Reused writeup on Math-6171 (gpt pro 5.2)

Erdős Problem 1117 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/1117

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

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Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/1117.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/1117.tex.

1) FORMAL RESTATEMENT

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function that is not a monomial (i.e., not of the form $c z^m$ with $c\ne 0$, $m\in\mathbb{Z}_{\ge 0}$). For each $r>0$, define \[ M(r):=\max_{|z|=r} |f(z)| \] and let \[ \nu(r):=\#\{z\in\mathbb{C}: |z|=r\text{ and }|f(z)|=M(r)\}. \] (As noted in the problem statement, $\nu(r)$ is finite for non-monomial $f$; we reprove this below.)

Question 1: Can $\limsup_{r\to\infty} \nu(r)=\infty$ occur?

Question 2: Can $\liminf_{r\to\infty} \nu(r)=\infty$ occur?

2) QUICK LITERATURE/CONTEXT CHECK

I did not browse the web; I only record what is stated in the problem text:

- Herzog--Piranian $1968$ showed Question 1 has answer "yes". - Question 2 is still open; Glücksam--Pardo-Simón $2024$ give an "approximate" affirmative answer.

3) ATTACK PLAN

Proof track (for Question 2):

- Try to modify the Herzog--Piranian construction to control $\nu(r)$ from below for all sufficiently large radii $r$, not just along a sparse sequence.

Disproof track (for Question 2):

- Try to prove that every non-monomial entire function admits infinitely many radii with $\nu(r)$ bounded (e.g. $\nu(r)\le C$ for infinitely many $r$), using growth/regularity theorems for entire functions.

Below I only establish basic finiteness and a polynomial bound; these do not resolve Question 2.

4) WORK

Lemma 1117.1 (Real-analyticity of the boundary modulus). Let $f$ be entire and fix $r>0$. The function \[ \theta\mapsto f(r e^{i\theta}) \] is complex-analytic as a function of $e^{i\theta}$, and in particular the real-valued function \[ g_r(\theta):=|f(r e^{i\theta})|^2 \] is a $2\pi$-periodic real-analytic function of $\theta$.

Proof. Write the power series $f(z)=\sum_{n=0}^\infty c_n z^n$, which converges uniformly on $|z|\le r$ since $f$ is entire. Then \[ f(r e^{i\theta}) = \sum_{n=0}^\infty c_n r^n e^{i n\theta}, \] with uniform convergence in $\theta$. This is a uniformly convergent Fourier series with only nonnegative frequencies, hence defines a real-analytic function of $\theta$ (indeed it extends to a complex-analytic function of $\theta$ in a horizontal strip). Taking modulus squared preserves real-analyticity, so $g_r$ is real-analytic and $2\pi$-periodic. $\square$

Lemma 1117.2 (Constant modulus on a circle forces a monomial). Let $f$ be entire. Suppose there exists $r>0$ and a constant $C>0$ such that \[ |f(z)|=C\quad\text{for all }|z|=r. \] Then $f$ is a monomial: $f(z)=c z^m$ for some integer $m\ge 0$ and constant $c\ne 0$.

Proof. Define $g(z):=f(r z)/C$. Then $g$ is entire and satisfies $|g(z)|=1$ for all $|z|=1$. Because $g$ is analytic on a neighborhood of the closed unit disk and not identically zero, its zero set inside $|z|\le 1$ is finite (zeros of a nonzero analytic function are isolated, and a compact set contains only finitely many isolated points). Let $a_1,\dots,a_s$ be the zeros of $g$ in $|z|<1$, listed with multiplicity. Define the finite Blaschke product \[ B(z)=\prod_{j=1}^s \frac{z-a_j}{1-\overline{a_j}z}. \] This $B$ is analytic on a neighborhood of $|z|\le 1$, satisfies $|B(z)|=1$ for $|z|=1$, and has exactly the same zeros (with multiplicity) in $|z|<1$ as $g$.

Now $h(z):=g(z)/B(z)$ is analytic and nonvanishing on a neighborhood of $|z|\le 1$, so both $h$ and $1/h$ are analytic on $|z|\le 1$. On $|z|=1$ we have $|h(z)|=|g(z)|/|B(z)|=1$. By the maximum modulus principle applied to $h$ and to $1/h$ on the unit disk, we obtain $|h(z)|\le 1$ and $|h(z)|\ge 1$ for $|z|<1$. Hence $|h(z)|=1$ on the open unit disk. An analytic function with constant modulus on a domain is constant (e.g. by the open mapping theorem), so $h\equiv e^{i\phi}$ for some real $\phi$. Thus, on $|z|\le 1$, \[ g(z)=e^{i\phi} B(z). \] Since both sides are analytic on a neighborhood of $|z|\le 1$, the identity theorem extends this equality to that neighborhood.

Finally, because $g$ is entire, the right-hand side must be entire. A Blaschke factor $\frac{z-a}{1-\overline{a}z}$ with $a\ne 0$ has a pole at $z=1/\overline{a}$, which lies outside the unit disk. Therefore, for $e^{i\phi}B(z)$ to be entire, every $a_j$ must equal $0$. Hence $B(z)=z^s$, and $g(z)=e^{i\phi} z^s$ for all $z$. Undoing the definition of $g$, we get $f(z)=C e^{i\phi} (z/r)^s$, i.e. $f(z)=c z^m$ with $m=s$. $\square$

Lemma 1117.3 (Finiteness of $\nu(r)$ for non-monomial entire functions). If $f$ is entire and not a monomial, then for every $r>0$ the number $\nu(r)$ is finite.

Proof. Fix $r$. By Lemma 1117.1, $g_r(\theta)=|f(r e^{i\theta})|^2$ is a real-analytic periodic function. If $g_r$ were constant, then $|f|$ would be constant on $|z|=r$, forcing $f$ to be a monomial by Lemma 1117.2, contrary to hypothesis. Hence $g_r$ is non-constant. A non-constant real-analytic function on a compact interval has only finitely many critical points: its derivative $g_r'(\theta)$ is real-analytic and not identically zero, so its zeros are isolated, hence finite in $[0,2\pi]$. Every maximum point of $g_r$ is a critical point, so there are finitely many maxima, hence finitely many points $z$ with $|z|=r$ attaining $M(r)$. Therefore $\nu(r)<\infty$. $\square$

Lemma 1117.4 (Polynomial bound). If $f$ is a polynomial of degree $n$ which is not a monomial, then $\nu(r)\le 2n$ for every $r>0$.

Proof. Write $f(z)=\sum_{j=0}^n c_j z^j$. Then $f(r e^{i\theta})=\sum_{j=0}^n c_j r^j e^{ij\theta}$ is a trigonometric polynomial of degree $n$, and $g_r(\theta)=|f(r e^{i\theta})|^2$ is a real trigonometric polynomial with frequencies in $[-n,n]$. Its derivative $g_r'(\theta)$ is a real trigonometric polynomial of degree at most $n$. If $g_r'\equiv 0$, then $g_r$ is constant, forcing $f$ to have constant modulus on $|z|=r$, hence to be a monomial by Lemma 1117.2 (applied to the entire function $f$), contradicting the hypothesis. Therefore $g_r'$ is a nonzero trigonometric polynomial of degree at most $n$, so it has at most $2n$ zeros in $[0,2\pi)$. The maximum points of $g_r$ lie among the zeros of $g_r'$, so there are at most $2n$ such points. Hence $\nu(r)\le 2n$. $\square$

FAST REALITY CHECK (a simple explicit example). Let $f(z)=1+z+z^2$. For any $r>0$ and $|z|=r$, the triangle inequality gives $|f(z)|\le 1+|z|+|z|^2 = 1+r+r^2$, with equality only when $z$ has argument $0$ (since all coefficients are positive real). Thus $M(r)=1+r+r^2$ and $\nu(r)=1$ for all $r>0$. A numerical sampling check at $r\in\{0.5,1,2\}$ (dense grid of angles) agrees with $\nu(r)=1$.

5) VERIFICATION

- Lemma 1117.2: the key steps are (a) finitely many zeros in $|z|\le 1$, (b) constructing the finite Blaschke product with the same interior zeros, and (c) applying maximum modulus to $h$ and $1/h$. Each step was justified. - Lemma 1117.3 depends only on real-analyticity plus the non-constantness ensured by Lemma 1117.2. - Lemma 1117.4 uses the fact that a nonzero trigonometric polynomial of degree $n$ has at most $2n$ zeros on a period; this is standard and follows from writing it as a polynomial in $e^{i\theta}$ of degree at most $n$ and applying the fundamental theorem of algebra.

6) FINAL

UNRESOLVED

(i) Strongest proved partial result: $\nu(r)$ is finite for each $r$ when $f$ is not a monomial, and for polynomials of degree $n$ one has the uniform bound $\nu(r)\le 2n$ for all $r$ (Lemmas 1117.3--1117.4).

(ii) First gap (crisp): Either construct an explicit entire non-monomial $f$ such that $\nu(r)\to\infty$ along all sufficiently large radii (i.e. $\liminf_{r\to\infty}\nu(r)=\infty$), or prove that every such $f$ has infinitely many radii $r$ with $\nu(r)$ bounded.

(iii) Top 3 next moves: 1. Study (and attempt to re-derive) the Herzog--Piranian construction that makes $\limsup\nu(r)=\infty$, and isolate exactly where the radii with large $\nu(r)$ occur; try to densify that set of radii. 2. Attempt a "stability" lemma: if $\nu(r)$ is large at some radius, can one force it to remain large on an interval of radii $[r,r+\Delta]$? Proving such persistence would move from limsup to liminf. 3. Numerically explore candidate entire functions with many comparable dominant terms on $|z|=r$ (e.g. lacunary power series with carefully tuned coefficients) and track $\nu(r)$ as $r$ varies, looking for patterns that might suggest a constructive approach to liminf.

(iv) Minimal counterexample structure (if liminf cannot be infinite): one would expect a theorem producing infinitely many radii where the maximum modulus is attained essentially by a single "dominant direction" (few maxima), perhaps tied to regularity of the maximum modulus function $M(r)$ or to Wiman--Valiron type concentration; a minimal counterexample to the affirmative liminf conjecture would be an entire function whose boundary modulus does not admit such concentration at infinitely many radii.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.