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The first 128 digits of an autoconvolution inequality

Andrew Rechnitzer

TL;DR

The paper delivers the first 128-digit rigorous bounds on the autoconvolution constant $ν_2^2 = \inf_f \|f \ast f\|_2^2$ over unit-mass, nonnegative $f$ in $L^1(-1/2,1/2)$, a quantity tied to Sidon-set bounds in additive combinatorics. It advances the methodology with two ansätze: (i) a near-Fourier-coefficient-based parameterization and (ii) a structured basis in terms of $(1-4x^2)^{j-1/2}$, enabling rigorous evaluation via Bessel function expansions, Kummer transforms, and ball-arithmetic. A rigorous lower bound is derived via Hölder’s inequality and a carefully constructed dual function $G$, with near-optimal $F$ yielding a near-constant triple-convolution that tightens the bound further. The culmination is a two-sided bound whose width is less than $1.2\times 10^{-129}$, with the first 128 digits identical, and a detailed exploration of the asymptotics of the coefficient sequences suggesting rich structure but no simple closed form yet. This work showcases how high-precision, rigorous numerics can resolve deep questions in additive combinatorics and paves the way for even more digits given sufficient computational resources.

Abstract

Using rigorous high-precision floating point arithmetic we compute very tight rigorous bounds on the auto-convolution constant \[ ν_2^2 = \inf_f \|f \ast f\|_2^2 = \inf_f \int_{-1}^1 (f \ast f)^2 \] where the infimum is taken over all unit mass functions $f \in L^1(-1/2,1/2)$. This quantity arises in additive combinatorics, particularly in the study of Sidon sets. Our bounds give the first 128 digits of $ν_2^2$, and so substantially improve previous bounds on this quantity due to White, Green, and Martin & O'Bryant.

The first 128 digits of an autoconvolution inequality

TL;DR

The paper delivers the first 128-digit rigorous bounds on the autoconvolution constant over unit-mass, nonnegative in , a quantity tied to Sidon-set bounds in additive combinatorics. It advances the methodology with two ansätze: (i) a near-Fourier-coefficient-based parameterization and (ii) a structured basis in terms of , enabling rigorous evaluation via Bessel function expansions, Kummer transforms, and ball-arithmetic. A rigorous lower bound is derived via Hölder’s inequality and a carefully constructed dual function , with near-optimal yielding a near-constant triple-convolution that tightens the bound further. The culmination is a two-sided bound whose width is less than , with the first 128 digits identical, and a detailed exploration of the asymptotics of the coefficient sequences suggesting rich structure but no simple closed form yet. This work showcases how high-precision, rigorous numerics can resolve deep questions in additive combinatorics and paves the way for even more digits given sufficient computational resources.

Abstract

Using rigorous high-precision floating point arithmetic we compute very tight rigorous bounds on the auto-convolution constant where the infimum is taken over all unit mass functions . This quantity arises in additive combinatorics, particularly in the study of Sidon sets. Our bounds give the first 128 digits of , and so substantially improve previous bounds on this quantity due to White, Green, and Martin & O'Bryant.
Paper Structure (9 sections, 1 theorem, 60 equations, 7 figures)

This paper contains 9 sections, 1 theorem, 60 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A first ansatz
  3. A second ansatz and rigorous upper bounds
  4. Lower bound
  5. Results and explorations
  6. Main result
  7. Explorations
  8. Ansatz coefficients
  9. Python code to compute upper and lower bounds

Key Result

Theorem 1

The quantity $\nu_2^2 = \inf_{f \in \mathcal{F}} \| f\ast f\|_2^2$, where the infimum is taken over all functions $f \in L^1(-1/2,1/2)$ with $\int f=1$, is bounded by where $\left| c_u-c_\ell \right| \leq 1.2 \times 10^{-129}$, and We have underlined first 128 digits that are common to the upper and lower bounds. Consequently, if $f:[-1/2,1/2] \to \mathbb{R}^+$ is a non-negative function with $

Figures (7)

  • Figure 1: The near-optimal $f(x)$ as computed via the ansatz in equation \ref{['eqn_ansatz_1']}. We also plot $1/f(x)^2$ to show the nature of the singularity as $x \to \pm 1/2$. This second plot is strongly suggestive that $1/f(x)^2 \approx \text{const}\cdot(1-4x^2)$.
  • Figure 2: A plot of $f(x) \cdot \sqrt{1-4x^2}$ with a straight line at $y=2/\pi$; note the vertical range is $[0.62,0.65]$. In the second figure we plot $f(x)-h(x)$ with $h(x)$ given by equation \ref{['eqn_approx_f_2']} and find that it is very close to $0$; note the vertical range is $[-0.001,0.001]$.
  • Figure 3: A plot of $(H \ast H \ast H)(x)$ based on $H(x)$ from equation \ref{['eqn_approx_f_2']} for $x \in (-1/2,1/2)$. Notice that the function is fairly constant on this range and takes a value close to $1$. The second plot shows the same function but for $x \in (-1,1)$.
  • Figure 4: A plot of $(H \ast H \ast H)(x)$from equation \ref{['eqn_approx_f_2']}, but now shifted by 1 to highlight the non-constant region. The second plot shows that this function is well approximated by equation \ref{['eqn_h_cube_approx']}.
  • Figure 5: Plots of the very-nearly optimal $F(x)$ and $G(x)$ used to prove Theorem \ref{['thm main']}.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1