Improving OpenAI's bound on the exact discrete Fourier transform below n log n

A research draft published on OpenAI Problem #130 proposes an all-length bound of T(n) = O(n(log n)^(1−δ)) with δ = 7.3×10⁻⁵ for computing the exact discrete Fourier transform, a 730-million-fold increase in the exponent saving over OpenAI's published δ = 10⁻¹³. The draft builds on Swapnil Jain's round-six complex network for Problem #109 and its cited predecessors, transferring those advances from integer multiplication to the Fourier setting. A sixth update to OpenAI Problem #109 reports κ > 2⁻¹⁵, tightened from κ = 2⁻¹⁸², with an exact witness of 3.667 × 10⁻⁵, about 2.4-fold over the previous 1.548 × 10⁻⁵ and a 2¹⁶⁷-fold improvement over the original OpenAI result.

We’re publishing a research draft on OpenAI Problem 130: computing the exact discrete Fourier transform below n log n. Our draft proposes an all-length bound of T n = O n log n ^ 1−δ , with δ = 7.3×10⁻⁵. That’s a 730-million-fold increase in the exponent saving over OpenAI’s published δ = 10⁻¹³. Building on Swapnil Jain’s round-six complex network for Problem 109 and its cited predecessors, we propose transferring those advances from integer multiplication to the Fourier setting. Sixth update to OpenAI problem 109 integer multiplication : we are now past 2^-15. κ 2⁻¹⁵ tightened from κ = 2⁻¹⁸² The exact witness is 3.667 × 10⁻⁵, about 2.4 fold over our previous 1.548 × 10⁻⁵, and a 2¹⁶⁷ fold improvement over the original OAI result. Both