A mathematical construction supplies a counterexample to a question about omitted values and bounded type in a half-plane. The paper constructs a real meromorphic function F on the complex plane whose preimages of 0, 1 and infinity are all contained in the real axis, while F is not of bounded type in either half-plane.
The construction also reaches every other value on the sphere. For each such value, the points where F takes it—the paper's point divisor—fail the Blaschke condition in both half-planes. In the paper's terminology, these are non-Blaschke divisors.
The exception is built around three values
The central question is whether a meromorphic function on the complex plane that omits three distinct values in a half-plane must be of bounded type there. The reported construction answers no for the normalized triple 0, 1 and infinity.
The three designated point sets—the preimages associated with 0, 1 and infinity—are each infinite, even though they remain on the real axis. Thus the example combines infinitely many points over each designated value with their exclusion from the two open half-planes.
The paper states that Möbius postcomposition extends the normalized example to any prescribed triple of distinct sphere values. An affine change of variable gives the analogous statement for every Euclidean half-plane.
A geometric route through the proof
The proof uses a modular-lambda universal covering of the sphere with 0, 1 and infinity removed. Its deck group is Γ(2), and the construction identifies the real locus with open Farey edges.
It then builds a comparison domain with a map written as ψ(z)=z+εC(z). Line-integration estimates provide bi-Lipschitz control for the map, keeping the relevant distances comparable between the domains.
Under the height and boundary-gap conditions stated in the lemma, the comparison-domain Green function has the lower bound GΩ0(p,w) ≥ y/[12(1+x²)].
Boundary extension of the conformal map supplies real values on Farey-edge intervals, and Schwarz reflection extends the mapped function across those intervals.
Counting the points that force the failure
To establish the non-Blaschke result, the proof identifies a subset of the a-point divisor in the upper half-plane. At each level m, it counts φ(2m) selected points and maps them to distinct simple a-points in H+.
That selected subset already fails the Blaschke condition, so the stronger conclusion does not depend on accounting for every point in the full divisor. The result is a theorem-level construction about a specifically built function, rather than a statistical estimate from observations.
A proof-based result with a defined scope
The supplied document is an arXiv version 1 preprint dated 26 August 2026. The reported conclusions rest on the stated mathematical proof, with no statistical uncertainty attached to them.
Its scope is existential: it presents the constructed counterexample and its associated point sets, rather than a classification of all meromorphic functions that omit three values in a half-plane.
The paper discloses AI generation of the core construction and proof, followed by human checking, mathematical auditing, revision and author review.
The paper also reports that a timestamped candidate-proof archive was preserved on 24 August 2026, before the first arXiv submission of the independently developed He–Zhang preprint on 25 August 2026.
No funding statement is reported in the supplied author-information block.
Paper data and sources
Original title: Three omitted values and non-Blaschke point divisors in half-planes
Authors: Quanyu Tang, Bokai Cui, Wei He et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text