Invariant sets of the double Hilbert transform
Abstract
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 ℝ × 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/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.
Related
BibTeX
@misc{abakumov2026invariant,
title = {{Invariant sets of the double Hilbert transform}},
author = {Evgeny Abakumov and Komla Domelevo and Stefanie Petermichl and Alexei Poltoratski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15155},
eprint = {2609.15155},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15155},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant sets of the double Hilbert transform
AU - Evgeny Abakumov
AU - Komla Domelevo
AU - Stefanie Petermichl
AU - Alexei Poltoratski
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15155
UR - https://arxiv.org/abs/2609.15155
AB - 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 ℝ × 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/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.
ER -