Invariant sets of the double Hilbert transform

We present an example that shows that the double Hilbert transform \H can have the characteristic function of an open set of finite measure in $\mathbb{R}\times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of \H, the double dyadic shift. We further construct non-diagonal sets whose indicators are fixed by \H – strips bent along the Boole curve $x=y-1\mathrm{/}y$ and, more generally, along level curves of real Herglotz functions with purely singular measures – and study the class of invariant sets which they generate. At positive density we construct an example built on three families of parallel lines, related to the carry cocycle $\{u\}+\{v\}-\{u+v\}$, and the fans – alternate sectors of an odd number of concurrent lines – which are the simplest members of the invariant cones, a class that we describe completely. We then study the classification of the invariant sets whose boundary lies on countably many lines. Its combinatorial half is settled completely: a locally finite arrangement of lines in which no point lies on exactly two lines is a family of parallel lines, a pencil of concurrent lines, or an affine image of the triangular lattice. The classification is further connected, via projective duality, with configurations of Sylvester–Gallai type, and, via a change of variables, with two-valued superpositions of traveling waves.