Bohnenblust–Hille constant on the Boolean cube

Degree at most d functions f:lbrace ± 1rbrace^n→ℝ have Fourier–Walsh expansion f(x)=Σ_S⊆ [n], |S|≤ d widehat f(S) x^S, x^S:=Π_i∈ Sx_i, [n]:=lbrace 1,…,nrbrace. For d∈ℕ set p_d:=2d/d+1.

Current state (tracked automatically)

Best known lower bound for constant 26a: 2 ([ADGP2025], Degree d address function achieves the bound 2^d-1d).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 26a)	2 [ADGP2025] — Degree d address function achieves the bound 2^d-1d.	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

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Cite

@misc{cairn-constant-26a-bohnenblust-hille-constant-on-the-boolean-cube,
  title        = {Bohnenblust–Hille constant on the Boolean cube},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-26a-bohnenblust-hille-constant-on-the-boolean-cube}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

Degree at most $d$ functions $f : {\pm 1}^{n} \to R$ have Fourier--Walsh expansion

$f (x) = S \subseteq [n], ∣ S ∣ \leq d \sum f (S) x^{S}, x^{S} := i \in S \prod x_{i}, [n] := {1, \dots, n} .$

For $d \in N$ set $p_{d} := \frac{2 d}{d + 1}$ . The (degree $\leq d$ ) Bohnenblust--Hille inequality asks for the smallest constant $C$ such that for every $n$ and every function $f : {\pm 1}^{n} \to R$ of degree at most $d$ ( $d \leq n$ ),

$S \subseteq [n] ∣ S ∣ \leq d \sum ∣ f (S) ∣^{p_{d}}^{1/ p_{d}} \leq C ∥ f ∥_{\infty}, ∥ f ∥_{\infty} := x \in {\pm 1}^{n} max ∣ f (x) ∣.$

Let $BH_{{\pm 1}}^{\leq d}$ denote this best constant $C$ (which depends on $d$ ). We define

$C_{26 a} := d \geq 1 sup BH_{{\pm 1}}^{\leq d} .$

Equivalently, $C_{26 a}$ is the smallest constant for which the above inequality holds simultaneously for all degrees $d$ (with the exponent $p_{d} = \frac{2 d}{d + 1}$ depending on $d$ as above).

Known upper bounds

Bound	Reference	Comments
$\infty$	Trivial	the best general estimate currently available is subexponential growth: $BH_{{\pm 1}}^{\leq d} \leq C^{d l o g d}$ for an absolute constant $C > 1$ [DMP2019].

Known lower bounds

Bound	Reference	Comments
$2$	[ADGP2025]	Degree $d$ address function achieves the bound $2^{\frac{d - 1}{d}}$ . At present, chasing incremental improvements of lower bounds seems less compelling than establishing any finite uniform upper bound. That said, exhibiting a construction that forces the constant to exceed $100$ would already be a genuinely interesting result.

Additional comments and links

The exponent $p_{d} = \frac{2 d}{d + 1}$ is best possible (cannot be increased), even if the constant is allowed to depend on $d$ . [Bl2001]
The paper [DMP2019] proves the subexponential upper bound $BH_{{\pm 1}}^{\leq d} \leq C^{d l o g d}$ arXiv:1706.03670
One application is to computational learning theory: quantitative bounds on the Bohnenblust--Hille constants for functions on $(\pm 1)^{n}$ yield improved upper bounds on the randomized query complexity for learning bounded degree- $d$ functions from random queries; see [EI2022].

References

[ADGP2025] Arunachalam, S.; Dutt, A.; Escudero Gutiérrez, F.; Palazuelos, C. A cb-Bohnenblust–Hille inequality with constant one and its applications in learning theory. Math. Ann. 392 (2025), 3367–3396. doi:10.1007/s00208-025-03142-5.
[BH1931] Bohnenblust, H. F.; Hille, E. On the absolute convergence of Dirichlet series. Ann. of Math. 32 (1931), no. 3, 600--622.
[Bl2001] Blei, R. Analysis in Integer and Fractional Dimensions. Cambridge Univ. Press, 2001.
[DMP2019] Defant, Andreas; Mastyło, Mieczysław; Pérez, Antonio. On the Fourier spectrum of functions on Boolean cubes. Math. Ann. 374 (2019), no. 1--2, 653--680. arXiv:1706.03670
[EI2022] Eskenazis, Alexandros; Ivanisvili, Paata. Learning Low-Degree Functions from a Logarithmic Number of Random Queries. Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC '22), 2022. arXiv:2109.10162. doi:10.1145/3519935.3519981.

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.