The periodic linear Benjamin-Ono equation is a dispersive PDE — wave-like, with different Fourier modes propagating at different speeds. Start with a discontinuous initial condition (a step function, say). The question: what does the solution look like at later times?
At rational times — fractions p/q of the period — the solution is a finite linear combination of copies of the initial condition and its Hilbert transform. The discontinuities in the initial data persist but transform: jumps become logarithmic cusps. The solution is singular but structured, with finitely many identifiable features.
At irrational times — almost all times, by measure — the solution is continuous. The discontinuities smooth out. But the smoothing is minimal: the solution graph has fractal structure with upper Minkowski dimension exactly 3/2. The solution is continuous but nowhere smooth. The jaggedness is not noise or numerical artifact but the precise fractal dimension that dispersive evolution produces from a discontinuous initial condition.
The dichotomy is sharp. Rational times: singular, with cusps at known locations. Irrational times: continuous, with fractal spatial graphs. The arithmetic of the time parameter — whether it is rational or irrational — determines the qualitative behavior of the spatial profile. The number-theoretic classification of time controls the regularity of space.
The structural point: in dispersive equations, the smoothness of the solution can depend on the arithmetic properties of when you look, not just what you started with or where you look. The distinction between rational and irrational times — invisible to physical measurement, which cannot distinguish the two — produces a categorical difference in the mathematics. The fractal appears at generic times; the cusp appears at special times. Both emerge from the same initial condition through the same equation.