The exponent a staircase shares with its set

A staircase with no steps built the Cantor function — continuous, rising from nought to one, and flat on every interval removed in making the middle-thirds set — and ended with a remark it could not explain. The function is Hölder continuous with exponent $\log 2/\log 3 \approx 0.6309$: there is a constant $C$ with

$$|F(x) - F(y)| \le C\,|x - y|^{\log 2/\log 3}$$

for all $x$ and $y$, and no larger exponent works. And $\log 2/\log 3$ is exactly the Hausdorff dimension of the middle-thirds set, which a dimension that is not a whole number computed by counting boxes. The essay called this “a coincidence that is not one” and left it there.

The explanation is short, and it is worth having because it is the prototype of one of the most useful arguments about fractal sets. Both numbers are read off one inequality about the measure the function is the distribution of. When the inequality is tilted — when the construction splits its mass unevenly — the two numbers come apart, and the way they come apart shows what each was really measuring.

A funnel round every point of the set #

A Hölder condition with exponent $\alpha$ says that near any point the function stays inside a funnel shaped like $|x - c|^{\alpha}$. For $\alpha = 1$ the funnel has straight sides and the condition is Lipschitz continuity: the function’s slope is bounded. For smaller $\alpha$ the funnel opens with vertical tangents at its tip, and the function is allowed to rise infinitely steeply — but only so steeply.

The Cantor function needs the room. At a point of the middle-thirds set such as $\tfrac14$, whose ternary expansion is $0.020202\ldots$, the function rises by $2^{-k}$ over an interval of width about $3^{-k}$ on either side, at every scale $k$. That is an average slope of $(3/2)^k$, which grows without bound, so the function has no derivative there and no Lipschitz bound. But $2^{-k} = (3^{-k})^{\log 2/\log 3}$ exactly, so the rise over a width $r$ is $r^{0.6309}$ at every scale, and the funnel of that exponent contains the function precisely.

Where the exponent is read #

The exponent can be read without any funnel. The rise of the function across an interval is the mass the measure puts in that interval, and the intervals of the $k$-th stage of the construction, of width $3^{-k}$, each carry mass $2^{-k}$. The ratio of rise to width raised to a power $\alpha$ is therefore $(3^{\alpha}/2)^k$, which is constant in $k$ exactly when $3^{\alpha} = 2$, that is when $\alpha = \log 2/\log 3$.

With a smaller exponent the ratio falls: the function is Hölder with that exponent too, with room to spare. With a larger one it grows geometrically, and no constant $C$ bounds it. So the best exponent is the one at which the mass of an interval scales like its width to a fixed power — and that power is a property of the measure, not of the function.

The same inequality proves the dimension #

Now read the same inequality the other way. Suppose a measure $\mu$ on a set $E$ satisfies $\mu(U) \le C\,(\operatorname{diam} U)^s$ for every small set $U$. Cover $E$ by any countable collection of sets $U_i$. Their masses add to at least the total mass of $E$, so

$$\sum_i (\operatorname{diam} U_i)^s \ge \frac{1}{C}\sum_i \mu(U_i) \ge \frac{\mu(E)}{C}.$$

Every cover, however fine, has $\sum (\operatorname{diam} U_i)^s$ bounded below by a fixed positive number, and so the $s$-dimensional Hausdorff measure of $E$ — the smallest such sum over covers by small sets — is positive. As infinite on one side and nought on the other showed, a positive Hausdorff measure in dimension $s$ means the dimension is at least $s$. This is the mass distribution principle, and it is the standard way lower bounds on dimension are proved: construct a measure on the set that does not concentrate too much, and read off the exponent.

For the middle-thirds set the measure is the natural one — the Cantor function’s — and the inequality holds with $s = \log 2/\log 3$. Counting boxes gives an upper bound of the same value. So the dimension is $\log 2/\log 3$, and the lower half of that proof is the same statement as the Hölder continuity of the function.

That is the coincidence explained. The Hölder exponent of a distribution function is the exponent $s$ in $\mu(\text{interval}) \le C\,(\text{length})^s$, and the mass distribution principle turns that same exponent into a lower bound for the dimension of any set carrying the measure. For the Cantor function the bound is also sharp, and so the two numbers agree.

The upper half, and why it is easy #

A lower bound on dimension needs a statement about every cover; an upper bound needs only one good cover. The construction supplies it. At the $k$-th stage the middle-thirds set is covered by $2^k$ intervals of width $3^{-k}$, so the sum $\sum (\operatorname{diam} U_i)^s$ for that cover is $2^k 3^{-ks} = (2/3^s)^k$. At $s = \log 2/\log 3$ this is exactly one at every stage, and for any larger $s$ it goes to nought as $k$ grows. So the $s$-dimensional Hausdorff measure is at most one at the critical exponent and nought above it, and the dimension is at most $\log 2/\log 3$.

The two halves together give the dimension exactly, and they are asymmetric in a way that is general. The upper bound used the construction’s own covers, and any sensible cover would have done nearly as well. The lower bound used the measure, and it needed the measure’s inequality for every small set, including sets that cut across the construction’s intervals awkwardly. That is where self-similarity earns its keep: a set of diameter $r$ meets at most two construction intervals of the stage whose width is closest to $r$, so its mass is at most twice theirs, and the inequality passes from the construction’s intervals to all sets with only a factor of two lost.

An equivalence, not a coincidence #

The mass distribution principle has a converse, and it turns the explanation into a characterisation. Otto Frostman proved in 1935 that a compact set has positive $s$-dimensional Hausdorff measure exactly when it carries a nonzero measure with $\mu(U) \le (\operatorname{diam} U)^s$ for every small set $U$. So the dimension of a compact set is the supremum of the exponents $s$ for which some measure on the set spreads its mass that evenly.

Read through distribution functions, Frostman’s lemma says the dimension of a set on the line is the best Hölder exponent achievable by a continuous increasing function that rises only on that set. The Cantor function achieves the best exponent for the middle-thirds set, and the skewed functions below achieve less. No function rising only on the middle-thirds set can be Hölder with an exponent above $\log 2/\log 3$, because such a function would be the distribution of a measure spreading its mass more evenly than the set’s dimension allows.

Tilting the weights #

The argument suggests a test. The set does not change if the construction splits its mass unevenly — giving a fraction $p$ of each interval’s mass to the left third and $1 - p$ to the right — and each such measure has its own distribution function, rising on exactly the same set of dimension $\log 2/\log 3$. The three drawn are continuous and flat on every removed interval, like the Cantor function. But the uneven ones rise in sharper bursts, because the heavy side’s weight compounds from stage to stage.

The largest mass among the stage-$k$ intervals is now $0.7^k$, at the interval reached by always going left, so the largest rise across a width $3^{-k}$ is $0.7^k = (3^{-k})^{\log(1/0.7)/\log 3}$. The Hölder exponent is $\log(1/0.7)/\log 3 \approx 0.325$ — barely half of the set’s dimension. With $p = 0.9$ it is $0.096$.

So the Hölder exponent is not the dimension of the set in general. It is the exponent at the point where the measure is most concentrated, the worst point for continuity. The mass distribution principle, applied with that exponent, proves the dimension is at least $0.325$, which is true and weak. The coincidence for the Cantor function happened because its measure is equally concentrated everywhere: every point of the set is as good, and as bad, as every other.

Three exponents that meet only once #

Between the worst point and the whole set sits a third exponent, the one a typical point has. Pick a point at random according to the measure: at each stage it goes left with chance $p$ and right with chance $1 - p$, and the mass of the stage-$k$ interval containing it is a product of $k$ such factors. By the law of large numbers its logarithm is about $k$ times the average, $p\log p + (1-p)\log(1-p)$, so the local exponent at a typical point is the entropy of the split divided by $\log 3$. That is the measure’s information dimension: $0.556$ at $p = 0.7$. The figure checks it by following one random point through four thousand ternary digits, which gives $0.548$.

The three exponents are ordered, Hölder exponent below information dimension below the set’s dimension, and they coincide only when the weights are even. The inequalities have a reading. The measure lives on a set of dimension $0.631$, but at $p = 0.7$ almost all of its mass sits on a smaller subset of dimension $0.556$ — the points whose digits go left about 70% of the time — and the worst points, where it piles up most, form a still smaller set. A dimension for every rate of crowding followed that spectrum of subsets all the way; what matters here is only its two ends and the middle, and the fact that the Cantor function is the one member of the family in which the spectrum collapses to a point.

What the derivative does meanwhile #

The Hölder exponent measures the function at its worst; the derivative describes it almost everywhere, and the two stories are very different. The Cantor function’s derivative is nought at every point off the middle-thirds set, since the function is constant on each removed interval, and the removed intervals have total length one — so the derivative is nought almost everywhere, which is the property a staircase with no steps was built to exhibit. At points of the set the derivative does not exist, and the rise over a width $r$ is of order $r^{0.63}$, which is infinitely steep.

So the function is flat on a set of full length and infinitely steep on a set of length nought, and the Hölder exponent is a single number measuring how steep the steep part is. That the steep part has length nought is what covering a set from outside proved of the middle-thirds set; that it nevertheless supports all of the rise is what makes the function continuous rather than a jump; and how steep it has to be to carry the whole rise on so small a set is the exponent.

Where the same argument is used #

The mass distribution principle is how most lower bounds on dimension are proved, because upper bounds come from covers, which are easy to write down, and lower bounds need a statement about every cover, which is hard. Building a measure turns the hard statement into an easy one: control the mass of small sets, and every cover is controlled at once.

The graphs of a curve with a corner at every point and the jagged functions of the room a jagged graph takes up use the same connection in the other direction. A function Hölder with exponent $\alpha$ has a graph of box dimension at most $2 - \alpha$, since each column of width $r$ needs at most about $r^{\alpha}/r$ boxes; for the Takagi and Weierstrass functions that bound is attained. There the smoothness of the function bounds the size of its graph; here the concentration of a measure bounds the size of its support from below. Both are the same exchange between how fast something varies and how much room it takes.

The measures do more than bound dimensions; they are the working tool whenever a question about a fractal set’s shadows or intersections has to be answered. Marstrand’s projection theorem of 1954 says that a planar set of dimension $s \le 1$ projects onto almost every line as a set of dimension $s$, and its modern proofs put a Frostman measure on the set and show that the projected measure still spreads its mass as evenly, for almost every direction, by averaging an energy integral over angles. A dust that almost every line misses showed the other side of that theorem: a set of dimension exactly one whose projections have length nought in almost every direction, the case where dimension alone cannot decide the size of the shadow. Every one of these arguments starts by choosing the measure, and the Cantor function is the first such measure anyone meets.

What the figures cannot show #

The functions are computed from ternary digits to thirty places, which resolves every stage of the construction far below a pixel. The Hölder checks are made at the points drawn and at the stage intervals up to the twelfth stage; the inequalities they illustrate are theorems about every point and every interval, proved by the self-similarity that makes every stage look like the first.

The typical local exponent is measured at one random point, and a different point gives a slightly different value; the law of large numbers says the value converges to $0.556$ for almost every point chosen by the measure, and not for every point. For the points that are not typical — like the one reached by always going left — the local exponent is anything between the Hölder exponent and the largest local exponent, $\log(1/0.3)/\log 3 \approx 1.10$.

And “dimension” in this essay means Hausdorff dimension throughout. For these self-similar sets and measures, box dimension, Hausdorff dimension and the other common definitions agree; for general sets they need not, and the mass distribution principle is specifically a tool for the Hausdorff kind.

Still open: measures that are not built from pieces #

For measures built by a self-similar rule, every exponent here is computable and the spectrum of local exponents is known in closed form. For measures that arise from dynamics without such a rule — the invariant measure of a chaotic map, the harmonic measure on the boundary of a fractal domain — the local exponents are well defined almost everywhere but their values are known only in special cases. Harmonic measure is the sharpest example. Nikolai Makarov proved in 1985 that the harmonic measure of any simply connected planar domain lives on a set of dimension exactly one, however wild the boundary, and Jean Bourgain proved in 1987 that in higher dimensions it lives on a set of dimension strictly less than the space’s; the exact bound in three and more dimensions is not known.

One inequality, read twice #

The Cantor function’s exponent and its set’s dimension are equal because they are two readings of one sentence: an interval of width $r$ carries mass at most a constant times $r^{0.6309}$. Read about the function, the sentence is a Hölder condition. Read about covers of the set, it is a lower bound on dimension. For the Cantor function the sentence is sharp at every point at once, and so both readings give the same number.

Tilt the construction and the sentence is sharp only at the worst point, the Hölder exponent drops, and the dimension of the set does not move. The number that tracks the set is then the dimension; the number that tracks the function is the worst concentration; and the number that tracks almost all of the mass lies between them. The coincidence was the special case in which there is nothing to choose.