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ERDőS #1117 · PARTIAL
Community database OPENas of 2025-12-29tags analysisproblem page · erdosproblems.com/1117

Erdős problem #1117 — exact finite-regime obstruction to upgrading approximate maxima

Date: 2026-07-31 Live page: <https://www.erdosproblems.com/1117> Discussion page: <https://www.erdosproblems.com/forum/discuss/1117>

Claim labels

0. Mandatory live-page gate

The page was fetched through the Bright Data browser path on 2026-07-31. Direct datacenter curl was not used as the authority.

Verbatim live statement

Let \(f(z)\) be an entire function which is not a monomial. Let \(\nu(r)\) count the number of \(z\) with \(\lvert z\rvert=r\) such that \(\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert\). (This is a finite quantity if \(f\) is not a monomial.)

Is it possible for \[ > \limsup \nu(r)=\infty? > \] Is it possible for \[ > \liminf \nu(r)=\infty? > \]

Everything else shown by the live page

to Erdős.

affirmatively.

affirmative answer by Glücksam and Pardo-Simón [GlPa24].

and Pardo-Simon paper can also be found on arXiv: https://arxiv.org/abs/2208.11154”. It was posted by ebarschkis at 14:56 on 14 February 2026.

Thus neither mandatory stop condition fired: the live page shows no claimed proof/solved/falsified status and no current worker.

1. Primary-source literature audit

1.1 Herzog–Piranian and the first question

(b) F. Herzog and G. Piranian, The counting function for points of maximum modulus, Proceedings of Symposia in Pure Mathematics 11 (1968), 240–243, exists with DOI 10.1090/pspum/011/0257359. The AMS volume metadata and DOI were verified. The original four-page article was behind the AMS volume access wall in this run, so I do not claim a direct reading of its proof.

(b) The precise historical claim \(v_f(n)=n\) for every positive integer \(n\) is independently stated in both the 2024 Glücksam–Pardo-Simón paper and the 2026 Pardo-Simón–Sixsmith preprint below. This proves the limsup part but does not control radii outside the integers.

1.2 The 2024 approximate result

(b) A. Glücksam and L. Pardo-Simón, An approximate solution to Erdős' maximum modulus points problem, arXiv:2208.11154v2, is the author-accepted manuscript of J. Math. Anal. Appl. 531 (2024), article 127768, DOI 10.1016/j.jmaa.2023.127768. The arXiv identifier, authors, versions, journal metadata, and PDF contents were all checked.

For an entire function \(g\), write

\[ M_g(r)=\max_{|z|=r}|g(z)|, \]

and let

\[ \begin{aligned} V_g(r,\varepsilon) &=\#\pi_0\{t\pmod {2\pi}:|g(re^{it})|>M_g(r)-\varepsilon\},\\ W_g(r,\varepsilon) &=\#\pi_0\{t\pmod {2\pi}:|g(re^{it})|<\varepsilon\}. \end{aligned} \]

(b) Their Theorem 1.2 constructs one transcendental entire \(g\) such that, for every \(\varepsilon>0\),

\[ \lim_{r\to\infty}V_g(r,\varepsilon) =\lim_{r\to\infty}W_g(r,\varepsilon)=\infty. \]

The paper explicitly says that its approximation method does not show how many near-maximum components actually contain a true maximum-modulus point; that number could, on the estimates given there, remain uniformly bounded. The constructed function has infinite lower order.

1.3 Finite lower order and the newest result found

(b) I. I. Marchenko, On the maximum modulus points of entire and meromorphic functions and a problem of Erdős, Matematychni Studii 38 (2012), 212–215, DOI 10.30970/ms.38.2.212-215, states as Conjecture 1 that every non-monomial entire function of finite lower order should satisfy

\[ \liminf_{r\to\infty}\nu(r)<\infty. \]

This is a conjecture, not a theorem.

(b) A very recent source, postdating the live page, is L. Pardo-Simón and D. J. Sixsmith, A finite-order answer to a problem of Erdős on maximum modulus points, arXiv:2607.09462v1, submitted 10 July 2026. Its Theorem 1.1 constructs a finite-order Eremenko–Lyubich-class entire function with \(\limsup_{r\to\infty}\nu(r)=\infty\). Its introduction explicitly says the liminf question remains open. Thus this new preprint strengthens the known answer to the first question and is consistent with Marchenko's liminf conjecture; it does not change the live page's answer to the second question.

(b) Searches by exact problem phrase, exact title, “liminf maximum modulus points”, arXiv, DOI chains, and the references of the 2024 and 2026 papers also found work on separated maximum-modulus points, including Marchenko (2022) and Kowalski–Marchenko (2025). Those results count components of superlevel sets which contain maxima; they do not claim a bound or construction for the raw number \(\nu(r)\).

(c) A literature search cannot certify nonexistence, but I found no primary source claiming a solution of the liminf question. The newest directly relevant primary source found, arXiv:2607.09462v1, expressly calls it open.

2. New exact finite-regime result

The following construction is not a solution of the liminf problem. It is an exact counterexample to the most tempting local bridge from the 2024 theorem: many alternating near-maximal and near-zero arcs do not force even two true maximum points.

Lemma 2.1: component bound for a trigonometric polynomial

(a) If \(N\geq1\) and \(T\) is a nonzero real trigonometric polynomial of degree at most \(N\), then each of

\[ \{t\pmod {2\pi}:T(t)>0\},\qquad \{t\pmod {2\pi}:T(t)<0\} \]

has at most \(N\) connected components.

Proof. If either sign set is empty or the whole circle, the assertion is immediate because \(N\geq1\). Otherwise, a nonzero trigonometric polynomial of degree \(N\) has at most \(2N\) zeros on the circle, counted with multiplicity: after multiplying its Laurent form by \(z^N\), one obtains a nonzero ordinary polynomial of degree at most \(2N\). Each component of \(\{T>0\}\) has two boundary incidences. A sign-changing boundary zero supplies one incidence and has odd multiplicity at least one; a zero with \(T>0\) on both sides supplies two incidences and has even multiplicity at least two. Hence \(2\,\#\pi_0\{T>0\}\leq 2N\). Apply the same argument to \(-T\). \(\square\)

Theorem 2.2: exact near-peak/true-peak separation

(a) For every integer \(N\geq2\), define

\[ p_N(z)=1+z^N+\frac{z}{100},\qquad \varepsilon=\frac1{10}, \]

and

\[ I_N=\left[1-\frac1{25N},\,1+\frac1{25N}\right]. \]

Then:

  1. for every \(r>0\),

\[ \nu_{p_N}(r)=1; \]

  1. for every \(r\in I_N\),

\[ V_{p_N}(r,\varepsilon) =W_{p_N}(r,\varepsilon)=N. \]

Consequently, for any scale \(R>0\), the polynomial

\[ p_{N,R}(z)=p_N(z/R) \]

has exactly the same phenomenon on the arbitrarily large annulus \(r/R\in I_N\).

Proof of the unique true maximum

(a) Put \(\delta=1/100\). On \(|z|=r\),

\[ |p_N(re^{it})| \leq 1+r^N+\delta r. \]

Equality holds at \(t=0\). Equality in the triangle inequality anywhere else would require the three nonzero summands \(1\), \(r^Ne^{iNt}\), and \(\delta re^{it}\) to have the same argument. The constant term fixes that argument as zero, and \(e^{it}=1\) then forces \(t=0\pmod{2\pi}\). Therefore

\[ M_{p_N}(r)=1+r^N+\delta r \quad\text{and}\quad \nu_{p_N}(r)=1 \]

for every \(r>0\).

Exact rational bounds on the annulus

(a) For \(r\in I_N\), Bernoulli's inequality gives

\[ r^N\geq\left(1-\frac1{25N}\right)^N\geq\frac{24}{25}. \]

For the upper bound, use \(\binom Nk\leq N^k\):

\[ \left(1+\frac1{25N}\right)^N \leq\sum_{k=0}^{N}\left(\frac1{25}\right)^k <\frac{25}{24}. \]

Also \(r\leq51/50\). Thus

\[ |1-r^N|\leq\frac1{24}, \qquad \delta r\leq\frac{51}{5000}. \]

The exact constants needed below are

\[ \frac1{24}+\frac{51}{5000} =\frac{389}{7500}<\frac1{10}, \qquad 2\delta r\leq\frac{51}{2500}<\frac1{10}, \]

and

\[ 1+r^N-\delta r \geq 1+\frac{24}{25}-\frac{51}{5000} =\frac{9749}{5000}>\frac1{10}. \]
\(N\) alternating witnesses

(a) Let

\[ \alpha_j=\frac{2\pi j}{N},\qquad \beta_j=\frac{(2j+1)\pi}{N}, \qquad 0\leq j<N. \]

At a peak witness \(\alpha_j\), \(e^{iN\alpha_j}=1\), so

\[ |p_N(re^{i\alpha_j})| \geq1+r^N-\delta r >M_{p_N}(r)-\varepsilon. \]

The last inequality follows because the deficit from \(M_{p_N}(r)\) is at most \(2\delta r<\varepsilon\). These points are also outside the near-zero set.

At a separator \(\beta_j\), \(e^{iN\beta_j}=-1\), so

\[ |p_N(re^{i\beta_j})| \leq |1-r^N|+\delta r \leq\frac{389}{7500}<\varepsilon. \]

Moreover,

\[ M_{p_N}(r)-\varepsilon \geq1+\frac{24}{25}-\frac1{10} =\frac{93}{50}, \]

so every \(\beta_j\) is outside the near-maximum set.

The \(\alpha_j\)'s and \(\beta_j\)'s alternate around the circle. Therefore the near-maximum set has at least \(N\) components, one containing each \(\alpha_j\), and the near-zero set has at least \(N\) components, one containing each \(\beta_j\).

Finally,

\[ t\longmapsto |p_N(re^{it})|^2 \]

is a real trigonometric polynomial of degree \(N\), since the exponent support is \(\{0,1,N\}\). Subtracting either squared threshold leaves a nonzero trigonometric polynomial of degree at most \(N\). Lemma 2.1 bounds each component count by \(N\), so both lower bounds are exact. \(\square\)

Corollary 2.3: no circle-by-circle upgrade lemma

(a) There is no universal implication of the form

\[ \min(V_g(r,\varepsilon),W_g(r,\varepsilon))\geq N \quad\Longrightarrow\quad \nu_g(r)>1 \]

valid for all entire \(g\), even with the two families of arcs alternating and even when the hypotheses hold throughout a nondegenerate interval of radii. Theorem 2.2 supplies a counterexample for every \(N\), and scaling moves that interval to any prescribed size.

(a) More generally, no lower bound for \(\nu_g(r)\) tending to infinity with \(\min(V_g,W_g)\) can follow from those two component counts alone: Theorem 2.2 keeps \(\nu=1\) while making both counts equal to arbitrary \(N\).

3. Reproducible exact computation

The standalone standard-library checker is:

runs/erdos1117_wavew053_reverify.py

SHA-256:

5a3747528d2e5def91afe60393d79d6702a118de59de283ebe8a62a2e60874ea

Run it from the repository root with:

python runs/erdos1117_wavew053_reverify.py

It performs three independent checks.

  1. (a) fractions.Fraction arithmetic recomputes the universal constants

\(389/7500\), \(51/2500\), \(9749/5000\), and \(93/50\).

  1. (d) Direct exact endpoint powers, exponent-support gcd, and Fourier

degree are recomputed for \(2\leq N\leq80\).

  1. (d) A rational polynomial/Sturm implementation written from scratch

constructs \[ S_N(x)=2+\delta^2+2T_N(x)+2\delta x+2\delta T_{N-1}(x), \quad x=\cos t, \] and counts the roots of the two threshold equations at \(r=1\). For every \(2\leq N\leq24\), each equation has exactly \(N\) simple roots in \((-1,1)\), hence \(2N\) circle crossings and exactly \(N\) components.

The saved run completed with:

elapsed=0.59 sec maxrss=12892 KB
FINAL: ALL CHECKS PASSED

Selected exact Sturm rows were:

\(N\)roots in \((-1,1)\), near zeroroots in \((-1,1)\), near maxcircle components of each kind
2222
3333
4444
8888
12121212
16161616
20202020
24242424

(d) A separate SymPy rational root-count cross-check for \(2\leq N\leq24\), not used by the standalone checker, also passed.

4. What this resolves and what remains

(a) Theorem 2.2 is a closed-form, uniform-in-\(N\), rigorously checked counterexample to deriving true maxima from the stated conclusion alone of the approximate theorem. This makes the approximation gap precise rather than merely saying that near-maximal arcs “might” fail to contain maxima.

(a) It does not answer Erdős's second question. The polynomial changes with \(N\), and each finite-degree polynomial has only finite asymptotic complexity. The construction gives arbitrarily large finite-scale obstructions, not one entire function with \(\nu(r)\to\infty\).

(c) An affirmative upgrade of the 2024 construction needs information not encoded by \(V_g\) and \(W_g\): it must force many different peak branches to have exactly equal global height on every sufficiently large circle. Uniform approximation controls the height errors but does not make them equal; a generic small perturbation splits equal peaks. The missing statement is therefore an exact equioscillation/branch-locking lemma with uniformity in the continuum parameter \(r\).

(c) The 2026 tract-model paper illustrates the same boundary from another side: its shooting parameters equalize many channels at selected real parts, which proves limsup divergence. To prove liminf divergence by that route, one would need tract geometry or a continuum-in-\(r\) locking mechanism that equalizes an increasing number of channels on every vertical line. Neither the cited theorem nor the present calculation supplies that mechanism.

(b) On the negative side, Marchenko-type theorems bound separated maximum-modulus components for finite lower order. They do not bound the raw number of exact maxima that may lie in one component, and in any event the full Erdős question permits infinite lower order. Thus those theorems do not close the problem.

(c) Larger brute force is not the bottleneck. For a fixed degree and fixed radius, exact angular threshold/root isolation is inexpensive, as the checker demonstrates. No finite truncation or radius grid supplies the missing all-radii and infinite-tail argument. The obstruction is an analytic uniformity lemma, not an unrun finite computation, so assigning core-hours to a larger scan would be misleading.

PARTIAL: Explicitly proved and exactly checked that \(p_N(z)=1+z^N+z/100\) has one true maximum point on every circle but exactly \(N\) alternating near-maximum and near-zero components on an explicit annulus, ruling out any local component-count upgrade of the known approximate theorem while leaving the liminf problem open.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← our attempt log · problem #1117