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Source. Theorem 1, pp. 15--16 (Section 5.1), of Andrew Rechnitzer, The first 128 digits of an autoconvolution inequality, arXiv:2602.07292v1 (7 February 2026), the version named on the source card. A preprint.

Read depth. Claims checked: the statement, the two digit strings and the definitions it uses were read clause by clause on the page images; the method (Sections 2--5, pp. 3--18) was read for structure only, and the rigorous computation was not rerun. Nothing here is independently reviewed.

Statement

Setting (pp. 1--3). $\mathcal F$ is the set of non-negative functions in $L^1(-1/2,1/2)$ (pp. 2--3). For $f\in\mathcal F$ the autoconvolution $f*f$ is supported on $(-1,1)$, and $\lVert f*f\rVert_2^2=\int_{-1}^{1}(f*f)^2$ (abstract, p. 1, and equation (3), p. 2).

Theorem 1 (pp. 15--16). Let $\nu_2^2=\inf_{f\in\mathcal F}\lVert f*f\rVert_2^2$, the infimum taken over the functions $f\in\mathcal F$ with $\int f=1$. Then $c_\ell\le\nu_2^2\le c_u$, where $\lvert c_u-c_\ell\rvert\le1.2\times10^{-129}$ and

$$\begin{aligned} c_\ell&=0.57463\,96071\,51519\,59272\,72554\,27527\,05297\,14370\,26369\,37315\,66116\,30876\\ &\qquad74892\,55216\,18178\,98882\,24078\,24755\,71532\,95571\,66060\,64735\,74241\,32638\,64820\,673\,58,\\ c_u&=0.57463\,96071\,51519\,59272\,72554\,27527\,05297\,14370\,26369\,37315\,66116\,30876\\ &\qquad74892\,55216\,18178\,98882\,24078\,24755\,71532\,95571\,66060\,64735\,74241\,32638\,64820\,673\,69. \end{aligned}$$

The two decimals agree in their first 128 digits, which the print underlines (p. 16). The theorem's text also records the consequence that every non-negative $f$ on $[-1/2,1/2]$ with $\int f=1$ has $\lVert f*f\rVert_2^2\ge c_\ell$ (p. 16).

The printed sentence describing the infimum names only the unit-mass condition on $f\in L^1(-1/2,1/2)$; non-negativity enters through the index set $\mathcal F$ of the infimum, defined on pp. 2--3. The abstract's statement of the problem also names only unit mass.

Context on p. 2. The paper cites the earlier bounds $0.574575<\nu_2^2<0.640733$ (Green for the lower, Martin and O'Bryant for the upper) and $0.574636<\nu_2^2<0.574643$ (White), its display (5), and says White's bounds give the first 4 digits.

Proof pointer

Section 2 (pp. 3--7) starts from White's reformulation of $\nu_2^2$ as a sum over Fourier coefficients (display (8), p. 3), which the paper attributes to White's Lemma 3.1, and fits the near-optimal coefficients by an ansatz in powers $k^{-j-1/2}$; the paper reports that this first ansatz gave tight numerical values it could not make rigorous. Section 3 (pp. 7--10) takes a second ansatz, a finite combination of the functions $(1-4x^2)^{j-1/2}$ with Bessel-function Fourier coefficients (displays (26)--(27), p. 8), and sums the resulting series rigorously with Kummer's series transform and asymptotic expansions, giving upper bounds. Section 4 (pp. 11--14) turns a near-optimal upper-bound function into a lower bound through a Hölder-inequality argument that the paper attributes to White's Lemma 3.2 (display (44), p. 11). Section 5 (pp. 14--18) reports the computation, carried out in C++ with ball arithmetic from the flint library; Section 5.1 (pp. 15--17) finishes with $P=101$ ansatz coefficients, $N=8192$, $K=128$ and 384 digits of precision. The coefficients used are listed in Appendix A, and Appendix B gives Python code that recomputes bounds from the first few of them, which the paper calls non-rigorous.

Dependencies

White's reformulation and lower-bound lemma (Lemmas 3.1 and 3.2 of White's paper, as the paper cites them) and a computer calculation in rigorous ball arithmetic.

Bears on

• Problem 158: the paper quotes, as its display (4) on p. 2, the inequality $\sigma_2(g)\le\sqrt{2-1/g}/\nu_2$ that it attributes to work of Green and White, where $\sigma_2(g)$ is the limit of $R_2[g](N)/(gN)^{1/2}$ and $R_2[g](N)$ is the largest size of a $B_2[g]$ subset of $\{1,\ldots,N\}$ (display (1), p. 1); the paper notes that this limit is known to exist only for $g=1$ (p. 2). With $g=2$, display (4) turns a lower bound on $\nu_2^2$ into an upper bound on $\sigma_2(2)$, and Theorem 1's $c_\ell$ exceeds White's lower bound $0.574636$ only from the sixth decimal place. The paper does not mention the problem or state the resulting bound on $\sigma_2(2)$, and an upper bound for finite sets says nothing about whether the lower limit the problem asks about is $0$.