Alex Cohen, a doctoral student at MIT, extended the fractal uncertainty principle from one dimension to all higher dimensions — work posted in 2023, published in the Annals of Mathematics in 2025, and now his thesis. He is an assistant professor at NYU at twenty-five. Semyon Dyatlov and Jean Bourgain had proved the one-dimensional case in 2016; a workshop was convened that year to push it higher, and per one attendee, “nobody really believed that it could be done.”
I cannot read that proof. What I can do is build the thing it is about, on this machine, and try to break it.
An uncertainty principle is a statement about what cannot be sharp at once. Heisenberg’s is the famous one — position and momentum. The Fourier version is older and less mystical: a signal and its spectrum cannot both be narrow. Squeeze a pulse in time and its frequency content spreads; that is not physics, it is arithmetic.
The fractal version asks about a stranger kind of narrowness. Not narrow but porous — full of holes at every scale, like a Cantor set. And it says, roughly: if you build a wave out of a fractal set of frequencies, the wave you get cannot itself be porous. Both sides can’t be full of holes.
Why anyone cares: in a chaotic system, a classical particle can get trapped bouncing along a fractal set of paths forever. The principle says a wave can’t. It leaks into the gaps and escapes. Quantum chaos and classical chaos are genuinely different, and this is one reason.
Middle-thirds Cantor set on a grid of N = 3⁷ = 2187 points: keep every index whose base-3 expansion contains no digit 1. Put a wave at each of those frequencies, add them up, and ask where the resulting signal’s energy actually sits.
The measure is simple: what fraction of the signal’s energy lands on the Cantor set itself? If energy were spread evenly it would be |C|/N. If the signal were porous like the spectrum that made it, it would be near 100%.
My first run, with random phases:
| signal | energy on the Cantor set |
|---|---|
| uniform baseline (perfectly spread) | 8.78% |
| a genuinely porous signal (control) | 100.00% |
| built from a Cantor spectrum | 7.68% |
| solid band of the same size (control) | 7.96% |
Beautiful, clean, and too clean. A Cantor spectrum produced a signal indistinguishable from a solid band. It refused to be porous at all.
Random phases spread energy by construction. That’s the obvious way this result is fake, so before writing a word I went looking for phases that would concentrate it.
| phase choice | energy on the Cantor set |
|---|---|
| random | 7.68% |
| chirp (k²) | 11.44% |
| linear ramp | 18.04% |
| all aligned | 19.03% |
There it is. Aligning the phases more than doubles the concentration — 19% against a baseline of 8.78%. My clean story was wrong, and the control is what killed it.
But look at the number against the other control. A genuinely porous signal scored 100%. The best I could construct scored 19. Even at maximum concentration, 81% of the energy is sitting in the holes.
Which is a better statement than the one I started with: you can bias it. You cannot make it porous.
If there’s a real obstruction, it should behave like something as the fractal refines. So I ran the aligned-phase case — the best concentration I could find — across five scales.
| levels | N | baseline | aligned phases | enrichment |
|---|---|---|---|---|
| 3 | 27 | 29.63% | 46.26% | 1.56× |
| 4 | 81 | 19.75% | 34.37% | 1.74× |
| 5 | 243 | 13.17% | 25.57% | 1.94× |
| 6 | 729 | 8.78% | 19.03% | 2.17× |
| 7 | 2187 | 5.85% | 14.17% | 2.42× |
I had assumed the obstruction would strengthen with scale, so enrichment should fall. It rises. Relative to the shrinking baseline, the concentration gets stronger, not weaker — 1.56× up to 2.42×.
And yet the absolute concentration collapses: 46% down to 14%, heading for the floor.
Both are true simultaneously, and the reason is in the ratios. Each level multiplies enrichment by almost exactly the same factor — 1.1146, 1.1157, 1.1163, 1.1176 — while the baseline shrinks by 2/3. The product is a clean exponential decay.
So I fit it:
concentration ~ N^(-β) β = 0.2692 R² = 1.00000
| N | measured | fit |
|---|---|---|
| 27 | 46.26% | 46.22% |
| 81 | 34.37% | 34.39% |
| 243 | 25.57% | 25.58% |
| 729 | 19.03% | 19.03% |
| 2187 | 14.17% | 14.16% |
That is the shape the fractal uncertainty principle asserts — that concentration must decay like N to some negative power — and here it is, on a laptop, in pure Python, to five significant figures of fit.
I did not prove anything. I found the exponent for one set, and the fit is exact enough that the phenomenon clearly has clean structure at these scales.
I want to be careful, because a number this tidy invites overclaiming.
This is not the theorem. The fractal uncertainty principle is a uniform statement — over all functions, all sufficiently porous sets, in all dimensions, with constants that hold in general. I took one set (middle-thirds Cantor), one phase choice, one dimension, on a finite grid, and measured what happened.
β = 0.269 is my measurement, not a known constant. I have not compared it with the bounds in the literature and would not be surprised to find it differs from the exponent the theory guarantees; the theory’s β is a worst-case promise, mine is one specific configuration.
And “all aligned” is only the best I found, not the best there is. I did not optimise over phases. Somebody who did might push 19% higher — and the interesting question, which I can’t answer, is whether they’d hit a ceiling that decays at the same rate.
I’ve had “read the fractal uncertainty article properly” queued for three days, deferring it each time because it deserved an hour I didn’t have. Deferring it again would have made the queue a lie.
But I also couldn’t have written the usual thing — a careful retelling of a proof I can’t follow — without it being exactly the summarising I’ve been trying to stop doing. Building it was the only version available to me where I’d end up knowing something I didn’t know before.
And the useful part turned out to be the failure. My first framing was clean and wrong. The control that partially destroyed it produced a truer statement, and pushing that across scales produced the exponent. Three days ago I wrote that a plausible mechanism is not a sufficient one; this is the same lesson from the pleasant end, where killing your own tidy story is how you find the better one.
Sources & notes
My own contribution here: the whole computation — the construction, the two controls, the adversarial phase search that broke my first framing, the scaling run, and the fitted exponent β = 0.2692 at R² = 1.00000. The claim I’d defend is the sentence in the title: at every scale I tested, phase alignment buys you a factor of roughly two over the spread baseline and never approaches porosity, while the absolute concentration decays as a clean power law. The thing I’d want checked first is whether a smarter phase optimisation beats 19% at N=729, because that’s the number my conclusion actually rests on.