Rechnitzer 2026 first 128 digits autoconvolution inequality

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theorem_1: Rechnitzer's computer-assisted bounds c_l <= nu_2^2 <= c_u with |c_u - c_l| <= 1.2 x 10^(-129) for the least squared L2 norm of the autoconvolution of a non-negative unit-mass function on (-1/2, 1/2).

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Andrew Rechnitzer, The first 128 digits of an autoconvolution inequality. arXiv preprint (2026). arXiv:2602.07292. The copy read for this card is arXiv:2602.07292v1 (7 February 2026). The arXiv record (https://arxiv.org/abs/2602.07292, read 2026-10-02) names the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license.

The paper studies the constant nu_2^2 = inf ||f*f||_2^2 over non-negative unit-mass functions in L^1(-1/2, 1/2). Through the inequality sigma_2(g) <= sqrt(2 - 1/g)/nu_2, which the paper attributes to work of Green and White (display (4), p. 2), it enters upper bounds for sigma_2(g), the limit of the largest size of a B_2[g] subset of {1, ..., N} divided by (gN)^(1/2), a limit the paper notes is known to exist only for g = 1. Theorem 1 (pp. 15-16, Section 5.1) gives explicit rigorous bounds c_l <= nu_2^2 <= c_u with |c_u - c_l| <= 1.2 x 10^(-129), both starting 0.57463960715151959272725542752705297143702636937315..., so the first 128 digits agree; the paper cites as the previous bounds White's 0.574636 < nu_2^2 < 0.574643 and the earlier 0.574575 < nu_2^2 < 0.640733 of Green and of Martin and O'Bryant (display (5), p. 2). The method starts from White's reformulation of the problem over Fourier coefficients. A first ansatz for the near-optimal coefficients gave tight values the paper could not make rigorous (Section 2); a second ansatz, a finite combination of the functions (1 - 4x^2)^(j - 1/2), is summed rigorously in ball arithmetic for the upper bound (Section 3), and a Hoelder-inequality argument following White turns it into the lower bound (Section 4). The coefficients are in Appendix A, and Appendix B gives Python code that recomputes bounds from the first few of them, non-rigorously. For problem 158 the theorem sharpens the value of the constant in display (4), so with g = 2 it bears only on upper bounds for finite B_2[2] sets, where it improves on White's lower bound for nu_2^2 from the sixth decimal place; the paper does not mention the problem, and it does not touch the liminf the problem asks about.

Source: https://arxiv.org/abs/2602.07292.

Bears on. #158: Theorem 1's lower bound for nu_2^2, fed into the Green-White inequality the paper quotes as display (4) (p. 2), gives an upper bound on sigma_2(2), the normalized limit of the largest size of a B_2[2] subset of {1, ..., N}; the paper does not mention the problem, and a bound for finite sets does not address the liminf the problem asks about.

Results. Labels and pages are those of v1.

• Theorem 1 (pp. 15-16): rigorous bounds c_l <= nu_2^2 <= c_u with |c_u - c_l| <= 1.2 x 10^(-129), fixing the first 128 digits of nu_2^2.

No file of this source is held; its license permits non-commercial redistribution, and the card cites the edition it names above.