minkowski-content.md

Take any set in the line, the plane or space, and a small radius eps. Collect every point within eps of the set. That fattened copy is the tube, or eps-neighbourhood, of the set, and its length, area or volume is written V(eps). Around a single point on the line the tube is an interval of length 2 eps. Around a segment of length L on the line it is L + 2 eps. Around the same segment in the plane it is a sausage of area 2 L eps + pi eps^2.

As eps shrinks the tube shrinks onto the set, and the rate tells you how big the set is. The segment's sausage falls like eps, one power, because the plane has two dimensions and the segment one: two minus one. A point in the plane has a disc of area pi eps^2, two powers, two minus zero. A set of dimension D inside a space of n dimensions has a tube that falls like eps^(n - D), and turning that round defines D: the Minkowski dimension is n - lim log V(eps) / log eps when the limit exists (Wikipedia). It agrees with box counting, the fractal dimension the tree uses everywhere.

The dimension is the exponent. The content is the number in front of it: divide the tube by eps^(n - D) and let eps go to zero. The upper Minkowski content is the largest value that ratio keeps coming back to, the lower content the smallest (Wikipedia). When the two agree the set is Minkowski measurable, and the common value is its Minkowski content. For the segment in the plane the ratio is 2 L + pi eps, which settles on 2 L, and once the standard factor of 2 is divided out that is the length itself. For smooth curves and surfaces the content is the length or area you already know.

Fractals are where it gets interesting. The Cantor set has dimension D = log 2 / log 3, about 0.631, and its tube can be written down exactly. The tube reaches eps past each end of the set. A gap no longer than 2 eps is swallowed whole, and a longer gap is entered eps from each side. The gaps are one of length 1/3, two of length 1/9, four of length 1/27 and so on, so V(eps) = 2 eps plus the sum over all gaps of the smaller of the gap and 2 eps. The reading to watch is M(eps) = V(eps) / eps^(1 - D).

Divide eps by 3. For any radius below 1/2 the new tube is two copies of the old one shrunk by 3, so V is multiplied by 2/3, and eps^(1 - D) is multiplied by exactly the same 2/3. The reading comes back to where it was: M(eps / 3) = M(eps). A reading that repeats like that can only have a limit if it is constant, and it is not. At eps = 1/9 it is exactly 2.5. At eps = 1/18, where the gaps of length 1/9 have just been swallowed, it is 2^(2 - D), about 2.583, its highest. Its lowest, near eps = 0.0975, is about 2.495. The reading swings through the same wave each time the radius shrinks by a factor of 3, for ever, so the upper and lower contents differ and the Cantor set is not Minkowski measurable.

The figure shows both halves of that. The upper panel draws the tube at seven radii, from 0.1 down to about 0.0037, each row's radius a factor of sqrt 3 below the one above, with the set itself as a thread through the middle; every row is two copies of the row two above it, shrunk by 3. The lower panel is the reading M(eps) as the radius shrinks from 1/9 to 1/729 left to right, with a faint rule at each factor of 3: the same wave four times.

The wave comes from a rhythm. Every piece of the Cantor set is shrunk by the same ratio 1/3, so the set looks the same at the scales 1, 1/3, 1/9, ... and at no scale in between, and its tube inherits that beat. Call a self-similar set lattice when the logarithms of all its shrinking ratios are whole multiples of one number, as when every ratio is a power of 1/3, and nonlattice otherwise, as with the two ratios 1/2 and 1/3, whose logarithms have an irrational quotient. A nonlattice set has no favourite scale; the beats of its different ratios drift out of step and wash out, and the reading settles.

Lapidus made this a dichotomy for self-similar fractal strings, the gaps of such a set on the line: a lattice string is never Minkowski measurable and a nonlattice one always is (Lapidus and van Frankenhuijsen 2006). The wave has an exact source there. The zeta function of the gap lengths of a lattice string has poles off the real line, at D + 2 pi i k / log 3 for the Cantor set, one for every whole number k; these are its complex dimensions, and each one adds a term that oscillates in log eps with period log 3. A nonlattice string has no complex dimension on that vertical line except D itself, so nothing oscillates at the leading order.

For self-similar sets on the line the picture is complete. Falconer 1995 proved that the nonlattice ones are Minkowski measurable, and Kombrink and Winter 2020 finished the lattice side: every nontrivial lattice self-similar set on the line fails to be. In the plane and in space the nonlattice half holds in every dimension (Gatzouras 2000), while the lattice half is known only under extra conditions on the shape of the holes (Kombrink, Pearse and Winter 2016).

In the tree

Complex dimensions computes the complex dimensions of the one-dimensional designs, watches them surface as the wave in the box count, and works through measurability for the Cantor design, the carpet and the sponge. The sponge paper carries the lattice half to the Menger sponge in space. The sponge's tube demo drags the radius across a slice of the sponge's tube and watches the reading climb onto its wave, and the tube demo does the same for plane designs such as the Sierpinski carpet.