SOME NOTES ON MINKOWSKI AND HAUSDORFF DIMENSION

TERENCE TAO

Abstract. We describe the properties of Minkowski and Hausdorff dimension which seem to be most useful to harmonic analysts. Many of these results are exploited in [2].

1. Notation

We define a dyadic number δ to be any number of the form 2^{−k} for some positive integer k. We define a hyper-dyadic number of spacing ε to be any number of the form 2^{−(1+ε)^{k}} for some positive integer k. Note that if δ is a hyper-dyadic number of spacing ε, then the next smallest hyper-dyadic number is δ^{1+ε} (as opposed to δ/2 for dyadic numbers).

Traditionally the dimension of sets has been analyzed by a dyadic range of scales; however this can easily be refined to the much sparser hyper-dyadic regime of scales without any significant loss, and this is occasionally useful.

Throughout this note n ≥ 1 is an integer and 0 ≤ α ≤ n is a real number. All implicit constants are allowed to depend on the parameter 0 < ε ≪ 1.

Define a (δ, α) set to be any union of a finitely overlapping collection of δ^{−α} δ-balls.

2. Minkowski dimension

Let E be a compact subset of R^{n}. Let N_{δ}(E) denote the δ-neighbourhood of E. We define the upper Minkowski dimension \overline{dim}_{M}(E) by

\overline{dim}_{M}(E) = lim sup_{δ→0} log_{1/δ} \frac{N_{δ}(E)}{δ^{n}}

and the lower Minkowski dimension \underline{dim}_{M}(E) by

\underline{dim}_{M}(E) = lim inf_{δ→0} log_{1/δ} \frac{N_{δ}(E)}{δ^{n}}

Clearly \overline{dim}_{M}(E) ≥ \underline{dim}_{M}(E). In practice we seek upper and lower bounds on these quantities; by untangling the definitions we thus obtain the following characterizations.

The following are equivalent:

• \overline{dim}_{M}(E) ≥ α.

• For all ε > 0, N_{δ}(E) ≳ δ^{n−α+ε} for infinitely many dyadic δ.

• For all ε > 0, for infinitely many dyadic δ, E cannot be contained in a (δ, α−ε) set.

• For all ε > 0, N_{δ}(E) ≳ δ^{n−α+ε} for infinitely many hyper-dyadic δ of spacing ε.

• For all ε > 0, for infinitely many hyper-dyadic δ of spacing ε, E cannot be contained in a (δ, α − ε) set.

The following are equivalent:

• \overline{dim}_{M}(E) ≤ α.

• For all ε > 0, for all 0 < δ ≪ 1, N_{δ}(E) ≲ δ^{n−α−ε}.

• For all ε > 0, for all 0 < δ ≪ 1, E can be contained in a (δ, α + ε) set. The following are equivalent:

• \underline{dim}_{M}(E) ≥ α.

• For all ε > 0, for all 0 < δ ≪ 1, N_{δ}(E) ≳ δ^{n−α+ε}.

• For all ε > 0, for all 0 < δ ≪ 1, E cannot be contained in a (δ, α − ε) set. The following are equivalent:

• \underline{dim}_{M}(E) ≤ α.

• For all ε > 0, N_{δ}(E) ≲ δ^{n−α−ε} for infinitely many dyadic δ.

• For all ε > 0, for infinitely many dyadic δ, E can be contained in a (δ, α + ε) set.

• For all ε > 0, N_{δ}(E) ≲ δ^{n−α−ε} for infinitely many hyper-dyadic δ of spacing ε.

• For all ε > 0, for infinitely many hyper-dyadic δ of spacing ε, E can be contained in a (δ, α + ε) set.

Minkowski dimension is also known as box counting dimension or packing dimension.

3. Hausdorff dimension

If {B} are a finite collection of balls, we define the α-cost c_{α}({B}) of the collection by

c_{α}({B}) = ∑_{B} r(B)^{α},

where r(B) is the radius of B. We define the Hausdorff content h_{α}(E; δ) of a compact set E by

h_{α}(E; δ) = inf_{{B}} c_{α}({B})

where {B} ranges over all finite collections of balls of radius at most δ which cover E. We define the Hausdorff dimension dim(E) to be the unique number α_{0} such that

lim_{δ→0} h_{α}(E; δ) = 0 for α > α_{0}

and

lim_{δ→0} h_{α}(E; δ) = ∞ for α < α_{0}

Proposition 3.1. The following are equivalent:

• dim(E) ≤ α.

• For every ε and every dyadic δ we can find a (δ, α + ε) set X_{δ} such that every point in E is contained in infinitely many X_{δ}.

• For every ε and every hyper-dyadic δ of spacing ε we can find a (δ, α + ε) set X_{δ} such that every point in E is contained in infinitely many X_{δ}.

Proof (iii) ⟹ (ii) is trivial. Now suppose that (ii) holds. Then for any δ_{0} and any ε > 0, E is contained in ⋃_{δ<δ_{0}} X_{δ}, which implies that h_{α+2ε}(E; δ_{0}) ≲ δ_{0}^{ε}, which implies (i).

Now suppose that (i) holds. Fix ε. For each hyper-dyadic δ_{0} ≪ 1 of spacing ε, we can find a covering {B} of E by balls of radius at most δ_{0} such that c_{α+ε}({B}) ≤ 1. By enlarging the balls if necessary we can assume that each ball has a radius which is a hyper-dyadic number of spacing ε, although the price we pay for this is that we can now only assume that c_{α+Cε}({B}) ≲ 1. If we define X_{δ,δ_{0}} to be the union of all the balls of radius δ, we see that each X_{δ,δ_{0}} is an (δ, α + Cε) set and that each point in E is contained in at least one X_{δ,δ_{0}}. The claim then follows by setting X_{δ} = ⋃_{δ≤δ_{0}} X_{δ,δ_{0}} and redefining ε.

We remark that we can improve the above proposition by making the (δ, α) sets X_{δ} satisfy an additional dimensional property, namely that

|X_{δ} ∩ B(x, r)| ≲ δ^{n}(r/δ)^{α}

for all balls B(x, r) and δ ≤ r ≤ 1. The point is that if the X_{δ} do not satisfy this property, then they cluster in balls of a larger scale and part of X_{δ} can be redistributed more efficiently to the higher scale. We leave as an exercise to the reader the job of making this rigorous and avoiding an infinite cascade of redistribution that may endanger the property of being contained in infinitely many X_{δ}. (For a solution, see [2].

As a corollary we see that dim(E) ≤ \underline{dim}_{M}(E) ≤ \overline{dim}_{M}(E).

Another consequence is

Proposition 3.2. Suppose μ is a compactly supported probability measure such that for any ε > 0, we have μ(X_{δ}) ≲ log log(1/δ)^{−1−ε} for all 0 < δ ≪ 1 and all (δ, α − ε) sets X_{δ}. Then the support of μ has Hausdorff dimension at least α.

Proof Suppose for contradiction that the support of μ had dimension less than α − ε for some ε > 0. Define X_{δ} as in Proposition 3.1(iii). The measures μ(X_{δ}) are summable as δ ranges over the hyper-dyadic numbers δ of spacing ε. From the Borel-Cantelli lemma we thus see that almost every (with respect to μ) point is contained in only finitely many X_{δ}, a contradiction.

Without the introduction of hyper-dyadic numbers, one would only be able to prove Proposition 3.2 with the double log replaced by a single log. This turns out to be important when exploiting the delicate combinatorial results involving the existence of arithmetic progressions in a set, see e.g. [1].

Define a Frostman measure μ of dimension at least α to be any compactly supported probability measure on R^{n} such that μ(B(x, r)) ≲ r^{α} for all balls B(x, r).

Proposition 3.3. • dim(E) ≥ α.

• For every ε, it is not possible to associate a (δ, α − ε) set X_{δ} for every dyadic δ such that every point in E is contained in infinitely many X_{δ}.

• For every ε, it is not possible to associate a (δ, α − ε) set X_{δ} for every hyperdyadic δ of spacing ε such that every point in E is contained in infinitely many X_{δ}.

• For every ε, there exists a Frostman measure of dimension at least α−ε which is supported on E.

Proof The equivalence of (i)-(iii) is essentially the contrapositive of Proposition 3.1. The implication (iv) ⟹ (i) is an easy consequence of Proposition 3.2, since if μ is a Frostman measure of dimension at least α − ε, then μ(X_{δ}) ≲ δ^{ε} for all (δ, α − 2ε) sets X_{δ}.

It remains to show (i) ⟹ (iv). Fix ε > 0. We may assume that E is contained in the unit cube Q_{0}. Define the dyadic Hausdorff content c(Q) by

c(Q) = inf_{{Q′}} ∑_{Q′} l(Q′)^{α−ε}

where {Q′} ranges over all collections of dyadic cubes in Q which cover E ∩ Q, and l(Q) is the side-length of Q. Since the α − ε Hausdorff measure of E is infinite, we see that c(Q_{0}) cannot vanish. Also, we have c(Q) ≲ l(Q)^{α−ε} for all Q. Finally, we see that if Q_{1}, …, Q_{2^{n}} are the immediate sub-cubes of Q, then

∑_{i=1}^{2^{n}} c(Q_{i}) ≥ c(Q).

Thus c is almost what we want; it is like a measure except that it is superadditive instead of additive, and it obeys the dimension bound, and it assigns a non-zero but finite mass c(Q_{0}) to the whole cube Q_{0}. To fix this up we just have to trim c slightly so that it becomes additive.

We shall define a non-negative quantity μ(Q) for each dyadic cube in Q_{0} such that μ(Q_{0}) = c(Q_{0}), μ(Q) ≤ c(Q) for all Q ⊆ Q_{0}, and such that μ(Q) = ∑_{i=1}^{2^{n}} μ(Q_{i}), where Q_{1}, …, Q_{2^{n}} are the immediate sub-cubes of Q. The fact that this determines a measure then follows from the Riesz representation theorem, since every continuous function is dyadically Riemann integrable. The fact that μ/c(Q_{0}) is then a Frostman measure of dimension at least α − ε is then easily verified, since

μ(Q)/c(Q_{0}) ≲ c(Q)/c(Q_{0}) ≲ l(Q)^{α−ε}/c(Q_{0}) ≲ l(Q)^{α−ε}.

We use downward induction on the size of the cubes. Suppose we have already assigned μ(Q) for all cubes Q of side-length 2^{−k}. For each such cube Q, we now wish to assign μ(Q_{i}) for all the immediate sub-cubes Q_{i} of Q. But this is easily achieved by defining

μ(Q_{i}) = \frac{c(Q_{i})}{∑_{i=1}^{2^{n}} c(Q_{i})}μ(Q);

the required estimates follow since ∑_{i=1}^{2^{n}} c(Q_{i}) ≥ c(Q).

References

[1] J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), no. 2, 256–282.

[2] N. Katz, T. Tao, Some connections between the conjectures of Falconer, Furstenburg, and Erdös, in preparation.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu