We construct a real meromorphic function $F$ on $\mathbb C$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna's 1925 work that had remained open for over a century. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.
Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket q...
Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{\xi_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ \xi_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $\zeta \in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz ang...
Let $\mathcal{A}\to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $\mathcal{X}\subseteq\mathcal{A}$ be a positive-dimensional subvariety. For every integer $t>0$, we prove that, outside the $t$-degeneracy locus of $\mathcal{X}$, at most finitely many points lie in flat grou...
Fabrizio Barroero, L. Capuano, Tang-Li Ge et al.· 0 citations
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty....
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit...
Friedlander and Iwaniec proved that the number of points of the orbit $\{\gamma i \colon \gamma\in\SL_2(\Z)\}$ lying at a distance $p-2$ from the origin $i$ of the upper half-plane, with $p\le x$ prime, is of order $x/\log x$; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning...
A. Sedunova· 0 citations
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