Jointly optimised piecewise-linear kernels and a certified lower bound C₁ₐ ≥ 1.30758 for the supremum of autoconvolutions

Ali Suleman

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Mathematics – Classical Analysis · October 2026

Jointly optimised piecewise-linear kernels and a certified lower bound C₁ₐ ≥ 1.30758 for the supremum of autoconvolutions

Ali Suleman

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abstract

Let C₁ₐ be the largest constant with ‖f∗f‖∞ ≥ C₁ₐ(∫f)² for every nonnegative f supported in [−¼, ¼]. We prove C₁ₐ ≥ 65379/50000 = 1.30758. The best refereed lower bound we are aware of is 1.28 (Cloninger–Steinerberger, 2017), and a 2026 public repository claims 1.292. The proof uses the Fourier-analytic method of Martin–O’Bryant and Matolcsi–Vinuesa without change. What is new is the choice of its two free functions: a class of continuous piecewise-linear kernels on a grid commensurate with the period, for which positivity of the kernel’s Fourier transform at all relevant frequencies reduces to finitely many exact checks; a joint convex optimisation of kernel and multiplier; and a kernel support δ ≈ 0.21, about 1.5 times wider than in earlier work. The certificate (a kernel with 240 and a multiplier with 800 rational coefficients) is verified in exact rational and ball arithmetic in about one minute, was re-verified by an independent implementation in exact integer arithmetic, and the final algebraic deduction is formalised in Lean 4. Diagnostics indicate that this method saturates near 1.308; further progress needs a new idea.

Autoconvolution · Sidon sets · Fourier analysis · Computer-assisted proof · Convex optimisation · Interval arithmetic

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