Real ultraflat Littlewood polynomials and unbounded binary merit factors
OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Constructs polynomials with N consecutive coefficients in \{-1,1\} whose modulus is (1+o(1))\sqrt N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.
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Ultraflat real Littlewood polynomials
For every ε∈(0,1) and every sufficiently large integer N, there is a polynomial of length N with coefficients in \{-1,1\} whose modulus lies between (1-ε)\sqrt N and (1+ε)\sqrt N everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be…
Full source abstract
For every ε∈(0,1) and every sufficiently large integer N, there is a polynomial of length N with coefficients in \{-1,1\} whose modulus lies between (1-ε)\sqrt N and (1+ε)\sqrt N everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.
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Nearly minimal maxima and positive minima of Littlewood polynomials
For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in \{-1,1\} whose modulus lies between \sqrt N/16 and (1+\eta)\sqrt N everywhere on the unit circle.
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Asymptotically minimal maxima of real Littlewood polynomials
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in \{-1,1\} is (1+o(1))\sqrt N, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor…
Full source abstract
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in \{-1,1\} is (1+o(1))\sqrt N, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.
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