Erdős #119 — max modulus of unimodular-root products

Statement

Let $z_i$ ($i\geq 1$) be an infinite sequence of complex numbers with $\lvert z_i\rvert=1$ for all $i$, and for $n\geq 1$ let $p_n(z)=\prod_{i\leq n}(z-z_i)$. Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Three nested questions, in increasing order of strength: 1. Is it true that $\limsup_n M_n=\infty$? 2. Is it true that there exists $c>0$ such that $M_n>n^c$ for infinitely many $n$? 3. Is it true that there exists $c>0$ such that, for all large $n$, $\sum_{k\leq n}M_k > n^{1+c}$?

Facts

- Prize \$100; site status OPEN, and the site explicitly notes "this is open, and cannot be resolved with a finite computation" (erdosproblems.com/119, last edited 23 Jan 2026 — reflects the site owner's belief). - Falsifiable: no — all three questions are $\limsup$/for-all-large-$n$ asymptotic statements about an infinite object (an infinite unimodular sequence $\{z_i\}$); none reduce to a finite counterexample search. Needs a genuine proof either way. - Origin: attributed to Erdős as Problem 4.1 in Hayman's problem list [Ha74]; restated in [Er57], [Er61], [Er64b], [Er82e], [Er90], [Er97f], [Va99, 2.38]. - Known results (both weaker sub-questions are already fully resolved in the literature): - Q1 (limsup $=\infty$): proved by Wagner [Wa80] ("On a problem of Erdős in Diophantine approximation," Bull. LMS 12 (1980), 81–88), who in fact shows the stronger quantitative bound $M_n > (\log n)^c$ for some $c>0$, infinitely often. - Q2 (power growth i.o.): proved by Beck [Be91] ("The modulus of polynomials with zeros on the unit circle: A problem of Erdős," Annals of Math. 134 (1991), 609–651), who shows $\max_{n\leq N}M_n > N^c$ for some $c>0$. - Matching upper-bound constructions exist, showing Q2's exponent cannot be pushed to "for all $n$": Erdős himself (see [Ha74]) constructed a sequence with $M_n \leq n+1$ for every $n$; Linden [Li77] improved this to $M_n \ll n^{1-c}$ for some $c>0$, for a suitable sequence. So $M_n$ individually *can* be kept polynomially small along the whole sequence — the open Q3 is specifically about whether the *cumulative sum* $\sum_{k\leq n}M_k$ must still be forced super-linear even though the pointwise $M_n$ is not. - Q3 (the actual $\$100$ open problem): no progress found anywhere in the online literature since Beck (1991); erdosproblems.com states plainly "the third question seems to remain open," and no arXiv/OpenAlex/Semantic-Scholar/Google-Scholar hit post-dating Beck (1991) or Erdélyi's related 2008 refinement addresses it. - No AI-system attempt on #119 is logged in github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erdős-problems (checked directly, no match for "119" among the ~40+ logged AI-assisted Erdős-problem resolutions as of Jun 2026). - Related problems: erdos/228 erdos/230 erdos/525 — three sibling "extremal modulus of a unimodular-object polynomial on the unit circle" problems, all solved (see Related).

Literature state

Two of the three nested sub-questions are fully resolved (Wagner 1980, Beck 1991); the strongest form (Q3, the sum bound, which is the substance of the remaining \$100) is genuinely open with zero traceable progress since 1991. This makes the site's blanket "OPEN" status slightly coarse-grained — it is accurate for the prize-bearing claim (Q3) but the problem as a research object is *partially resolved*.

- Wagner's and Beck's proofs both sit in the discrepancy-theory / Erdős–Turán tradition: the classical Erdős–Turán theorem (see K. Soundararajan, "Equidistribution of zeros of polynomials," arXiv:1802.06506, full text read) bounds the angular discrepancy $D(P)$ of a monic polynomial's zeros by $O(\sqrt{N\cdot h(P)})$ where $h(P)=\frac1{2\pi}\int\log^+|P(e^{i\theta})|\,d\theta/\sqrt{|a_0|}$. Since here $\lvert a_0\rvert=\prod\lvert z_i\rvert=1$ exactly and $h(p_n)\leq\log M_n$, small $M_n$ forces the (already-on-the-circle) points $z_1,\dots,z_n$ to be angularly equidistributed. Wagner's actual argument (per WebSearch of secondary sources) runs this in reverse: it uses classical discrepancy *lower* bounds (van der Corput/Roth-type $\Omega(\log N)$ irregularity for any $N$-point set on the circle) to force $M_n\gg(\log n)^c$ infinitely often — i.e., a set that stays too "regular" for too long is impossible. Beck's 1991 Annals paper strengthens this using his own combinatorial/potential-theoretic discrepancy machinery (Beck is a discrepancy-theory specialist; the paper title explicitly frames it as an Erdős problem) to get power-law (not just log-law) growth, i.o. Full technique details are behind a paywall (Annals of Math 1991); only the theorem statements were independently verified via erdosproblems.com's remarks + secondary abstracts. - T. Erdélyi, "An improvement of the Erdős–Turán theorem on the distribution of zeros of polynomials," C. R. Acad. Sci. Paris / Comptes Rendus Math. 346 (2008), gives a related but distinct sharpening (bounding zero *multiplicity* via a one-sided Erdős–Turán self-improvement) — same toolkit family, not a direct advance on Q3. - No paper found addressing the aggregate/sum form (Q3) specifically. Search covered: arXiv full-text search, OpenAlex (rate-limited, retried), Semantic Scholar (rate-limited), Google-Scholar-via-WebSearch, the erdosproblems.com forum thread for #119 (0 comments, confirmed empty), and the teorth/erdosproblems AI-contributions wiki (no entry for 119). - Sibling family context (D.S. Lubinsky, "A survey of Erdős–Szekeres products," 2023, full PDF read): the closely related Erdős–Szekeres problem on $\prod_{j\le n}(1-z^{s_j})$ (roots restricted to roots of unity, a special case of the #119 setup) shows that *this* general family of "how small can a unimodular-root product be kept" problems has gone through decades of technique escalation: Erdős–Szekeres (1959, elementary Diophantine-approximation estimate) → Atkinson 1961 (Fourier-series/log-sine-series estimate) → Odlyzko 1982 (random trigonometric polynomials + nonnegative-cosine-polynomial estimates) → Kolountzakis (refined random construction) → Belov–Konyagin 1996 (nonnegative-trigonometric-polynomial theory) achieving $f(n)=\exp(O((\log n)^4))$ — i.e. *sub-polynomial*, which actually falsifies Erdős's own conjecture that $f(n)$ should grow faster than every power of $n$ for that restricted-root variant. This is a cautionary/instructive analog: in this problem family, "must grow polynomially" conjectures have been proven true in some variants (Q1, Q2 of #119) and *false* in others (Erdős–Szekeres roots-of-unity variant) — so Q3's truth value is not a foregone conclusion.

Attack surface

- Mode: literature-resolution (mostly exhausted — confirmed no post-1991 progress on Q3) + derivation+formalization (the only realistic path: extend/strengthen Beck's 1991 method). - Concrete first experiment: (a) obtain and closely read Beck [Be91] (Annals of Math. 134 (1991), 609–651) to extract the exact combinatorial/potential-theoretic mechanism giving $\max_{n\leq N}M_n>N^c$, and check whether it is a "single witness in $[1,N]$" argument or already gives some persistence/density of large-$M_k$ indices; (b) numerically explore, for small-to-moderate $n$ (say $n\leq 200$), the extremal sequences minimizing $\sum_{k\leq n}M_k$ via direct numerical optimization (gradient descent / basin-hopping over $n$ points on the circle, computing $M_k=\max_{|z|=1}|p_k(z)|$ by dense sampling + local refinement for each prefix $k\leq n$) to get empirical growth-rate data — this cannot prove anything (falsifiability is not-finite) but can suggest whether the extremal cumulative sum looks like $\Theta(n\log n)$ (i.e. Q3 false-ish) or genuinely super-linear-by-a-power. - Oracle: none mechanical for the actual open question (not finite-checkable); for the numerical-exploration sub-experiment, $M_k$ for a candidate finite sequence is directly computable (compact optimization on the circle), giving a lower bound on the true extremal sum that can sanity-check any proposed proof strategy. - Feasibility: hard, but well-scoped as a derivation target — the natural move is to try to "localize" Beck's 1991 discrepancy/potential-theoretic argument from "there exists a large-$M_k$ witness somewhere in $[1,N]$" to "every sufficiently long dyadic block of indices contributes enough to force $\sum M_k \gg n^{1+c}$," i.e. a persistence/amortization strengthening of an already-published, highly technical (60-page Annals) argument. This is squarely a case where the win condition is a genuine new theorem, not a computation; realistic only with either (i) a close technical read of Beck's proof to see if the method already secretly gives more, or (ii) importing a different technique (e.g. the nonnegative-trigonometric-polynomial / random-construction toolkit that both proved and disproved analogous questions in the Erdős–Szekeres sibling family) adapted to the aggregate-sum setting.

Related

- erdos/228 — Littlewood's flat-polynomial conjecture: $\pm1$-coefficient degree-$n$ polynomial with $\delta\sqrt n\leq|P(z)|\leq\Delta\sqrt n$ on $|z|=1$. PROVED by Balister, Bollobás, Morris, Sahasrabudhe, Tiba (Annals of Math. 192 (2020); arXiv:1907.09464) via an explicit iterative/decoupling probabilistic construction. Sibling "extremal modulus of a unimodular-object polynomial on the circle" problem, solved by heavy probabilistic-combinatorics machinery decades after being posed — a template for what it might take to crack #119's Q3. - erdos/230 — Erdős–Newman ultraflat conjecture: is $\max_{|z|=1}|P(z)|\geq(1+c)\sqrt n$ for all $\pm1$-coefficient $P$? DISPROVED: Kahane [Ka80] constructed ultraflat $\pm$unimodular-coefficient polynomials with $P(z)=(1+o(1))\sqrt n$ uniformly on $|z|=1$ (improved by Bombieri–Bourgain [BoBo09] to an explicit error term $O(n^{7/18}(\log n)^{O(1)})$). Cautionary sibling: shows a "must be large" conjecture in this exact problem family can be *false*, via a clever random-phase (Steinhaus-variable) Fourier construction — relevant caution for Q3 of #119. - erdos/525 — Littlewood's conjecture on the *minimum* modulus of a random $\pm1$ (Kac/Rademacher) polynomial on the unit circle. PROVED: Kashin [Ka87] (existence of $m(f)=o(1)$ a.s.), sharpened by Konyagin [Ko94] to $m(f)\leq n^{-1/2+o(1)}$ a.s., matched by Konyagin–Schlag [KoSc99], limiting distribution found by Cook–Nguyen (2021). Technique lineage: small-ball/anti-concentration probability — a different, purely probabilistic toolkit than Beck's discrepancy-theoretic one, worth trying against #119's Q3. - solved/erdos-szekeres-product-growth — sibling problem (roots restricted to roots of unity) on $f(n)=\inf\|\prod_{j\le n}(1-z^{s_j})\|_{L^\infty(|z|=1)}$: resolved (in the *disproving* direction of Erdős's own conjectured super-polynomial growth) via Odlyzko 1982 / Kolountzakis / Belov–Konyagin 1996 ($f(n)=\exp(O((\log n)^4))$, using nonnegative-trigonometric-polynomial theory). Direct technique-transfer candidate: nonnegative trig/cosine polynomial constructions. - solved/erdos-szekeres-liminf-irrational — sibling limsup/liminf question for $\prod(z-e^{2\pi i j\alpha})$, $\alpha$ irrational: resolved by Avila, Jitomirskaya, Marx using ergodic-theory/quasiperiodic-Schrödinger-operator methods (the "Ten Martini"-adjacent toolkit) — shows this problem family has already been cracked once by techniques totally outside discrepancy/potential theory. - concept/erdos-turan-inequality — the discrepancy-vs-log-modulus inequality (Soundararajan, arXiv:1802.06506) underlying both Wagner's and (likely) Beck's proofs; the core bridge between "$M_n$ small" and "zeros equidistributed." - concept/discrepancy-theory — van der Corput/Roth-type lower bounds on point-set discrepancy on the circle; Beck's home field, used to force $M_n$ growth. - concept/logarithmic-potential-theory — capacity/transfinite-diameter extremal-polynomial framework plausibly underlying Beck's 1991 argument. - concept/nonnegative-trigonometric-polynomials — Odlyzko/Kolountzakis/Belov–Konyagin's toolkit for constructing small-product sequences in the sibling Erdős–Szekeres problem; candidate transfer technique for Q3 upper-bound constructions. - concept/anti-concentration — small-ball probability methods (Kashin, Konyagin, Konyagin–Schlag, Cook–Nguyen) that solved the sibling min-modulus problem erdos/525; candidate alternate toolkit for #119. - concept/probabilistic-polynomial-construction — Kahane's random-phase (Steinhaus) construction and BBMST's iterative decoupling construction, both used to build extremal unimodular polynomials on the circle in the sibling problems.

PRIOR-ART SCAN (2026-07-03)

Verdict: HARD-OPEN (Q1 and Q2 are RESOLVED-IN-LITERATURE — confirmed independently again below; Q3, the actual \$100 question, has zero traceable progress since Beck 1991 and no evidence of current active work on it specifically).

This is a re-verification pass, prompted by the #64 lesson (we once claimed novelty for something already sitting in a public repo). This time the repo check came up clean: google-deepmind/formal-conjectures has a Lean formalization of #119, and it independently confirms the "Q1/Q2 solved, Q3 open" status rather than contradicting it.

What was checked

- Direct re-fetch of erdosproblems.com/119 (curl with browser UA, 200 OK) — statement, remarks (Wagner [Wa80], Beck [Be91], Erdős/Linden [Li77] constructions), "0 comments on this problem," last edited 23 Jan 2026. Matches what's already recorded in this page's Facts section verbatim. - GitHub repo scan (the #64-lesson check): github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/119.lean — a faithful Lean 4 formalization of all three sub-questions, added in PR #1225 ("Erdos 119", 2025-11-27, fixes #360), mechanically renamed in a later PR (2026-01-06), and touched again in a bulk status-audit commit fd086a79 (2026-02-20, "Went through all of these by hand and updated statuses accordingly" — diff inspected directly, confirms this was a pure identifier rename with no status change). Current tags in the file: erdos_119.parts.i and .ii are @[category research solved] (both proved as answer(True) ↔ ... with sorry proof bodies, citing Wagner and Beck exactly as erdosproblems.com does); erdos_119.parts.iii is @[category research open] with answer(sorry). No additional references, no hidden proof sketch, no third-party comment thread on the PR suggesting new progress. This independently corroborates (not contradicts) the "Q3 open" status as of Feb 2026 — the closest thing to a live prior-art trap here, and it checked out clean. - Checked github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erdős-problems directly (fetched, grepped) — no entry for problem 119 among the logged AI-assisted attempts. - WebSearch sweep for 2023–2026 arXiv/Scholar activity: arXiv:2501.10333 ("Resolution of Erdős' problems about unimodularity," Stijn Cambie) initially looked promising by keyword match but on inspection is about *unimodality of an arithmetic density sequence* (a completely different "unimodal/unimodular" sense) and addresses erdosproblems.com #690/#692, not #119 — false positive, ruled out. arXiv:2407.15306 (Mithun Kumar Das, "Distribution of the zeros of polynomials near the unit circle," extends Borwein–Erdélyi–Littmann 2008) and arXiv:2109.11006 ("The Sharp Erdős–Turán Inequality") plus arXiv:2104.00105 ("Hilbert transforms and the equidistribution of zeros of polynomials") are genuine 2021–2024 advances in the same discrepancy/Erdős–Turán toolkit that underlies Wagner's and Beck's proofs — but none of them touch the aggregate-sum form (Q3), and none cite or claim to extend Beck 1991 in that direction. No paper found anywhere addressing $\sum_{k\le n}M_k$ specifically. - Forum thread erdosproblems.com/forum/thread/119 re-confirmed empty (0 comments) directly on the fetched page.

Sources actually read this pass

https://www.erdosproblems.com/119 (curl fetch, 2026-07-03) · https://raw.githubusercontent.com/google-deepmind/formal-conjectures/main/FormalConjectures/ErdosProblems/119.lean · https://api.github.com/repos/google-deepmind/formal-conjectures/commits?path=FormalConjectures/ErdosProblems/119.lean (commit history) · https://api.github.com/repos/google-deepmind/formal-conjectures/commits/fd086a79 (diff of the bulk status-audit commit) · https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erdős-problems · arXiv:2501.10333 (ruled out as off-topic) · arXiv:2407.15306, arXiv:2109.11006, arXiv:2104.00105 (tangential discrepancy-theory advances, none resolving Q3) · WebSearch sweeps for "Erdos problem 119," "Beck 1991 modulus polynomials citing papers," "Wagner 1980 Diophantine approximation citations," "github erdosproblems 119 lean formalization."

No concrete first experiment is added here per the task's own rule (only required for FRESH-AND-TRACTABLE); the existing Attack surface section above already lays out the realistic path (close technical read of Beck 1991 + numerical exploration of extremal cumulative sums) and nothing found in this pass changes that assessment.