Goldberg 1979 asymptotic curves entire functions finite order

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spiral_barriers: Polynomial approximation on a winding arc creates barriers that force every escaping asymptotic path to have unbounded length divided by radius.

theorem_1: For every function tending to infinity, there is an entire function, of order zero in the construction, whose logarithmic maximum modulus is at most a constant times that function times the square of the logarithm, with no asymptotic path to infinity of length O(r) inside the disc of radius r.

theorem_2: For every order from zero to infinity inclusive there is an entire function of that order with no asymptotic path to infinity of length O(r) inside the disc of radius r.

theorem_4: Every order at least one half admits an entire function with asymptotic value zero but no path to that value with length bounded by a constant times the radius.

theorem_5: There are entire functions, growing arbitrarily slowly beyond the logarithmic square, whose modulus exceeds one only on angular sets of measure tending to zero, along sets of radii of upper density one.

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A. A. Gol'dberg and A. E. Eremenko, On asymptotic curves of entire functions of finite order, Math. USSR-Sbornik 37 (1980), no. 4, 509–533. DOI: 10.1070/SM1980v037n04ABEH001989.

Source identity

The copy read for this card is the 25-page English translation by H. T. Jones, published in 1980, of the 1979 Russian article in Mat. Sb. (N.S.) 109(151), no. 4, 555–581. The card keeps the 1979 source identity of the Russian original. The manuscript was received 20 September 1977. All result-page references here use the English translation's printed pages and labels.

That copy is the one downloaded from the author's page on 2026-09-05. The publisher record and author bibliography confirm the translation. No correction or later replacement was found in those records in this search. That copy prints "© American Mathematical Society 1980" on its first page (printed p. 509), and the author's papers page from which it was downloaded states no copyright, license or terms (https://www.math.purdue.edu/~eremenko/papers.html, read 2026-10-02), every other right reserved.

Read status: claims checked. The statements of Theorems 1, 2, 4 and 5, the spiral-barrier ingredients and the Winkler example were read clause by clause on the page images of the translation; the proofs were read in outline, and none of it is independently reviewed.

Path-length counterexamples

In a 1960 lecture Hayman conjectured that every entire function of finite order has an asymptotic path to infinity whose length inside the disc of radius $r$ is $O(r)$ (p. 509). Erdős repeated it as Hayman's Problem 2.41 (1974), adding a finite-asymptotic-value variant. For order zero, Hayman went further, conjecturing length $r+o(r)$.

• Theorem 1, pp. 510–513, constructs an order-zero counterexample with $\log M(r,f)=O(\phi(r)(\log r)^2)$ for any given $\phi(r)\to\infty$. This makes the growth threshold in Hayman's positive ray theorem sharp: the introduction, p. 509, cites Hayman's Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75–84, for rays to infinity at almost every angle when $f$ is nonconstant and $\log M(r,f)=O((\log r)^2)$.
• Theorem 2, pp. 513–516, gives counterexamples of every prescribed order $0\le\rho\le\infty$: for $0<\rho<1$ the construction adds rescaled Mittag-Leffler factors, larger finite orders follow by $z\mapsto z^m$, and infinite order uses Carleman approximation on a spiral.
• The common spiral-barrier construction uses Runge approximation to make successive scaled winding arcs small-value barriers. Crossing their annuli while avoiding them forces arbitrarily large ratios of path length to radius. Infinite products preserve the barriers; zero counting and sparse growth estimates identify the limiting function as a canonical product.

Theorems 1 and 2 give a negative answer to the linear-length question in Problem 1115. They do not provide an optimal length bound for any growth class. The result pages give the statements with proof sketches written here; the external approximation and value-distribution theorems they use are cited, not proved.

Further results and proof scope

Theorem 4, pp. 524–529, treats finite asymptotic values for every $1/2\le\rho\le\infty$. Its finite-order functions also obstruct short paths to infinity and have lower order equal to order. The method is different: a winding semistrip, conformal and quasiconformal maps, and a Cauchy integral construction. The result page gives the statement and a sketch. Theorem 3 (p. 517, a conformal-mapping theorem for such semistrips) and Lemmas 1–3 (pp. 517–524) are auxiliary to Theorem 4, bear on no problem here, and have no pages of their own. The historical normal-type question at that endpoint is not resolved by this construction.

Theorem 5, pp. 529–530, makes the angular measure of $|f|>1$ tend to zero on sets of radii of upper linear density one. It gives slow-growth and prescribed-order versions separately; an arbitrary slow-growth bound cannot simultaneously be imposed on a positive prescribed order. The result page includes the statement and source sketch, not a full reconstruction.

The unnumbered example on pp. 531–532 answers the first part of Winkler's Problem 2.42. For an integer $n\ge2$, let

$$f(z)=\int_0^z\frac{\sin(\zeta^n)}{\zeta^n}\,d\zeta,\qquad \gamma_k=\{re^{\pi i k/n}:r>0\},\quad0\le k<2n.$$

The removable singularity at zero is filled in. The distinct asymptotic values on these rays are $a_k=a_0e^{\pi i k/n}$, with $a_0=\Gamma(1/n)\cos(\pi/(2n))/(n-1)>0$. The numbers of $a_k$-points in the whole disc and on the corresponding ray satisfy

$$n(r,a_k,f)\sim\frac{2n}{\pi}r^n,\qquad n(r,a_k,f;\gamma_k)\sim\frac1\pi r^n.$$

Thus their ratio tends to $1/(2n)$. The source derives the first estimate from completely regular growth with indicator $|\sin(n\theta)|$, using Levin [23], and the second from alternating, strictly decreasing oscillation increments on the positive ray and rotation symmetry. This is a statement and sketch only. The paper's remark that Winkler's second question remained open is historical; Hayman–Lingham's 2018 Update 2.42 reports subsequent work by Barsegyan. That separate problem is not compiled here.

Later literature and remaining coverage

The existing Hayman–Lingham survey, arXiv:1809.07200v2 (21 September 2018), Update 2.41, printed p. 38, records this resolution. Its Update 2.7, p. 25, points to Toppila's independent three-page proof, On the length of asymptotic paths of entire functions of order zero, Ann. Acad. Sci. Fenn. A I Math. 5 (1980), 13–15, doi:10.5186/aasfm.1980.0525. A separate reconstruction of that proof remains useful.

The same update cites K. H. Chang, Asymptotic values of entire and meromorphic functions, Sci. Sinica 20 (1977), 720–739, for an upper bound with exponent $1+\rho/2+\varepsilon$. The survey defines its length there as length to the first circle intersection; primary-source verification is needed before using it for total in-disc length. Update 2.57, p. 44, points to J. M. Anderson, Asymptotic values of meromorphic functions of smooth growth, Glasgow Math. J. 20 (1979), 155–162, doi:10.1017/S0017089500003876, for nearly radial paths to a deficient value under the extra hypothesis $T(2r,f)\sim T(r,f)$. A further primary lead is Toppila's On the length of asymptotic paths of meromorphic functions of order zero, Ann. Acad. Sci. Fenn. A I Math. 9 (1984), 79–87, doi:10.5186/aasfm.1984.0912. These are follow-up sources, not fully checked bounds in this unit.

Searches covered the author bibliography, publisher record, this existing survey, primary-paper title searches, and a search for E1115 announcements on X. No new accepted replacement of the counterexample theorem was identified. This limited search does not establish an exhaustive current optimum for the wider quantitative question.

Bears on. #1115, which asks whether every entire function of finite order has a rectifiable path on which $f\to\infty$ whose length in $|z|<r$ is $\ll r$: Theorem 1 gives an entire function, of order zero by its construction, with $\log M(r,f)=O(\varphi(r)(\log r)^2)$ for any prescribed $\varphi(r)\to\infty$ and no such path, and Theorem 2 gives one of every order $0\le\rho\le\infty$, so the answer to that question is negative; Theorem 4 gives the same negative answer, for every order $\rho\ge1/2$, for paths to the finite asymptotic value $0$, the variant added in Problem 2.41 of Hayman's 1974 collection. Theorem 5 is related growth theory and does not concern path length.

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