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Source. Theorem 2, p. 2, proof p. 6, with Remark 7, p. 7, of P. Chojecki, A note on an Erdős path problem for transcendental entire functions, note, ulam.ai, 2026, 7 pp., the edition named on the source card.

Statement

Setting (p. 1). $M(r,f)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ is the maximum modulus.

Theorem 2 (p. 2). There is a transcendental entire function $f$ with the following property. For every unbounded connected set $E\subset\mathbb C$ and every $\varepsilon>0$ there is a sequence $w_k\in E$ with $\lvert w_k\rvert\to\infty$ and

$$\lvert f(w_k)\rvert\le M(\lvert w_k\rvert,f)^{\varepsilon} \qquad(k\to\infty).$$

Consequently no fixed exponent $\varepsilon>0$ has the property that every transcendental entire function has a path to infinity on which $\lvert f(z)\rvert\ge M(\lvert z\rvert,f)^{\varepsilon}$ eventually.

The single function $f$ works for every $E$ and every $\varepsilon$; the sequence depends on both. A path to infinity has unbounded connected image, so it is one such $E$.

Remark 7 (p. 7). The note says Theorem 2 excludes every universal lower bound $M(r,f)^\varepsilon$ with fixed $\varepsilon>0$ but does not settle whether some much slower universal comparison function of $M(r,f)$ can still be forced along a suitable path; the abstract (p. 1) places that broader formulation outside the note's scope.

Proof pointer

Pp. 5--6. Langley's Theorem 1.4 (J. K. Langley, Complex flows, escape to infinity and a question of Rubel, Ann. Fenn. Math. 47 (2022)) gives a transcendental entire $G$ such that every unbounded connected plane set $E$ contains $w_n$ with $\lvert w_n\rvert\to\infty$ and $(-1)^n\operatorname{Re}G(w_n)\le\lvert w_n\rvert^{1/2}$; the even terms have $\operatorname{Re}G(w_n)\le\lvert w_n\rvert^{1/2}$. The note sets $f=e^G$, so $M(r,f)=e^{B_G(r)}$ with $B_G(r)=\max_{\lvert z\rvert=r}\operatorname{Re}G(z)$. Lemma 6 (p. 5) states that $B_G(r)/r^\alpha\to\infty$ for every transcendental entire $G$ and every $\alpha>0$; its proof (p. 6) bounds $M(r,G)$ by $2B_G(2r)+3\lvert G(0)\rvert$ through the Borel--Carathéodory theorem and applies Lemma 3. With $\alpha=1/2$, for large $n$ one gets $\lvert w_n\rvert^{1/2}\le\varepsilon B_G(\lvert w_n\rvert)$, hence $\lvert f(w_n)\rvert\le M(\lvert w_n\rvert,f)^\varepsilon$.

Read depth

Claims checked: Theorem 2, Lemma 6, Remark 7 and the proofs on pp. 5--6 were read clause by clause on the page images of the note. Langley's Theorem 1.4 is quoted, not proved, in the note and was not checked against Langley's paper. Nothing here is independently reviewed.

Dependencies

• Lemma 6 (p. 5): for transcendental entire $G$, $B_G(r)/r^\alpha\to\infty$ for every $\alpha>0$.
• Lemma 3 (p. 3), used in the proof of Lemma 6: $M(r,F)/r^\alpha\to\infty$ for transcendental entire $F$ and every $\alpha>0$.
• J. K. Langley, Complex flows, escape to infinity and a question of Rubel, Ann. Fenn. Math. 47 (2022), no. 2, 885--894, Theorem 1.4.

Bears on

• Problem 514: the third question asks for a path along which $\lvert f(z)\rvert$ grows faster than a fixed function of $M(r)$, giving $M(r)^\epsilon$ as an example. Theorem 2 answers that power example no: one transcendental entire $f$ has, for every $\varepsilon>0$, no path to infinity along which $\lvert f(z)\rvert\ge M(\lvert z\rvert,f)^\varepsilon$ eventually. By Remark 7 the note leaves the question with a general fixed comparison function open. The problem's claim page records the claim and its standing.