The signed third autocorrelation constant
Description of constant
$C_{1d}$ is the largest constant for which one has
\(\max_{t \in \mathbb{R}} \left|\int_{\mathbb{R}} f(t-x) f(x)\ dx\right| \geq C_{1d} \left(\int_{-1/4}^{1/4} f(x)\ dx\right)^2\)
for all real-valued (not necessarily non-negative) $f \colon \mathbb{R} \to \mathbb{R}$ supported on $[-1/4,1/4]$.
| This is the maximum of $ |
f*f |
$ over $t$, for signed $f$. It is not the quantity $ |
\max_t f*f |
$, whose records are smaller and which is a different constant; the two are sometimes conflated. |
Known upper bounds
| Bound |
Reference |
Comments |
| $1.4688$ |
[GGSWT2025] |
Problem 6.4 of [GGSWT2025], signed variant |
| $1.463071582607559*$ |
[CERT2026] |
Step function with $300$ equal cells on $[-1/4,1/4]$ and signed cell values $v_0,\dots,v_{299}$. Exact value of $\max_t \lvert f*f\rvert / (\int f)^2$ is the rational number $2039224160683882663565441886346340189798400 / 1393796574908163689489385839036226787118881$, which is $1.4630715826075593\dots$. Self-archived artifact; see the note on the asterisk below. |
Results should be arranged in chronological order, and can include bounds that are inferior to the previous bounds.
Known lower bounds
| Bound |
Reference |
Comments |
| $1$ |
(trivial) |
$(\int f)^2 = \int (f*f) \le \int \lvert f*f\rvert \le \max \lvert f*f\rvert$, since $f*f$ is supported on an interval of length $1$. |
-
| How the upper bound is checked. For a step function on a uniform grid with cell width $h = 1/600$, $f*f$ is piecewise linear with knots at integer multiples of $h$, so $\max_t |
f*f |
$ is attained at a knot, where $(f*f)(mh) = h\sum_{i+j=m-1} v_i v_j$. The ratio $\max |
f*f |
/ (\int f)^2$ with $\int f = h\sum_i v_i$ is therefore computed exactly in rational arithmetic. The verifier uses only the Python standard library and runs in under a second: python -I verify.py construction.json prints $C(f) = 1.4630715826075593$. The certificate and verifier are in the pull request description, with SHA-256 44f3f2b63b0d698dd5f279661d4be7077ca6703d62ea7ae6e2b2945e0f001e1c (construction.json) and 3e75c4ab0cdb40387765cbd086a572e46605f0c9fd8b8252783ed01aada80258 (verify.py). |
- Asterisk. The bound carries an asterisk under the convention in the README: the $300$ values are too many to check by hand and are available only as a self-archived artifact.
-
| Other variant. For the non-negative-part variant $ |
\max_t f*f |
$ the value $1.4523$ is reported in Appendix A.5 of arXiv:2606.10402; it is not a bound for the signed constant on this page. |
- AlphaEvolve repository page for the related autocorrelation constants and the pages for 1a and 1b.
- The file number is provisional; it was chosen as the first letter in family $1$ not claimed by an open pull request.
References
- [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
- [CERT2026] Step-function certificate for the signed third autocorrelation inequality ($300$ cells),
construction.json and verify.py, 12 Oct 2026, in the pull request description.
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