White 2024 optimal l2 autoconvolution inequality

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Ethan Patrick White, An optimal L2 autoconvolution inequality. Canadian Mathematical Bulletin 67 (2024), no. 1, 108-121. doi:10.4153/S0008439523000565.

For F the set of nonnegative densities on [-1/2,1/2], Theorem 1.1 determines the infimum of the squared L2 norm of f*f to within 4 x 10^{-6}, proving 0.574636066 <= mu_2^2 <= 0.574642912 and thereby advancing a problem of Ben Green; the paper also proves that a unique minimizer exists, by the direct method in the calculus of variations, and computes arbitrarily close approximations to it. Corollary 1.2 converts the new lower bound, via Green's additive-energy theorems, into improved asymptotic upper bounds on the B_h[g] constants sigma_2(g) for 2 <= g <= 4 and sigma_h(1) for h = 3, 4, the first improvement in the latter cases since 2001; Corollary 1.3 transfers the continuous bound to the discrete setting, showing that any nonnegative H on [N] with sum N has additive energy at least mu_2^2 N^3 for all sufficiently large N. Both bounds come from a convex quadratic program whose optimum converges to mu_2^2: the upper bound evaluates a computed near-optimal function, and the lower bound applies the inequality of Lemma 3.2 to a test function built from the same solution, both in variable-precision arithmetic; the paper contrasts this with earlier methods limited by long computation times. For problem 158 this is the modern finite-extremal source: through Green's method it improves the finite B_2[2] bound to F(2,N) <= (2.2848842 + o(1)) sqrt(N). It is a finite upper bound only and supplies no cross-scale decay for a single infinite sequence, so it does not by itself address the infinite version of the problem.

Source: https://doi.org/10.4153/S0008439523000565. The file prints on its first page a copyright line for the authors, 2023, published by Cambridge University Press on behalf of the Canadian Mathematical Society, followed by "This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)", the Creative Commons Attribution 4.0 license.

Bears on. #158

Results to transcribe.

• Theorem 1.1: 0.574636066 <= mu_2^2 <= 0.574642912 for the infimum of the squared L2 norm of an autoconvolution of a density on [-1/2,1/2].
• Uniqueness (Section 2): Existence and uniqueness of the minimizer of the autoconvolution L2 norm, by the direct method in the calculus of variations.
• Corollary 1.2: Improved bounds sigma_2(g) <= ((2-1/g)/0.574636066)^{1/2} for 2 <= g <= 4 and new bounds for sigma_h(1), h = 3, 4, where 0.574636066 is the lower bound on mu_2^2 from Theorem 1.1.
• Corollary 1.3: For H: [N] -> R_{>=0} with sum H(j) = N and N large, the additive energy sum over a+b=c+d of H(a)H(b)H(c)H(d) is at least mu_2^2 N^3.