Infinite on one side and nought on the other

The rung below defined dimension by counting boxes: cover a set with a grid of side $\varepsilon$, count the boxes that meet it, and read off how fast that count grows. The number that comes out is a growth rate, and it is the right number for every set on that page.

Hausdorff’s definition, which is older and is the one the subject is built on, returns something else. It returns a measure — a size — and the dimension appears as the exponent at which that size stops being infinite and starts being nought.

What the quantity is #

For a set $S$ and an exponent $s$, take a cover of $S$ by sets of small diameter and add up the $s$-th powers of those diameters. Take the smallest such total over all covers with pieces below a stated size, and then let that size shrink. What comes out is the $s$-dimensional Hausdorff measure of $S$.

At $s = 1$ that is a length: the pieces’ diameters add up, and the infimum over covers is what “total length” means for a set with no obvious length. At $s = 2$ it is an area, up to a constant. At $s = 0$ it counts points. The definition interpolates between the familiar measures and does not stop at whole numbers, which is the whole of what it buys.

For a set built from $m$ copies at ratio $r$, the natural cover at depth $k$ is the $m^k$ pieces of the construction, each of diameter $r^k$, and its total is

$$m^k \cdot (r^k)^s = (m\,r^s)^k .$$

The bracket is a number, and the whole behaviour follows from whether it is above or below one.

The pivot #

Set $d = \log m / \log(1/r)$, which is the dimension the rule forces. Then $m\,r^d = 1$ exactly, so the total is $1^k = 1$ at every depth.

Below $d$ the bracket exceeds one and the total grows like a geometric series with ratio above one, so refining the cover sends it to infinity. Above $d$ the bracket is below one and refining sends the total to nought.

So the measure is infinite for every $s < d$, nought for every $s > d$, and neither at $s = d$, and the dimension is defined to be that exponent. Nothing about the definition mentions a growth rate; it mentions a size that has exactly one exponent at which it is a real number.

That reading also explains a fact the rung below stated and did not explain: why the Cantor set has measure zero as a subset of the line and is still not a point. Its $1$-dimensional measure — its length — is nought, because $1 > 0.6309$, which is the fact that the discarded intervals account for the whole of $[0,1]$. Its $0$-dimensional measure — its count of points — is infinite, because $0 < 0.6309$. Both of the familiar measurements are degenerate and the one in between is finite, and that is what having a fractional dimension means.

What allowing any cover buys #

Box counting uses a grid of equal squares and Hausdorff’s definition allows a cover by sets of any sizes at all. That is the whole difference between the two, and it produces a difference in the answers.

The Hausdorff dimension is always at most the box dimension, because a grid is one available cover among many and taking an infimum over more covers cannot give a larger answer. Where they differ, they differ for a reason that is easy to state: a grid must resolve the set at one scale everywhere, and a general cover may use big pieces where the set is sparse and small ones where it is dense.

The standard example is the rationals in $[0,1]$. Their box dimension is $1$: any grid box containing a rational meets the set, and every box does. Their Hausdorff dimension is $0$: the set is countable, so it can be covered by a sequence of intervals whose lengths add to as little as wanted, and the $s$-dimensional total is then arbitrarily small for every $s > 0$.

That gap has a clean diagnosis. Box counting measures a set’s closure, since a box meeting a set meets its closure and conversely; the rationals and the whole interval have the same closure and no grid can tell them apart. Hausdorff’s definition is sensitive to countability and the grid is not.

The sensitivity has a name and a general form. Hausdorff measure is countably stable: the dimension of a countable union is the largest of the dimensions in it, so a countable set — a countable union of points, each of dimension nought — has dimension nought however densely it sits. Box dimension has no such property, and cannot: a countable union of sets each needing few boxes can need many, because the boxes are not shared between the pieces.

That single property is what makes Hausdorff dimension the one theorems are stated about. Almost every argument in the subject splits a set into countably many pieces and handles them separately, and the split is legitimate only for a notion that recombines. A quantity that is not countably stable cannot be reasoned about piecewise, and reasoning piecewise is nearly the whole of measure theory.

For every set the rung below drew the two agree, and that is not luck: a self-similar set satisfying a mild separation condition has equal box and Hausdorff dimensions, which is a theorem. Where they part company is the subject of the rung above.

The cost of the definition #

The definition is the right one and it is nearly impossible to use.

Computing it requires an infimum over all covers — the same shape of definition as the outer measure of a set, and inheriting the same difficulty. There are uncountably many, they are not parameterised by anything, and no algorithm enumerates them. For a self-similar set the natural cover happens to be optimal — that is a theorem, not an observation — and for a general set nobody knows what an optimal cover looks like.

So Hausdorff dimensions are proved rather than computed, and the proofs come in two halves that are completely different in difficulty. An upper bound needs one good cover, which is a construction and is usually available. A lower bound needs a statement about every cover, which is a universally quantified claim, and the standard tool for it is the mass distribution principle: put a measure on the set, show that no small ball carries too much of it, and conclude that no cover can be too economical.

That asymmetry is the reason the subject is hard. Upper bounds are exhibitions and lower bounds are arguments, which is the same asymmetry the piece counts of a dissection have one field away — and there the lower bounds are missing entirely, while here they exist and are difficult.

The mass distribution principle deserves its statement, since it is the only general tool for the hard half and it is short. Suppose a measure $\mu$ can be put on the set with $\mu(S) > 0$, such that every set $U$ of small diameter satisfies $\mu(U) \le C\,|U|^s$. Then for any cover, $\mu(S) \le \sum \mu(U_i) \le C \sum |U_i|^s$, so no cover’s total can fall below $\mu(S)/C$ — and the $s$-dimensional measure is bounded away from nought, giving a dimension of at least $s$.

The whole argument is a sum inequality applied twice, and its value is that it converts a claim about every cover into a claim about a single measure. Finding the measure is the work, and for a self-similar set it is the obvious one — give each of the $m$ pieces a $1/m$ share and repeat — which is why those sets’ dimensions are known exactly and a generic set’s is not.

The three properties that make it a measure #

Calling the quantity a measure is a claim with content, and it is worth checking, because the whole reason to prefer it to a growth rate is that it behaves like a size.

It is monotone. A subset’s total cannot exceed the whole set’s, since any cover of the whole covers the subset. So a smaller set has a smaller measure and therefore no larger a dimension, which is the statement that dimension does not increase on subsets — obvious for a size and not obvious for a growth rate.

It adds over separated pieces. Two sets at positive distance from each other have measures adding to the union’s, because a cover with small enough pieces cannot straddle the gap. That is the property that makes it a metric outer measure, and it is what Carathéodory’s construction was designed to guarantee.

And it scales the right way. Multiplying every distance by $\lambda$ multiplies every diameter by $\lambda$ and therefore the $s$-dimensional measure by $\lambda^s$. At $s = 1$ that is a length scaling like a length and at $s = 2$ an area scaling like an area, and at $s = 1.26$ it is the statement that the Koch curve’s measure scales by $\lambda^{1.26}$.

The third is the one that gives the whole subject its shape. A self-similar set is a union of $m$ copies scaled by $r$; the measure of each copy is $r^s$ times the whole; and if the copies are essentially disjoint the measures add, giving $M = m\,r^s M$. That equation has a positive finite solution $M$ only when $m\,r^s = 1$, which is the pivot again — derived this time from the properties of a measure rather than from a cover’s arithmetic.

The box count has none of these. It is not monotone in any useful way beyond the obvious, it does not add over separated pieces, and it does not scale like anything, because it is not a size at all. A growth rate is a number extracted from a limit and the machinery of measures is unavailable to it. That is the honest reason Hausdorff’s definition is the primary one and box counting is the one that can be computed.

Where the measure at the dimension itself matters #

The quantity at $s = d$ is a real number and it is not always finite, which is worth stating because the definition’s phrasing suggests otherwise.

For the sets on this page the pivot value is exactly $1$ at every depth, so the measure is finite and positive. That is what makes them $d$-sets — sets whose Hausdorff measure at their own dimension is neither nought nor infinite — and it is a strong property that the standard constructions happen to have.

A general set of Hausdorff dimension $d$ may have $d$-dimensional measure nought or infinite, and both happen. The dimension is defined as the exponent where the measure changes, and whether it is finite exactly there is a separate question with its own answer.

That last figure makes a point the rung below made in prose. The carpet’s area is nought and its dimension is $1.8928$; the two facts are the same curve read at two exponents, and the second is finer because the first is one value of it.

Where it needs care #

The pieces of a cover must be small. The definition takes an infimum over covers with pieces below a size $\delta$ and then lets $\delta$ shrink; without that, one huge piece covers everything and the total is bounded.

The natural cover need not be optimal. For the self-similar sets here it is, by a theorem; for a set whose pieces overlap substantially it is not, and the natural cover gives an upper bound only.

And the measure is not additive across dimensions. The $s$-dimensional measure of a union is the sum of the parts’ measures at the same $s$, and a set whose two halves have different dimensions has the larger of the two as its dimension and infinite measure at it. Dimension is a maximum over pieces, which is a fact worth having and is not obvious from the definition.

Where it came from #

Hausdorff introduced the measure in 1918, in a paper titled Dimension und äußeres Maß — dimension and outer measure — and the order of those two words is the argument. He was extending Carathéodory’s construction of an outer measure, which had been given for whole-number dimensions two years earlier, and the extension was to let the exponent be any positive number.

The paper is short and the idea is a single move: Carathéodory’s $s$-dimensional measure was defined for integer $s$ because that is where it had an interpretation, and Hausdorff noticed that the formula does not care.

Besicovitch spent the 1920s and 1930s turning that into a theory — the structure of sets of fractional dimension, the projections of such sets, the difference between sets that behave like curves and sets that do not — and the subject was a corner of measure theory for half a century before it had an audience. Mandelbrot’s contribution, fifty years later, was a claim about relevance rather than a theorem.

A definition arrived by noticing that a formula’s restriction was unnecessary, which is the same move promoting a coordinate expression to a definition makes in a different field, and it is one of the cheapest sources of new mathematics there is.

What the pictures cannot show #

The curves are the totals of the natural cover, computed in closed form and checked against the cover summed piece by piece. The measure is an infimum over all covers, and no picture contains an infimum over an uncountable family; what is drawn is one cover’s total, which is an upper bound for the measure at every exponent.

That the natural cover achieves the infimum is a theorem about self-similar sets, quoted here and not drawn. For any set the theorem does not cover, the curves would be too high and the crossing would be in the wrong place, and nothing in the drawing would say so.

And the pivot is a limit. Each curve is one depth; the measure is what the totals approach as the depth grows, and four curves are four terms of a sequence. The exactness at $s = d$ — where the total is $1$ at every depth rather than approaching something — is the one part of the picture that is not an approximation, and it is exact because the arithmetic is $1^k$.

The ladder from here #

Rungs above: the carpet whose two dimensions differ, where the grid and the general cover give different answers and both are correct. A dimension read off the stretching rates, which computes one without covering anything. The mass distribution principle, which is how a lower bound is actually proved. Packing dimension, defined by packing rather than covering, which is a third answer that agrees with the others on tame sets. And the dimension of a projection, where Marstrand’s theorem says that almost every direction preserves it — a statement about a whole family of shadows.

A restriction that was not doing anything #

The habit worth carrying is Hausdorff’s own, stated as a question to ask of any definition: which of its ingredients is the formula using, and which is only there because somebody had an interpretation in mind?

Carathéodory’s construction needed $s$ to be a whole number to be called a length, an area or a volume. The arithmetic needed nothing of the kind. Removing the restriction cost nothing, produced a quantity defined for every set and every exponent, and made “dimension” a number rather than a count of directions.

The test of whether such a removal is honest is whether the extended object still does what the original did. Here it does: at $s = 1$ the Hausdorff measure of a rectifiable curve is its length, at $s = 2$ it is area up to a constant, and the familiar cases are recovered exactly. An extension that changed the familiar answers would be a different definition wearing the same name, and checking that it does not is the first thing to do with any generalisation.