[#P2690] Flattest 32-term Littlewood polynomial on the unit circle

A neutral schematic of the objects and relations in the statement.

Problem. For signs \(\varepsilon_0,\ldots,\varepsilon_{31}\in\{-1,1\}\), determine \(\min_{\varepsilon}\max_{|z|=1}|\sum_{j=0}^{31}\varepsilon_jz^j|\).

1Context

The L2 norm gives a lower bound sqrt(32)=5.6568542495. The current incumbent has a certified coarse upper bound below 7.7651.

2Remarks

Remark 1. Multiplying every sign by -1 leaves the objective unchanged, so epsilon_0 may be fixed to 1.

Remark 2. The maximum is continuous over the full unit circle, rather than over a sampled grid.

3What counts as a solution

Give a sign vector and its rigorously isolated circle maximum, plus a complete symmetry-reduced exclusion certificate for every smaller peak.

1Status

Current status (The minimum peak lies between 1064^(1/4) and 7.7174713). Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.^[1]

1Packet records

Record	Kind	Assessment
	Result	Reproduced
claim · Computation 1 Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.^[1] Relevance to this problem For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-claim-certified-interval (“The minimum peak lies between 1064^(1/4) and 7.7174713”) records a bound, answer, status fact, or structural consequence. The record states: Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. Evidence ReproducedA computation someone reran from the artifact on this page. Record state established Scope all 32-term Littlewood polynomials on the complex unit circle Sources Reference 1 Details Write \[ C_k=\sum_{j=0}^{31-k}\varepsilon_j\varepsilon_{j+k}. \] The normalized fourth moment is \[ \\|P\\|_4^4=32^2+2\sum_{k=1}^{31}C_k^2. \] For odd \(k\), the integer \(C_k\) is odd. For even \(k\), it is even. The parity floor is therefore \(\sum C_k^2\geq16\). Equality would make every odd-shift correlation equal to \(\pm1\) and every even-shift correlation zero, which is a Barker sequence of length 32. Turyn proved that an even Barker length greater than four must have the form \(4u^2\). Since 32 has no such form, equality is impossible. The next possible increase in the integer energy is four, so \[ \sum_{k=1}^{31}C_k^2\geq20. \] Since \(\\|P\\|_\infty\geq\\|P\\|_4\), every polynomial in the candidate family satisfies \[ \\|P\\|_\infty\geq1064^{1/4}=5.7113057054\ldots. \] The interval artifact proves that the displayed 32-sign polynomial has circle maximum below \(7.7174713\). Hence the present certified result is \[ \boxed{1064^{1/4}\leq\min_P\\|P\\|_\infty<7.7174713}. \] The exact minimum remains open in this record. Open full record →
	Trace	Inconclusive
attempt · Route 1 Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.^[2][3][1] Relevance to this problem For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-attempt-symmetry-and-literature-audit (“Symmetry leaves 536,887,296 reversal orbits”) documents a concrete method, search boundary, or failed route. The record states: Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum. Evidence InconclusiveThe recorded search or audit ended without settling the question. Scope symmetries and primary literature relevant to the exact 32-term Littlewood circle-peak problem Sources Reference 2, Reference 3, Reference 1 What happened Global sign change, substitution \(z\mapsto-z\), and coefficient reversal preserve the circle peak. Global sign and alternation select a unique representative with \(\varepsilon_0=\varepsilon_{31}=1\), leaving \(2^{30}\) sequences. Reversal has \(2^{15}\) fixed representatives in this normalization. Burnside's lemma therefore leaves \[ \frac{2^{30}+2^{15}}2=536{,}887{,}296 \] reversal orbits. This count was checked by the artifact, though those orbits were not enumerated. The 500 seeded one-flip descents recorded in the candidate found the displayed incumbent. They carry no exclusion force. Balister, Bollobás, Morris, Sahasrabudhe, and Tiba prove the existence of uniformly flat Littlewood polynomials at every length, with absolute constants. Erdélyi surveys the sup-norm flatness problem and the Rudin-Shapiro construction. Turyn's even-Barker restriction supplies the finite lower bound used here. Focused searches for fixed-degree tables, length 32, degree 31, minimum circle maximum, and Littlewood sup norm found no primary-source table giving this exact optimum. A complete proof can continue with the 536,887,296 normalized reversal orbits, using interval lower bounds to discard every orbit below the incumbent. Related: Computation 1 Open full record →
	Artifact	Reproduced
artifact · Artifact 1 A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic. Relevance to this problem For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-artifact-fixed-point-circle-bound (“Exact fixed-point interval certificate for the incumbent”) supplies evidence or a replay used to check the packet. The record states: A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic. Evidence ReproducedA computation someone reran from the artifact on this page. Record state available Scope rigorous full-circle peak bounds for the displayed 32-term Littlewood polynomial Entry point `join source_lines with newline and run with python3` Runtime CPython 3.9 or later, standard library only Details The coefficient sequence, in increasing exponent order, is \[ (1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,1,-1,1,-1, 1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1). \] For its aperiodic correlations \(C_k\), the verifier uses the exact identity \[ F(t)=\|P(e^{it})\|^2=32+2\sum_{k=1}^{31}C_k\cos(kt). \] Cosines on a \(2^{18}\)-point grid are enclosed at scale \(10^{30}\). The program starts from explicit rational bounds on \(\pi\), reduces every angle to \([0,\pi/2]\), and evaluates 24 terms of the alternating cosine series with outward integer rounding. The next term bounds the remainder. Sines are obtained by a quarter-turn of the same cosine table. For a grid point \(t_m\), Taylor's theorem gives \[ F(t)\leq F(t_m)+\|F'(t_m)\|h+\tfrac12Bh^2, \qquad h=\pi/2^{18}, \] where the exact coefficient bound is \[ B=2\sum_{k=1}^{31}k^2\|C_k\|=21264. \] Every point of the circle is within \(h\) of a grid point. The resulting all-cell comparison proves \[ 7.7174710<\max_{\|z\|=1}\|P(z)\|<7.7174713. \] A separate 100-decimal Newton evaluation places the apparent maximizing angle at \(0.9689875113845837543182660733966\ldots\) and the peak at \(7.717471112388378843091998621392\ldots\), consistent with the certified enclosure. The integer certificate supplies the stated bound without relying on that Newton calculation. Related: Computation 1 Open full record →
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Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-25. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. An incumbent sign vector is [1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,1,-1,1,-1,1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1].

Computational notes

Five hundred seeded one-flip descents evaluated on a 32768-point FFT grid found maximum 7.717469866649506 for the displayed signs. The derivative bound adds at most 0.047553404424455 between grid points, and direct coefficient squaring verified L2 norm sqrt(32).

How the 3 records connectTyped relations and evidence flow

ProblemFlattest 32-term Littlewood polynomial on the unit circle

Computation 1The minimum peak lies between 1064^(1/4) and 7.7174713in this packetReproduced
Artifact 1Exact fixed-point interval certificate for the incumbentsupportsReproduced
Route 1Symmetry leaves 536,887,296 reversal orbitsinformsInconclusive

2See also

Bochner-Riesz conjecture in higher dimensionsharmonic analysis
Openness of convolution on l1 of the integersharmonic analysis
Sharp fourth-power norm of the cyclic Hilbert transform at order 31harmonic analysis

How to cite

TheoremDB contributors, “Flattest 32-term Littlewood polynomial on the unit circle,” TheoremDB research memory, snapshot of July 25, 2026. https://theoremdb.org/statements/littlewood-32-minimum-peak

This page as plain text: littlewood-32-minimum-peak.md

This problem includes 3 records joined by 2 typed links, sourced from doi.org^[2], current as of July 25, 2026.

1References

Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound; Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.The minimum peak lies between 1064^(1/4) and 7.7174713. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.Also cited at R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound.For Flattest 32-term Littlewood polynomial on the unit circle: The minimum peak lies between 1064^(1/4) and 7.7174713. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.
Packet source. Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.For Flattest 32-term Littlewood polynomial on the unit circle: Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.Source named by the research packet.
Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle. people.tamu.edu checked 2026-08-01. Author-hosted manuscript. ↗preprint · primary source · PDF checked 2026-07-25 · checked 2026-07-25Source use: original summary.Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.For Flattest 32-term Littlewood polynomial on the unit circle: Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.

Original CC0 fixed-length flat-polynomial optimization.

[#P2690] Flattest 32-term Littlewood polynomial on the unit circle

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance