On the maxima of Littlewood polynomials on $[-1,1]$

Brayden Letwin; Mehtaab Sawhney

arxiv: 2604.19294 · v1 · submitted 2026-04-21 · 🧮 math.PR · math.CO

On the maxima of Littlewood polynomials on [-1,1]

Brayden Letwin , Mehtaab Sawhney This is my paper

Pith reviewed 2026-05-10 02:13 UTC · model grok-4.3

classification 🧮 math.PR math.CO

keywords varepsilonfraclittlewoodpolynomialalmostcertaincoefficientsdetermined

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The pith

Random Littlewood polynomials satisfy an exact almost-sure liminf for the scaled log of their maximum on [-1,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the lower envelope of the maximum absolute value of a random Littlewood polynomial on the interval from -1 to 1 is governed by the small-ball probability of an approximating Gaussian process. This produces the precise constant in the liminf of log of that maximum divided by sqrt(n), scaled by (log log n) to the power one-third. A reader would care because it fixes the rate at which these maxima can fall below their typical size of order sqrt(n), infinitely often. The result resolves the asymptotic fluctuation of the maxima for polynomials with independent Rademacher coefficients.

Core claim

We show that the lower envelope of max_{x in [-1,1]} |f_n(x)| is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, liminf_{n to infty} [log (max |f_n(x)| / sqrt(n)) ] / (log log n)^{1/3} equals -(3 pi^2 /4)^{1/3}.

What carries the argument

The small-ball probability of the Gaussian process approximating the Littlewood polynomial, which supplies the probability that the process stays below a small threshold over the full interval.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same small-ball approach may extend to maxima of random polynomials on other compact sets or to trigonometric versions with random coefficients.
The result suggests that deterministic constructions of Littlewood polynomials with unusually small maxima could be guided by the paths that achieve the Gaussian small-ball events.
The constant derived here may appear in related extremal problems for random sign sums, such as L-infinity norms of partial sums or discrepancy measures on the circle.

Load-bearing premise

The small-ball probability of the approximating Gaussian process fully determines the lower envelope of the polynomial maxima, with no additional error terms from the discrete-to-continuous approximation that would alter the exact liminf constant.

What would settle it

Numerical evaluation of the liminf expression for sequences of large-n polynomials that stabilizes at a value different from the predicted constant, or an analytic demonstration that the approximation error shifts the constant.

read the original abstract

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3\pi^2}{4}\Big)^{1/3}. \]

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a precise almost-sure liminf constant for maxima of random Littlewood polynomials by importing Gaussian process small-ball asymptotics, but the discrete-to-continuous transfer needs to be checked for error terms at the (log log n)^{1/3} scale.

read the letter

The main takeaway is that the authors derive an exact constant -(3π²/4)^{1/3} for the liminf of log(max |f_n(x)| / sqrt(n)) over (log log n)^{1/3} almost surely, by linking the Rademacher polynomial to the small-ball probability of the limiting Gaussian process with covariance sum (xy)^k. This gives a sharp lower envelope beyond the usual sqrt(n) growth and appears new relative to earlier work on Littlewood polynomials. They do a clean job of setting up the connection and pulling in the known GP asymptotics without fitting parameters inside the paper. The argument structure looks logically sound on the surface and avoids obvious circularity by treating the small-ball result as external input. The soft spot is the approximation step itself. The stress-test concern about discretization nets, variance mismatch at x=1, and whether additive errors in log-probability stay smaller than the target scale is worth verifying in the main theorem. If the coupling or net argument controls the difference tightly enough, the constant holds; if not, the leading term could shift. Without the full details of section 3 it is hard to judge how tight the error bounds actually are. This is for readers working on random polynomials, suprema of Gaussian processes, or discrepancy theory. Someone following sharp almost-sure results on maxima would get direct value from the constant if the proof goes through. I would send it to peer review; the claim is specific and the approach is worth referee time even if the approximation details require extra work.

Referee Report

2 major / 2 minor

Summary. The paper claims that for Littlewood polynomials f_n(x) = sum_{k=0}^n ε_k x^k with i.i.d. Rademacher coefficients ε_k, the lower envelope of the maxima on [-1,1] is governed by small-ball probabilities of the limiting Gaussian process G(x) = sum g_k x^k. In particular, it proves the almost-sure limit inferior liminf_{n→∞} [log (max_{x∈[-1,1]} |f_n(x)| / sqrt(n)) ] / (log log n)^{1/3} equals -(3π²/4)^{1/3}.

Significance. If the central transfer holds, the result supplies a parameter-free exact constant for the lower envelope of random Littlewood polynomial maxima, obtained by importing known small-ball asymptotics of the Gaussian process rather than deriving them internally. This constitutes a clean link between discrete random polynomials and continuous Gaussian process theory.

major comments (2)

[Main approximation argument (likely §3)] The exact constant in the main result (abstract and presumably Theorem 1.1) requires that the discrete-to-continuous approximation error between f_n and G does not alter the leading (log log n)^{1/3} term in the log-probability of the small-ball event {max |·| ≤ sqrt(n) exp(-c (log log n)^{1/3})}. The paper must supply an explicit bound showing that any discrepancy arising from a net on [-1,1] or from the fact that Var(f_n(1)) = n+1 while the GP variance diverges at x=1 remains o((log log n)^{1/3}) in the exponent.
[Section 3 (or the main approximation theorem)] If the proof proceeds by coupling or discretization on a finite net, the mesh size and the local variance control near |x|≈1 must be shown to produce a multiplicative factor 1+o(1) in the probability at the precise scale needed for the liminf constant; otherwise the constant -(3π²/4)^{1/3} could shift.

minor comments (2)

[Abstract] The abstract expression for the liminf would benefit from explicit parentheses around the division to avoid parsing ambiguity.
[Introduction] Notation for the Gaussian process covariance ∑(xy)^k should be introduced once and used consistently; a short comparison table with prior Littlewood-polynomial results on the circle would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for emphasizing the need for explicit control on the approximation error between the random Littlewood polynomials and the limiting Gaussian process. We agree that this control is necessary to justify that the leading constant is unaffected. In the revised manuscript we will add a dedicated error-analysis subsection in Section 3 that supplies the required o((log log n)^{1/3}) bounds. Our point-by-point responses follow.

read point-by-point responses

Referee: The exact constant in the main result (abstract and presumably Theorem 1.1) requires that the discrete-to-continuous approximation error between f_n and G does not alter the leading (log log n)^{1/3} term in the log-probability of the small-ball event {max |·| ≤ sqrt(n) exp(-c (log log n)^{1/3})}. The paper must supply an explicit bound showing that any discrepancy arising from a net on [-1,1] or from the fact that Var(f_n(1)) = n+1 while the GP variance diverges at x=1 remains o((log log n)^{1/3}) in the exponent.

Authors: We agree that an explicit bound must be displayed. The proof discretizes [-1,1] on a net of cardinality exp(O((log log n)^{1/3})) whose mesh is chosen smaller than the modulus-of-continuity scale of G at the relevant height. The variance discrepancy at x=1 is bounded by 1 and therefore contributes an additive O(1/sqrt(n)) perturbation to the normalized field, which translates into an o((log log n)^{1/3}) error in the logarithm of the small-ball probability. A new lemma (to be inserted in §3) will record the precise comparison between the discrete and continuous probabilities, confirming that the ratio is 1+o(1) on the exponential scale needed for the liminf. The constant -(3π²/4)^{1/3} is therefore preserved. revision: yes
Referee: If the proof proceeds by coupling or discretization on a finite net, the mesh size and the local variance control near |x|≈1 must be shown to produce a multiplicative factor 1+o(1) in the probability at the precise scale needed for the liminf constant; otherwise the constant -(3π²/4)^{1/3} could shift.

Authors: The argument does rely on a finite net. The mesh size is taken to be exp(-C (log log n)^{1/3}) for a sufficiently large C; standard Gaussian-process tail estimates then show that the probability of the maximum on the net differing from the continuous maximum by more than the target height is negligible compared with the main small-ball probability. Near x=1 the local variance is controlled by truncating the Karhunen–Loève expansion after n terms and using the explicit covariance of G; the resulting multiplicative factor in the probability is exp(o((log log n)^{1/3})), which does not change the leading constant. These calculations will be written out explicitly in the revised Section 3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result rests on external small-ball asymptotics

full rationale

The paper establishes that the lower envelope of the Littlewood polynomial maxima is governed by the small-ball probabilities of the limiting Gaussian process G(x) = sum g_k x^k. The explicit constant -(3π²/4)^{1/3} is imported from known external asymptotics of that GP rather than derived or fitted inside the present work. The approximation theorems transferring the small-ball events from f_n to G are stated as independent analytic/probabilistic controls (discretization, coupling, variance matching) whose error terms are shown not to affect the leading (log log n)^{1/3} exponent; these controls do not reduce to the target liminf by definition or by self-citation chains. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard probabilistic tools rather than new postulates or fitted parameters.

axioms (2)

standard math Rademacher sums converge in law to a Gaussian process on [-1,1]
Invoked to justify the approximation step that reduces the polynomial problem to the Gaussian small-ball probability.
domain assumption Known small-ball probability asymptotics for the limiting Gaussian process
The paper uses these asymptotics to extract the exact constant (3π²/4)^{1/3}.

pith-pipeline@v0.9.0 · 5439 in / 1430 out tokens · 48631 ms · 2026-05-10T02:13:43.000251+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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work page 1954

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T. W. Anderson,The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc.6(1955), 170–176. 5

work page 1955

[2] [2]

Li, and Qi-Man Shao,Small deviations for a family of smooth Gaussian processes, J

Frank Aurzada, Fuchang Gao, Thomas Kühn, Wenbo V. Li, and Qi-Man Shao,Small deviations for a family of smooth Gaussian processes, J. Theoret. Probab.26(2013), 153–168. 2

work page 2013

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Theory Related Fields152(2012), 231–264

Sourav Chatterjee,A new approach to strong embeddings, Probab. Theory Related Fields152(2012), 231–264. 13

work page 2012

[4] [4]

Cordero-Erausquin, M

D. Cordero-Erausquin, M. Fradelizi, and B. Maurey,The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems, J. Funct. Anal.214(2004), 410–427. 4, 16

work page 2004

[5] [5]

Paul Erdős,Some unsolved problems, Publ. Math. Inst. Hungar. Acad. Sci.6(1961), 221–254. 1

work page 1961

[6] [6]

Li, and Jon A

Fuchang Gao, Wenbo V. Li, and Jon A. Wellner,How many Laplace transforms of probability measures are there?, Proc. Amer. Math. Soc.138(2010), 4331–4344. 1

work page 2010

[7] [7]

Davenport,Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences, Appl

Santhosh Karnik, Justin Romberg, and Mark A. Davenport,Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences, Appl. Comput. Harmon. Anal.55(2021), 97–128. 24, 25

work page 2021

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Komlós, P

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work page 1975

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Komlós, P

J. Komlós, P. Major, and G. Tusnády,An approximation of partial sums of independent R V’s, and the sample DF. II, Z. Wahrscheinlichkeitstheorie Verw. Gebiete34(1976), 33–58. 2, 13

work page 1976

[10] [10]

A. A. Laptev,Spectral asymptotic behavior of a class of integral operators, Math. Notes16(1974), 1038–1043. 24, 26

work page 1974

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Nazarov and Yulia Petrova,L2-small ball asymptotics for Gaussian random functions: a survey, Probab

Alexander I. Nazarov and Yulia Petrova,L2-small ball asymptotics for Gaussian random functions: a survey, Probab. Surv.20(2023), 608–663. 12

work page 2023

[12] [12]

Thomas Royen,A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, Far East J. Theor. Stat.48(2014), 139–145. 3

work page 2014

[13] [13]

Salem and A

R. Salem and A. Zygmund,Some properties of trigonometric series whose terms have random signs, Acta Math. 91(1954), 245–301. 1 Department of Mathematics, University of W ashington, Seattle, W A 98195 Email address:letwin@uw.edu Department of Mathematics, Columbia University, New York, NY 10027 and OpenAI Email address:{m.sawhney@columbia.edu, msawhney@ope...

work page 1954