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 $\lvert f*f \rvert$ over $t$, for signed $f$. It is not the quantity $\lvert \max_t f*f \rvert$, whose records are smaller and which is a different constant; the two are sometimes conflated.

Known upper bounds

Bound Reference Comments
$1.4993$ [MV2010] Cited as the previous bound for this variant in Problem 6.4 of [GGSWT2025]
$1.4688$ [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.
$1.460217284022874*$ [STEPS2026] Step function with $600$ equal cells on $[-1/4,1/4]$, obtained by refining the [CERT2026] function to $600$ cells and running a sequential linear-programming descent. Exact value of $\max_t \lvert f*f\rvert / (\int f)^2$ is the rational number $698746209786798359455483373677229695641600 / 478522078482569620433992386249161163190849$, which is $1.4602172840228739\dots$, checked with the same knot formula as below. Self-archived artifact; see the note on the asterisk below.
$1.459989046905384*$ [AER2026] Step function with $600$ equal cells on $[-1/4,1/4]$, labelled $C’_4$ there. Its $\max_t \lvert f*f\rvert / (\int f)^2$ is $1.4599890469053831\dots$, computed with the same knot formula as below, whether the printed decimals are read exactly or as binary64. The bound $1.4554649$ for $C_4$ in the same source is for the other variant below. Self-archived artifact in an unmerged pull request; see the note on the asterisk below.
$1.457854884921989*$ [SCIEL2026] Step function with $800$ equal cells on $[-1/4,1/4]$, found by SCiEL, a recursive self-improvement (RSI) system built on cultural transmission, running an open language model (Qwen3.8-27B) that writes and refines search programs scored by the exact bound. Exact value of $\max_t \lvert f*f\rvert / (\int f)^2$ is the rational number $38534420343892614563110610373646557184000 / 26432274393314964174848236787785831382961$, which is $1.4578548849219886\dots$, checked with the same knot formula as below with $h = 1/1600$; the cell values are the binary64 numbers in construction.json, each read as an exact rational. Self-archived artifact; see the note on the asterisk below.
$1.4630715826075594$ [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. Independently replayed [T2026]: the same fraction in two implementations; it is $1.46307158260755932746\ldots$, above $1.463071582607559$ (the bound first printed here), so the bound is printed rounded up.

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 $1$.

References

Contribution notes

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