Zeros and one-points on three rays

Definitions and mathematical statement

For a real angle $\theta$, write $R_\theta=\{re^{i\theta}:r>0\}$ for the open ray from the origin. A $1$-point of an entire function $f$ is a solution of $f(z)=1$, counted with multiplicity when needed. The intended problem concerns transcendental entire functions, with infinitely many zeros and infinitely many $1$-points.

Does there exist such an $f$ and an angle \[ \frac{\pi}{3}<\alpha<\frac{\pi}{2}, \qquad\alpha\ne\frac{2\pi}{5}, \] for which \[ \{z:f(z)=0\}\subset R_0,\qquad \{z:f(z)=1\}\subset R_\alpha\cup R_{-\alpha}? \] These are requirements on every zero and every $1$-point in the entire plane, not only on their limiting directions at large modulus. The two value sets may contain multiple points; no simplicity condition is imposed. Since the origin lies on none of the open rays, it is neither a zero nor a $1$-point.

The target is existence at any angle in the displayed range other than the distinguished angle, or a proof of nonexistence throughout that range. No finite-order growth assumption is imposed. Infinitely many preimages of each value make the radial distribution condition substantive: polynomials or functions omitting one of the two values would permit vacuous or elementary configurations unrelated to the intended three-ray problem.

Short English statement

Can infinitely many zeros and one-points of a transcendental entire function be confined to these three rays at another allowed angle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700015 (AMR-036-0015), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Radially distributed values, first question. Original source indexed by AMR-036-0015. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Radially distributed values. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/radial.pdf.