OpenAI PARTIALLY DISPROVES LITTLEWOOD'S 1961 Theorem

Abstract

We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on L^α-flat polynomials. The theorem asserts that no L^α-flat polynomials exist within the class of trigonometric polynomials of the form∑_m=1ⁿ a_m cos(mθ+α_m), n ∈ ℕ^*,satisfyingk n² ∑_m=1ⁿ a_m² ≤ ∑_m=1ⁿ m² a_m²,where k is an absolute constant (alternatively expressed on the right-hand side as K = 1/k). We show that OpenAI's counterexample is contained within the class of hyper-lacunary sequences. Consequently, it offers only a limited refutation, as such sequences fail to satisfy an implicit assumption of Littlewood's theorem. We also discuss broader questions surrounding this counterexample and provide a brief historical overview.

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BibTeX

@misc{abdalaoui2026openai,
  title = {{OpenAI PARTIALLY DISPROVES LITTLEWOOD'S 1961 Theorem}},
  author = {El Houcein El Abdalaoui},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05830397v1},
  note = {CC BY-NC 4.0},
}

RIS

TY  - UNPB
TI  - OpenAI PARTIALLY DISPROVES LITTLEWOOD'S 1961 Theorem
AU  - El Houcein El Abdalaoui
PY  - 2026
DA  - 2026/10/15
PB  - HAL
UR  - https://hal.science/hal-05830397v1
AB  - We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on L^α-flat polynomials. The theorem asserts that no L^α-flat polynomials exist within the class of trigonometric polynomials of the form∑_m=1^n a_m cos(mθ+α_m), n ∈ ℕ^*,satisfyingk n^2 ∑_m=1^n a_m^2 ≤ ∑_m=1^n m^2 a_m^2,where k is an absolute constant (alternatively expressed on the right-hand side as K = 1/k). We show that OpenAI's counterexample is contained within the class of hyper-lacunary sequences. Consequently, it offers only a limited refutation, as such sequences fail to satisfy an implicit assumption of Littlewood's theorem. We also discuss broader questions surrounding this counterexample and provide a brief historical overview.
ER  -

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