The first 128 digits of an autoconvolution inequality

TL;DR

This paper rigorously computes the first 128 digits of an autoconvolution constant related to additive combinatorics, significantly improving previous bounds and providing precise numerical insights into the problem.

Contribution

It provides the first 128-digit rigorous bounds on the autoconvolution constant using high-precision arithmetic, advancing understanding in additive combinatorics.

Findings

First 128 digits of the autoconvolution constant computed

Significant improvement over previous bounds

Methodology ensures rigorous and highly precise results

Abstract

Using rigorous high-precision floating point arithmetic we compute very tight rigorous bounds on the auto-convolution constant \[ \nu_2^2 = \inf_f \|f \ast f\|_2^2 = \inf_f \int_{-1}^1 (f \ast f)^2 \] 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 $\nu_2^2$, and so substantially improve previous bounds on this quantity due to White, Green, and Martin & O'Bryant.