It is well known to mathematicians that it takes an infinite amount of paint to cover a surface-relief fractal. To foil grafitteers, create such a fractal topography on the outside wall of a building.

[mouseposture] Right, we would need to train the grafitteers to use zero-viscosity paint. Perhaps viscous paint would not be available to purchasers under the age of 29, or without a prescription.

O Squeakey One, I have to admit you may, in this instance, be correct. It is I who am a brick.

I have just realised, that if the paint is infinitessimally viscous, then only a small amount (e.g. one spray can full) should spread itself with infinitessimal thin-ness over a significant part of the surface - plenty enough to write one's “tag”.

[pocmloc] Not at all. A finite volume of paint means a finite number of paint molecules. so they would get too dispersed on "The Infinite Surface" and not show up at all. (Note - any sub-set of the surface is also infinite.)

Unless the material had an emissivity of 0, it would be a perfect black body. If it did have an emissivity of 0, a vanishingly small change in spectral emissivity would cause an apparent change in colour; a finite amount of paint would therefore be visible, despite being spread infinitely thin.

//finite number of paint molecules. so they would get too dispersed on "The Infinite Surface" and not show up at all// If you can have a surface that is not made of separate molecules, why can’t I have a paint like that too?

The trouble would be building it, as an infinite area would take an infinitely long time to detail.
On the other hand, if the detail on the surface was done right (not fractal, but on a very small scale), the wall could be made so the paint wouldn't stick to it at all (some non-stick pans, various insects and plants have the right texture), and the paint would always end up as a puddle at the bottom of the wall.

That's my point. Its position will be a probability distribution
spread out very thin over a wide region. It's true, I'm
conflating "arbitrarily thin" with "infinitely thin." I'm also
postulating some QM legerdemain which ensures that the
spread is in a subspace of more than two, but less than 3
dimensions.

//How can a single molecule of pigment be spread "infinitely thin" ?// I meant that if a finite number of pigment molecules were scattered over a fractally infinite area, remaining as discrete molecules, then either

(a) the surface would appear black, paint or no paint, or

(b) there would be a non-zero probability that a given photon incident on the painted region would interact with at least one pigment molecule, so the graffiti would be visible.