Chojecki 2026 note erdos path problem transcendental entire

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theorem_1: Chojecki's theorem that every transcendental entire function f has a path to infinity on which log|f| divided by log|z| tends to infinity, so f outgrows every fixed power of z there, and whose initial segment up to radius R has length O(M(R,f)^eps) for every eps > 0.

theorem_2: Chojecki's theorem that some transcendental entire function f has, in every unbounded connected plane set and for every eps > 0, points w_k tending to infinity with |f(w_k)| at most M(|w_k|,f)^eps, so no fixed positive power of the maximum modulus is a lower bound along a path for every transcendental entire function.

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P. Chojecki, A note on an Erdős path problem for transcendental entire functions. erdosproblems.com forum note (thread 514) (2026).

The note answers the first two of the questions Erdős asked in problem 514 yes and the power-type example of the third no. Theorem 1 shows every transcendental entire function f admits a path to infinity gamma with log|f(gamma(t))| / log|gamma(t)| tending to infinity, hence |f(z)/z^n| tending to infinity along gamma for every fixed n, and moreover the initial segment up to radius R has length O(M(R,f)^eps) for every eps > 0. The proof applies Wu's Theorem B (quoted as Theorem 4) on subharmonic functions along paths to u = max{log|f|, -1}, using Lemma 3 (a Cauchy-estimate argument that M(r,F)/r^alpha tends to infinity for transcendental F) to verify B_u(r)/log r tending to infinity, and then the finiteness of the integral of e^{-eps u} along gamma to bound the length. Theorem 2 gives the negative answer to the power version: there is a transcendental entire f = e^G such that for every eps > 0 every unbounded connected set E contains w_k with |w_k| to infinity and |f(w_k)| <= M(|w_k|,f)^eps, so no fixed exponent eps > 0 works universally; it uses Langley's Theorem 1.4 to get G with (-1)^n Re G(w_n) <= |w_n|^{1/2} and Lemma 6 (via Borel-Caratheodory) to show B_G(r)/r^alpha tends to infinity. Remark 7 notes that Theorem 2 does not settle whether some much slower universal comparison function of M(r,f) can still be forced along a path; the abstract places that broader formulation outside the note's scope. In the forum thread, Nat Sothanaphan reports that the answer to Q1/Q2 reduces to Wu's 1985 subharmonic-path theorem and that they have worked through the note's proof of the first two questions and vouch for it; that is a thread post, not a review.

Source: https://www.erdosproblems.com/forum/thread/514, where Przemek Chojecki posted the note on 20 April 2026 with a link to https://www.ulam.ai/research/erdos514.pdf; the copy read is that file. The file is a seven-page note that prints no byline, no date and no copyright or license statement on any of its pages, so the attribution rests on the forum post. The forum thread shows no license, copyright or terms-of-use statement (read 2026-10-02), and the host's site (https://www.ulam.ai/, read 2026-10-07) carries the footer "© 2017-2026 ULAM" and no license or terms-of-use statement, every other right reserved.

Read status: claims checked. Theorems 1 and 2, Lemmas 3 and 6, Remark 7 and the proofs on pp. 3--6 were read clause by clause on the page images of the note; the quoted theorems of Wu and Langley were not checked against their papers.

Contents

• Theorem 1 (p. 2; proof pp. 4--5): every transcendental entire $f$ has a path to infinity $\gamma$ with $\log\lvert f(\gamma(t))\rvert/\log\lvert\gamma(t)\rvert\to\infty$, hence $\lvert f(\gamma(t))/\gamma(t)^n\rvert\to\infty$ for every fixed $n$, and with initial-segment length $\ell_\gamma(R)=O(M(R,f)^\varepsilon)$ as $R\to\infty$ for every $\varepsilon>0$.
• Theorem 2 (p. 2; proof p. 6), with Remark 7 (p. 7): one transcendental entire $f$ such that for every unbounded connected $E\subset\mathbb C$ and every $\varepsilon>0$ there are $w_k\in E$ with $\lvert w_k\rvert\to\infty$ and $\lvert f(w_k)\rvert\le M(\lvert w_k\rvert,f)^\varepsilon$; so no fixed power of $M(r,f)$ is a lower bound along a path for every transcendental entire function. Remark 7 leaves open whether a much slower universal comparison function of $M(r,f)$ can be forced.
• Lemma 3 (p. 3): for transcendental entire $F$ and every $\alpha>0$, $M(r,F)/r^\alpha\to\infty$, by Cauchy's estimate for a nonzero coefficient $a_m$ with $m>\alpha$. Used in both theorems.
• Theorem 4 (p. 3), Wu's Theorem B, quoted input: if $u$ is subharmonic on $\mathbb C$ with $B_u(r)/\log r\to\infty$, where $B_u(r)=\sup_{\lvert z\rvert=r}u(z)$, there is a path to infinity with $u(\gamma(t))/\log\lvert\gamma(t)\rvert\to\infty$ and $\int_\gamma e^{-\delta u}\lvert dz\rvert<\infty$ for every $\delta>0$, the path independent of $\delta$.
• Lemma 6 (p. 5; proof p. 6): for transcendental entire $G$, $B_G(r)=\max_{\lvert z\rvert=r}\operatorname{Re}G(z)$ satisfies $B_G(r)/r^\alpha\to\infty$ for every $\alpha>0$; proved from Borel--Carathéodory and Lemma 3.

Bears on. #514: Theorem 1 answers the first question yes for every transcendental entire function and the second yes when the length is read, as the note reads it, as the length of the initial segment up to radius $R$, bounded by $O(M(R,f)^\varepsilon)$ for every $\varepsilon>0$; Theorem 2 answers the third question's example $M(r)^\epsilon$ no and, by the note's Remark 7, leaves the question with a general fixed function of $M(r)$ open. The note is not refereed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.