) ∈ R^{n} × R^{n} :

lim inf_{h→0} h^{−u} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0}) > 0}

are what we need when considering intersection measures.

The following lemma is a modification of [4, Lemma 3.5].

Lemma 5.1. Let μ and ν be Radon measures on R^{n} with compact supports such that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t ≥ n. Let 0 < r_{1} < r_{2} < ∞. Assume that there exists u > 0 such that for μ × ν almost all (x, y) ∈ R^{n} × R^{n} there is 0 < c < ∞ such that

∫_{r_{1}}^{r_{2}}∫ μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r ≤ ch^{u}

for arbitrarily small h > 0. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] we have

dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u

for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0.

Proof. Let ε > 0. For h > 0 and (x, y) ∈ R^{n} × R^{n}, define
J_{h}(x, y) = ∫_{r_{1}}^{r_{2}}∫ μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r

provided that the right hand side is defined (see [4, Lemma 3.4]). If J_{h}(x, y) ≤ ch^{u}, then

\begin{matrix}θ_{n} × ℒ^{1}(\{(g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] : μ∩ & (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) ≥ h^{u−ε}\}) \\ ≤ h^{ε−u}J_{h}(x, y) ≤ ch^{ε}. \end{matrix}

Hence, for μ × ν almost all (x, y) ∈ R^{n} × R^{n} we have

θ_{n}×ℒ^{1}({(g, r) ∈ 𝒪_{n}×[r_{1}, r_{2}] : lim inf_{h→0} h^{ε−u}μ∩(τ_{x}∘g∘δ_{r}∘τ_{−y})_{♯}ν(B(x, h)) > 1}) = 0,

which gives, by Fubini’s theorem, that for θ_{n} ×ℒ^{1} almost all (g, r) ∈ 𝒪_{n} ×[r_{1}, r_{2}]

(5.2)  μ × ν(A_{g,r}) = 0,

where

A_{g,r} = {(x, y) ∈ R^{n} × R^{n} : lim inf_{h→0} h^{ε−u}μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) > 1}

is a Borel set. For all g ∈ 𝒪_{n} and r_{1} ≤ r ≤ r_{2} , define a Borel set

B_{g,r} = {(x, y) ∈ R^{n} × R^{n} : lim inf_{h→0} h^{ε−u}μ ∩ (τ_{x−y} ∘ g ∘ δ_{r})_{♯}ν(B(x, h)) > 1}.

Then (5.2) implies that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}]

μ × (g ∘ δ_{r})_{♯}ν(B_{g,r}) = μ × ν(A_{g,r}) = 0,

and so, using [9, Theorem 6.6] and (2.4), we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}]

(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(B_{g,r}) = 0,

for ℒ^{n} almost all z ∈ R^{n}, that is,

(5.3)  lim inf_{h→0} h^{ε−u}μ ∩ (τ_{x−y} ∘ g ∘ δ_{r})_{♯}ν(B(x, h)) ≤ 1

for (μ×(g ∘δ_{r})_{♯}ν)_{W,(z,−z)/2} almost all (x, y) ∈ R^{n} ×R^{n}. Since (x, y) ∈ W_{(z,−z)/2} if and only if x − y = z, inequality (5.3) gives for ℒ^{n} almost all z ∈ R^{n}

lim inf_{h→0} h^{ε−u}μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(B(x, h)) ≤ 1

for μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν almost all x ∈ R^{n}. Thus, for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] we have

dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u − 2ε

for ℒ^{n} almost all z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0, and the claim follows by taking a sequence (ε_{i}) tending to zero.

In the proof of the following lemma we need the estimates given in [4, Lemma 3.8].

Lemma 5.4. Let μ and ν be Radon measures on R^{n} with compact supports such that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t ≥ n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

ess inf{dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} ≥ d_{μ,ν}.

Proof. Consider (x, y) ∈ R^{n} × R^{n} such that ∫ |a − x|^{t−n} dμa < ∞ and ∫ |b − y|^{−t} dνb < ∞. Let 0 < r_{1} < r_{2} < ∞ and h > 0. By (2.3), Fatou’s lemma and Fubini’s theorem we have

\begin{aligned}∫_{r_{1}}^{r_{2}}∫ μ & ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r \\ & ≤ lim inf_{δ→0} α(n)^{−1}δ^{−n} ∫_{B(x,2h)×R^{n}} I_{δ}(a, b) d(μ × ν)(a, b), \end{aligned}

where

I_{δ}(a, b) = ∫_{{r∈[r_{1},r_{2}]:| |a−x|−r|b−y| |≤δ}} θ_{n}({g ∈ 𝒪_{n} : |a − x − rg(b − y)| ≤ δ}) dℒ^{1}r.

Hausdorff and packing dimensions, intersection measures 179

Define

A_{δ} = {(a, b) ∈ R^{n} × R^{n} : r_{1}|b − y| − δ ≤ |a − x| ≤ r_{2}|b − y| + δ}.

If (a, b) ∉ A_{δ}, then {r ∈ [r_{1}, r_{2}] : ||a − x| − r|b − y|| ≤ δ} = ∅. Hence,

\begin{aligned}∫_{r_{1}}^{r_{2}} ∫ μ & ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r \\ & ≤ lim inf_{δ→0} α(n)^{−1}δ^{−n} ∫_{(B(x,2h)×R^{n})∩A_{δ}} I_{δ}(a, b) d(μ × ν)(a, b). \end{aligned}

Let

A_{δ}^{1} = {(a, b) ∈ A_{δ} : |a − x| ≤ 2δ},
A_{δ}^{2} = {(a, b) ∈ A_{δ} : r_{1}|b − y| ≤ 2δ},

and

A_{δ}^{3} = {(a, b) ∈ A_{δ} : |a − x| > 2δ and r_{1}|b − y| > 2δ}.

Then A_{δ} = A_{δ}^{1} ∪ A_{δ}^{2} ∪ A_{δ}^{3} . Further, if (a, b) ∈ A_{δ}^{1} , then

δ^{−n} ≤ 2^{n−t}3^{t}r_{1}^{−t}|a − x|^{t−n}|b − y|^{−t},

and so,

lim sup_{δ→0} α(n)^{−1}δ^{−n} ∫_{A_{δ}^{1}} I_{δ}(a, b) d(μ × ν)(a, b) = 0.

Similarly,

lim sup_{δ→0} α(n)^{−1}δ^{−n} ∫_{A_{δ}^{2}} I_{δ}(a, b) d(μ × ν)(a, b) = 0.

Thus,

\begin{aligned}∫_{r_{1}}^{r_{2}} ∫ μ & ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r \\ & ≤ lim sup_{δ→0} α(n)^{−1}δ^{−n} ∫_{(B(x,2h)×R^{n})∩A_{δ}^{3}} I_{δ}(a, b) d(μ × ν)(a, b). \end{aligned}

If (a, b) ∈ A_{δ}^{3} , then \frac{1}{2}r_{1}|b − y| ≤ |a − x| ≤ 2r_{2}|b − y|, and so by [10, Lemma 3.8] there is a constant c_{1} depending only on n such that for any a ≠ x and b ≠ y

I_{δ}(a, b) ≤ c_{1}δ^{n−1}|a − x|^{1−n}ℒ^{1}({r ∈ [r_{1}, r_{2}] : ||a − x| − r|b − y|| ≤ δ})
≤ c_{1}δ^{n−1}|a − x|^{1−n}2δ|b − y|^{−1}
≤ 4c_{1}r_{2}δ^{n}|a − x|^{−n}.

Thus, for μ × ν almost all (x, y) ∈ R^{n} × R^{n} we have

(5.5)

∫_{r_{1}}^{r_{2}} ∫ μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r

≤ lim sup_{δ→0} 4c_{1}α(n)^{−1}r_{2} ∫_{(B(x,2h)×R^{n})∩A_{δ}^{3}} |a − x|^{−n} d(μ × ν)(a, b)

≤ lim sup_{δ→0} c_{2} ∫_{(B(x,2h)×R^{n})∩A_{δ}^{3}} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b)

≤ c_{2} ∫_{B(x,c_{3}h)×B(y,c_{3}h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b),

where c_{2} is a constant depending on n, r_{1} , and r_{2} and c_{3} = max{2, 4/r_{1}}. The last inequality follows from the fact that (B(x, 2h) × R^{n}) ∩ A_{δ}^{3} ⊂ B(x, 2h) × B(y, 4h/r_{1}).

Consider u ≥ 0 such that

lim inf_{h→0} h^{−u} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0

for μ × ν almost all (x, y) ∈ R^{n} × R^{n}. Then, by (5.5) and Lemma 5.1 we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}]

ess inf{dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} ≥ u.

The claim follows by taking a sequence (u_{i}) tending to d_{μ,ν}.

In order to prove that the opposite inequality holds in Lemma 5.4 we need the following version of [6, Lemma 2.4] concerning sections of the product measures μ × (g ∘ δ_{r})_{♯}ν.

Lemma 5.6. Let μ and ν be Radon measures on R^{n} with compact supports. If I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n such that s + t > n, then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) there exists for any ε > 0 a compact set C_{ε} ⊂ R^{n} × R^{n} with μ × (g ∘ δ_{r})_{♯}ν((R^{n} × R^{n}) \ C_{ε}) < ε and H_{ε} > 0 such that for ℋ ^{n} almost all (z, −z)/2 ∈ W^{⊥} we have

((μ × (g ∘ δ_{r})_{♯}ν) | C_{ε})_{W,(z,−z)/2}(B(x, h) × B(y, h)) ≤ ch^{(s+t−n)/2}

for all (x, y) ∈ W_{(z,−z)/2} and 0 < h ≤ H_{ε}. Here c is a constant depending only on s, t, and n.

Proof. Let q = s + t − n. By [9, Theorem 6.7] we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞)

(5.7)  I_{q}(μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν) < ∞

for ℒ^{n} almost all z ∈ R^{n}. Consider (g, r) ∈ 𝒪_{n} × (0, ∞) such that (5.7) holds and S_{g,r_{♯}}(μ×ν) ≪ ℒ^{n} (see [9, Theorem 6.6]). We will prove that for μ×(g∘δ_{r})_{♯}ν almost all (x, y) ∈ R^{n} × R^{n} we have

∫ |(x, y) − (a, b)|^{−q} d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(x−y,y−x)/2}(a, b) < ∞.

To show that this holds, define

E = {(x, y) ∈ R^{n}×R^{n} :∫ |(x, y)−(a, b)|^{−q}d(μ×(g∘δ_{r})_{♯}ν)_{W,(x−y,y−x)/2}(a, b) = ∞}.

If μ × (g ∘ δ_{r})_{♯}ν(E) > 0, then by (2.4) we have ℒ^{n}(F) > 0, where

F = {z ∈ R^{n} : (μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(E) > 0}.

For all z ∈ F we have

I_{q}(μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν)
= 2^{n}α(n)^{−2} ∫ ∫ |x − a|^{−q} d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(a, b)
× d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(x, y)
≥ 2^{n}α(n)^{−2} ∫_{E} ∫ |(x, y) − (a, b)|^{−q} d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(a, b)
× d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(x, y) = ∞,

which is a contradiction by (5.7).

Let ε > 0. For all positive integers i, define a Borel set

B_{i} = {(x, y) ∈ R^{n}×R^{n} :∫ |(x, y)−(a, b)|^{−q} d(μ×(g∘δ_{r})_{♯}ν)_{W,(x−y,y−x)/2}(a, b) ≤ i}.

Since

lim_{i→∞} μ × (g ∘ δ_{r})_{♯}ν((R^{n} × R^{n}) \ B_{i}) = 0,

we find a compact set C_{ε} ⊂ R^{n} ×R^{n} such that μ×(g ∘δ_{r})_{♯}ν((R^{n} ×R^{n})\C_{ε}) < ε and C_{ε} ⊂ B_{i_{ε}} for some positive integer i_{ε} (see [10, Theorem 1.10]).

Let H_{ε} = i_{ε}^{−2/q}/3. Consider (z, −z)/2 ∈ W^{⊥} such that (μ×(g∘δ_{r})_{♯}ν)_{W,(z,−z)/2} and ((μ × (g ∘ δ_{r})_{♯}ν) | C_{ε})_{W,(z,−z)/2} are defined. If (z, −z)/2 ∉ P_{W^{⊥}}(C_{ε}), then

((μ × (g ∘ δ_{r})_{♯}ν) | C_{ε})_{W,(z,−z)/2}(B(x, h) × B(y, h)) = 0

for all (x, y) ∈ W_{(z,−z)/2} and h > 0. This follows from (2.1) and from the fact that W_{(z,−z)/2}(δ) ∩ C_{ε} = ∅ for all small δ > 0, since C_{ε} is compact. If (z, −z)/2 ∈ P_{W^{⊥}}(C_{ε}) and (x, y) ∈ W_{(z,−z)/2} ∩ C_{ε}, then for any 0 < h ≤ 3H_{ε}, we have (5.8)

((μ × (g ∘ δ_{r})_{♯}ν) | C_{ε})_{W,(z,−z)/2}(B(x, h) × B(y, h))
≤ (μ × (g ∘ δ_{r})_{♯}ν)_{W,(x−y,y−x)/2}(B(x, h) × B(y, h))

≤ 2^{q/2}h^{q} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−q} d(μ × (g ∘ δ_{r})_{♯}ν)_{W,(x−y,y−x)/2}(a, b)

≤ 2^{q/2}i_{ε}h^{q} ≤ 2^{q/2}h^{q/2}.

Finally, let (z, −z)/2 ∈ P_{W^{⊥}}(C_{ε}) and (x, y) ∈ W_{(z,−z)/2} \ C_{ε}. Then there exists h_{x,y} > 0 such that (B(x, 2h) × B(y, 2h)) ∩ C_{ε} ∩ W_{(z,−z)/2} = ∅ for all 0 < h < h_{x,y} and (B(x, 2h) × B(y, 2h)) ∩ C_{ε} ∩ W_{(z,−z)/2} ≠ ∅ for all h ≥ h_{x,y}. If 0 < h < h_{x,y}, then by (2.1)

((μ × (g ∘ δ_{r})_{♯}ν) | C_{ε})_{W,(z,−z)/2}(B(x, h) × B(y, h)) = 0.

If h_{x,y} ≤ h ≤ H_{ε}, then B(x, h) × B(y, h) ⊂ B(a, 3h) × B(b, 3h) for some (a, b) ∈ C_{ε} ∩ W_{(z,−z)/2} , and (5.8) gives the claim.

Theorem 5.9. Let μ and ν be Radon measures on R^{n} with compact supports such that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t > n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

ess inf{dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν : z ∈ R^{n} with μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(R^{n}) > 0} = d_{μ,ν}.

Proof. Consider (g, r) ∈ 𝒪_{n}×(0, ∞) such that Lemma 5.6 holds and S_{g,r_{♯}}(μ× ν) ≪ ℒ^{n}. Let u ≥ 0 such that dim_{p} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u for ℒ^{n} almost all z ∈ R^{n} with μ∩(τ_{z} ∘g ∘δ_{r})_{♯}ν(R^{n}) > 0. Then for ℋ ^{n} almost all (z, −z)/2 ∈ W^{⊥} with (μ × (g ∘ δ_{r})_{♯})ν)_{W,(z,−z)/2} (R^{n} × R^{n}) > 0 we have

u ≤ (dim_{p}(μ × (g ∘ δ_{r})_{♯})ν)_{W,(z,−z)/2}.

Now we use a result given in [6, Theorem 5.16]. What is actually proved there is the following: if k and p are integers with 0 < p < k and if m is a Radon measure on R^{k} with compact support and with I_{p}(m) < ∞, then

ess inf{dim_{p} m_{V,a} : a ∈ V^{⊥} with m_{V,a}(R^{k}) > 0}

≤ sup{v ≥ 0 : lim inf_{h→0} h^{−v} ∫_{B(x,h)} |x − y|^{−p} dm(y) = 0 for m almost all x ∈ R^{k}}

provided that V ∈ G_{k,k−p} such that [6, Lemma 2.4] holds and P_{V^{⊥}}_{♯}m ≪ ℋ ^{p} | V^{⊥}. Note that the assumption that I_{p+d}(m) < ∞ for some d > 0 in [6, Theorem 5.16] is needed only to make sure that [6, Lemma 2.4] holds for almost all (k − p)-planes. If we know that this is the case for V ∈ G_{k,k−p}, then this assumption is unnecessary, since in the proof of [6, Theorem 5.16] we need only the fact that I_{p}(m) < ∞ when applying [6, Lemma 5.9]. Here we choose m = μ×(g∘δ_{r})_{♯}ν. Then I_{n}(m) < ∞, since I_{n}(μ × ν) < ∞. Further, P_{W^{⊥}}m ≪ ℋ ^{n} | W^{⊥} and [6, Lemma 2.4] holds for W. So, by the above result

u ≤ c_{μ×(g∘δ_{r})_{♯}ν},

where

c_{μ×(g∘δ_{r})_{♯}ν} = sup{v ≥ 0 : lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n}
× d(μ × (g ∘ δ_{r})_{♯}ν)(a, b) = 0
for μ × (g ∘ δ_{r})_{♯}ν almost all (x, y) ∈ R^{n} × R^{n}}.

We will show that c_{μ×(g∘δ_{r})_{♯}ν} ≤ d_{μ,ν}, which gives the claim by Lemma 5.4. If v ≥ 0 is such that

lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × (g ∘ δ_{r})_{♯}ν)(a, b) = 0

for μ × (g ∘ δ_{r})_{♯}ν almost all (x, y) ∈ R^{n} × R^{n}, then

(5.10)  lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h/r)} |(x, ry) − (a, rb)|^{−n} d(μ × ν)(a, b) = 0

for μ × ν almost all (x, y) ∈ R^{n} × R^{n}. Consider (x, y) ∈ R^{n} × R^{n} such that (5.10) holds. If r ≤ 1, then |(x, y) − (a, b)|^{−n} ≤ |(x, ry) − (a, rb)|^{−n}, and so,

lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b) = 0.

If r ≥ 1, then |(x, y) − (a, b)|^{−n} ≤ r^{n}|(x, ry) − (a, rb)|^{−n}, which gives

lim inf_{h→0}(h/r)^{−v} ∫_{B(x,h/r)×B(y,h/r)} |(x, y) − (a, b)|^{−n} d(μ × ν)(a, b)

≤ lim inf_{h→0} r^{n−v}h^{−v} ∫_{B(x,h)×B(y,h/r)} |(x, ry) − (a, rb)|^{−n} d(μ × ν)(a, b) = 0.

Hence c_{μ×(g∘δ_{r})_{♯}ν} ≤ d_{μ,ν}.

6. Upper packing dimension and intersection measures

In this chapter we prove analogues of the results of the previous chapter for the upper packing dimension. For this purpose we need the following lemma.

Lemma 6.1. Let μ and ν be Radon measures on R^{n} with compact supports such that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t ≥ n. Let 0 < r_{1} < r_{2} < ∞. Assume that there is u > 0 such that

μ × ν({(x, y) ∈ R^{n} × R^{n} : there is 0 < c < ∞ such that for arbitrarily

small h > 0 ∫_{r_{1}}^{r_{2}} ∫ μ ∩ (τ_{z} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r ≤ ch^{u}}) > 0.

Then for any ε > 0 we have for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}]

ℒ^{n}({z ∈ R^{n} : dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u − ε}) > 0.

Proof. Let ε > 0. For h > 0 and (x, y) ∈ R^{n} × R^{n}, define
J_{h}(x, y) = ∫_{r_{1}}^{r_{2}}∫ μ ∩ (τ_{z} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r

provided that the right hand side is defined (see [4, Lemma 3.4]). Define

A_{u} = {(x, y) ∈ R^{n} × R^{n} : there is 0 < c < ∞ such that J_{h}(x, y) ≤ ch^{u}

for arbitrarily small h > 0}.

Then μ × ν(A_{u}) > 0 and for all (x, y) ∈ A_{u} we have

lim inf_{h→0} h^{ε−u}μ ∩ (τ_{z} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) ≤ 1

for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}]. This together with Fubini’s theorem gives that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] we have μ × ν(B_{g,r}) > 0, where

B_{g,r} = {(x, y) ∈ R^{n} × R^{n} : lim inf_{h→0} h^{ε−u}μ ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) ≤ 1}.

Since μ × ν(B_{g,r}) = μ × (g ∘ δ_{r})_{♯}ν(C_{g,r}) for

C_{g,r} = {(x, y) ∈ R^{n} × R^{n} : lim inf_{h→0} h^{ε−u}μ ∩ (τ_{x−y} ∘ g ∘ δ_{r})_{♯}ν(B(x, h)) ≤ 1},

[9, Theorem 6.6] and the disintegration formula (2.4) imply that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] we have ℒ^{n}(D_{g,r}) > 0, where

D_{g,r} = {z ∈ R^{n} : (μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2}(C_{g,r}) > 0}.

For any z ∈ D_{g,r} we have

μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν({x ∈ R^{n} : lim inf_{h→0} h^{2ε−u}μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν(B(x, h)) = 0}) > 0,

that is,

dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u − 2ε,

and the claim follows.

Lemma 6.2. Let μ and ν be Radon measures on R^{n} with compact supports such that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t ≥ n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0} ≥ d_{μ,ν}^{∗}.

Proof. Let 0 < r_{1} < r_{2} < ∞. Consider v ≥ 0 such that μ × ν(A_{v}) > 0, where

A_{v} = {(x, y) ∈ R^{n} ×R^{n} : lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h)}|(x, y)−(a, b)|^{−n} dμ×ν(a, b) = 0}.

As in the proof of Lemma 5.4, we see that for μ × ν almost all (x, y) ∈ A_{v} we have for some constants c_{1} and c_{2}

\begin{aligned}∫_{r_{1}}^{r_{2}} ∫ μ & ∩ (τ_{x} ∘ g ∘ δ_{r} ∘ τ_{−y})_{♯}ν(B(x, h)) dθ_{n}g dℒ^{1}r \\ & ≤ c_{1} ∫_{B(x,c_{2}h)×B(y,c_{2}h)} |(x, y) − (a, b)|^{−n} dμ × ν(a, b) ≤ c_{1}(c_{2}h)^{v} \end{aligned}

for arbitrarily small h > 0. For any ε > 0 Lemma 6.1 gives that for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × [r_{1}, r_{2}] we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0} ≥ v − ε.

The claim follows by taking sequences (ε_{i}) tending to zero and (v_{i}) tending to d_{μ,ν}^{∗}

Theorem 6.3. Let μ and ν be Radon measures on R^{n} with compact supports. Assume that I_{s}(μ) < ∞ and I_{t}(ν) < ∞ for some 0 < s < n and 0 < t < n with s + t > n. Then for θ_{n} × ℒ^{1} almost all (g, r) ∈ 𝒪_{n} × (0, ∞) we have

sup{u ≥ 0 : ℒ^{n}({z ∈ R^{n} : dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0} = d_{μ,ν}^{∗}.

Proof. Consider (g, r) ∈ 𝒪_{n}×(0, ∞) such that Lemma 5.6 holds and S_{g,r_{♯}}(μ× ν) ≪ ℒ^{n}. Let u ≥ 0 such that ℒ^{n}({z ∈ R^{n} : dim_{p}^{∗} μ ∩ (τ_{z} ∘ g ∘ δ_{r})_{♯}ν ≥ u}) > 0. Then

ℋ ^{n}({a ∈ W^{⊥} : dim_{p}^{∗}(μ × (g ∘ δ_{r})_{♯}ν)_{W,(z,−z)/2} ≥ u}) > 0.

As in the proof of Theorem 5.9 we use a result from [6] to prove the claim. What we need here and what is actually proved in [6, Theorem 6.4] is as follows: if k and 186 Maarit Järvenpää

p are integers with 0 < p < k and m is a Radon measure on R^{k} with compact support and with I_{p}(m) < ∞, then

sup{v ≥ 0 : ℋ ^{p}({a ∈ V^{⊥} : dim_{p}^{∗} m_{V,a} ≥ v}) > 0}

≤ sup{v ≥ 0 : m({x ∈ R^{k} : lim inf_{h→0} h^{−v} ∫_{B(x,h)} |x − y|^{−p} dmy = 0}) > 0}

provided that V ∈ G_{k,k−p} such that [6, Lemma 2.4] holds and P_{V_{♯}^{⊥}}m ≪ ℋ ^{p} | V^{⊥}. In our case this result implies that

u ≤ c_{μ×(g∘δ_{r})_{♯}ν}^{∗},

where

c_{μ×(g∘δ_{r})_{♯}ν}^{∗} = sup{v ≥ 0 : μ × (g ∘ δ_{r})_{♯}ν({(a, b) ∈ R^{n} × R^{n} :

lim inf_{h→0} h^{−v} ∫_{B(x,h)×B(y,h)} |(x, y) − (a, b)|^{−n} dμ × (g ∘ δ_{r})_{♯}ν(a, b) = 0}) > 0}.

Hence it suffices to show that c_{μ×(g∘δ_{r})_{♯}ν}^{∗} ≤ d_{μ,ν}^{∗}. This can be verified in the same way as the fact that c_{μ×(g∘δ_{r})_{♯}ν} ≤ d_{μ,ν} was shown in the proof of Theorem 5.9.

References

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[5] Järvenpää, M.: Packing dimension, intersection measures, and isometries. - Math. Proc. Cambridge Philos. Soc. 122, 1997, 483–490.

[6] Järvenpää, M., and P. Mattila: Hausdorff and packing dimensions and sections of measures. - Mathematika (to appear).

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Received 14 July 1997