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

- ID: `P2690`
- Reference: `littlewood-32-minimum-peak`
- Page: https://theoremdb.org/statements/P2690
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Remarks

- **Remark.** Multiplying every sign by -1 leaves the objective unchanged, so epsilon_0 may be fixed to 1.
- **Remark.** The maximum is continuous over the full unit circle, rather than over a sampled grid.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Computation 1 (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.

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.

### Background and intake notes

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

- Original intake status: Novelty remains unverified. Littlewood flatness is heavily studied, and no primary-source status audit was completed for the exact length-32 optimum.
- There are 2^31 normalized sign vectors. FFT screening should be followed by certified root isolation for the derivative of |P(e^{it})|^2.
- A fine angular grid supplies a lower estimate for each polynomial, never a certified upper estimate by itself.
- The elementary derivative bound |P'|<=496 turns a 32768-point grid maximum into an upper bound with additive error at most 496*pi/32768.

- Recorded example: 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].

### Prior approaches

- **Route 1** (inconclusive): Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum. [2](#reference-2) [3](#reference-3) [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic.

### 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).

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `littlewood-32-minimum-peak`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>R. Turyn, “On Barker codes of even length”. Proceedings of the IEEE 51(9) (1963), 1256-1256. DOI 10.1109/PROC.1963.2526. 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 https://doi.org/10.1109/PROC.1963.2526
   - 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
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, “Flat Littlewood polynomials exist”. Annals of Mathematics 192(3) (2020). DOI 10.4007/annals.2020.192.3.6. 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 https://doi.org/10.4007/annals.2020.192.3.6
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>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 https://people.tamu.edu/~terdelyi/papers-online/power.pdf
   - website; reference source; PDF checked 2026-07-25; checked 2026-07-25
   - Source use: citation_only
   - 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.