Level B · Reproducible Analysis P-constant-1c-barnard-steinerberger-autocorrelation-constant

Barnard–Steinerberger autocorrelation constant

C_1c is the smallest constant for which one has min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C_1c (∫_ℝ f(x) dx)^2 for all non-negative f ∈ L^1(ℝ). Equivalently, in the notation of [K2021], τ = C_1c^-1/2 is the infimum of ∫ f over non-negative f with ∫ f(x) f(x+t) dx ≥ 1 for all t ∈ [0,1].

Current state (tracked automatically)

Best known bounds for constant 1c: 0.4103333 ≤ C ≤ 0.4107675. Upper bound ([BS2020], Here \_0 = -\_x xx = 0.2172336); lower bound ([Kos2026], **Construction:** a step function f = \_i w\_i 1\_[i/K,(i+1)/K) with K = 24576 and 24592 non-negative integer weights, the last 8 of them zero (support [0, 3073), not yet independently verified.
The best independently verified lower bound is 0.4103287 ([R2026], **Construction:** a step function f = \_i w\_i 1\_[i/K,(i+1)/K) with K = 6144 and 6148 non-negative integer weights (support [0, 1.0005))).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 1c)	0.4103333* [Kos2026] — Construction: a step function f = \_i w\_i 1\_[i/K,(i+1)/K) with K = 24576 and 24592 non-negative integer weights, the last 8 of them zero (support [0, 3073	Tao's optimization constants
Best lower bound with an explicit value, independently verified (constant 1c)	0.4103287 [R2026] — Construction: a step function f = \_i w\_i 1\_[i/K,(i+1)/K) with K = 6144 and 6148 non-negative integer weights (support [0, 1.0005)).	Tao's optimization constants
Best upper bound with an explicit value (constant 1c)	0.4107675 [BS2020] — Here \_0 = -\_x xx = 0.2172336.	Tao's optimization constants

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Cite

@misc{cairn-constant-1c-barnard-steinerberger-autocorrelation-constant,
  title        = {Barnard–Steinerberger autocorrelation constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-1c-barnard-steinerberger-autocorrelation-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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The problem

Description of constant

$C_{1 c}$ is the smallest constant for which one has $0 \leq t \leq 1 min \int_{R} f (x) f (x + t) d x \leq C_{1 c} (\int_{R} f (x) d x)^{2}$ for all non-negative $f \in L^{1} (R)$ .

Equivalently, in the notation of [K2021], $τ = C_{1 c}^{- 1/2}$ is the infimum of $\int f$ over non-negative $f$ with $\int f (x) f (x + t) d x \geq 1$ for all $t \in [0, 1]$ .

The constant was introduced in [BS2020], alongside the constant [ $C_{1 a}$ ](1a.html). It is Problem 6 of [GGSWT2025].

Known upper bounds

Bound	Reference	Comments
$1/2$	Trivial	The autocorrelation is even, and its integral over $R$ is $(\int f)^{2}$ .
$\frac{1}{2 ( 1 + θ _{0} )} = 0.4107675 \dots$	[BS2020]	Here $θ_{0} = - min_{x} \frac{s i n x}{x} = 0.2172336 \dots$ . Stated as $0.411$ in [BS2020]; the exact form is [MR2020, (1.2)]. [MR2020] show it is not sharp within a restricted class of functions (non-quantitatively), and that dual test-function arguments of this type cannot go below it [MR2020, Thm 4.1].

Known lower bounds

Bound	Reference	Comments
$0.37$	[BS2020]	Explicit example.
$0.4103287$	[R2026]	Construction: a step function $f = \sum_{i} w_{i} 1_{[i / K, (i + 1) / K)}$ with $K = 6144$ and $6148$ non-negative integer weights (support $[0, 1.0005)$ ). Exact value: $21072342208353792/51354782902387969 = 0.41032871754 \dots$ , computed as $K min_{0 \leq k \leq K} \sum_{i} w_{i} w_{i + k} / (\sum_{i} w_{i})^{2}$ . This is exact because the autocorrelation of an equal-step function is linear between consecutive multiples of $1/ K$ (as in the proof of [K2021, Thm 3.1]). Certificate: in [R2026], commit `19f3971`, SHA-256 of `certificate.json`: `62ac49ddbbfc9aa1053406363d9df9c912022ab85b72a447cf73c94bb9c1fd0e`. An earlier certificate with $K = 3072$ giving $0.4103021$ is kept in the same repository. Independently replayed [T2026]: both certificates give the stated values exactly ( $0.41032871754918741449 \dots$ and $0.410302117239 \dots$ ) in two implementations.
$0.4103333$ *	[Kos2026]	Construction: a step function $f = \sum_{i} w_{i} 1_{[i / K, (i + 1) / K)}$ with $K = 24576$ and $24592$ non-negative integer weights, the last 8 of them zero (support $[0, 3073/3072)$ ), scored by the same exact rule as [R2026]. Exact value: $134648548624804778248790016/328144294688458468928334649 = 0.41033335274 \dots$ , the minimum attained at $k = 21264$ . Found by splitting each cell of the [R2026] certificate in four (in two steps) and re-optimizing the weights numerically; the result keeps the shape of [R2026], including about 22% of the mass in the interval $[1/6144, 2/6144)$ , next to the left end. Certificate: in [Kos2026], commit `a4ee94ef29ff`, SHA-256 of `K24576_0.4103333.json`: `963cb7273062e7360fdfa460bb19c1cf37729364b8f708b09580b6655b783c9e`. A standard-library verifier there computes every lag exactly; the value was also reproduced by an independent verifier written from the definition alone.

Additional comments and links

Elementary examples. Difference bases already give step-function examples via [K2021, Thm 3.1]. For instance, the ruler $0, 1, 4, 6$ (all differences $1, \dots, 6$ ) gives exactly $3/8$ , which beats $0.37$ by a hand computation.
Relation to difference bases. [K2021] relates $τ$ to the asymptotics of $g$ -difference bases and states the bounds $1.560 \dots < τ \leq 1.643 \dots$ .
AlphaEvolve. [GGSWT2025] report that their attempt on this problem did not improve $0.37$ : the search exploited the numerical integration in the scoring function. They suggest that a search over step functions with fixed step sizes would work better.
Structure of near-extremizers. In the numerical near-extremizers of [R2026], about 21–22% of the mass sits in a single grid cell at one end, at both $K = 3072$ and $K = 6144$ , and it re-concentrates there after being spread out. This suggests that extremizers are unbounded at one end (compare [K2021, Question 7.2]). Numerics cannot distinguish a point mass from a very sharp singularity.
Averaged variant. [dDPM2021] study the variant in which the minimum over $t$ is replaced by an average.

References

[BS2020] Barnard, R. C.; Steinerberger, S. Three convolution inequalities on the real line with connections to additive combinatorics. J. Number Theory 207 (2020), 42–55. arXiv:1903.08731
[dDPM2021] de Dios Pont, J.; Madrid, J. On classical inequalities for autocorrelations and autoconvolutions. arXiv:2106.13873
[GGSWT2025] Georgiev, B.; Gómez-Serrano, J.; Tao, T.; Wagner, A. Z. Mathematical exploration and discovery at scale. arXiv:2511.02864
[K2021] Kravitz, N. Generalized difference sets and autocorrelation integrals. arXiv:2004.06611
[Kos2026] Koshta, Dennis, with Claude (Anthropic, Claude Opus 5.5). An improved step-function lower bound for the Barnard–Steinerberger autocorrelation constant. Certificate and verifiers: https://github.com/DennisKoshta/c1c-autocorrelation-certificate (commit a4ee94ef29ff, 23 Oct 2026).
[MR2020] Madrid, J.; Ramos, J. P. G. On optimal autocorrelation inequalities on the real line. arXiv:2003.06962
[T2026] Toledo, Carlos. Independent replay of the step-function certificates for $C_{1 c}$ [R2026] and $C_{1 d}$ [CERT2026]: the weights as exact rationals, every lag of the autocorrelation ( $C_{1 c}$ ) and of the autoconvolution ( $C_{1 d}$ ) summed exactly, in two implementations that share no code with each other or with the certificates' verifiers. Report (§7); code and record at commit d9a3942 (2026).
[R2026] rozetyp (GitHub). A lower bound for the Barnard–Steinerberger autocorrelation constant. Certificate and verifiers: <https://github.com/rozetyp/barnard-steinerberger-lower-bound>

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.