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Grokability in five inequalities

Grokability in five inequalities Paata Ivanisvili∗ Xinyuan Xie†

arXiv:2605.05193v1 [math.PR] 6 May 2026

Abstract In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in Rn , sharper L2 -L1 moment comparison inequalities on the Hamming cube {−1, 1}n , a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest g-Sidon sets in {1, . . . , n}, and an optimal balanced Szarek’s inequality.

1

Introduction

In this note, we record several results in analysis, probability, convex geometry, and additive combinatorics, obtained through conversations with Grok. These results, to the best of our knowledge, have not previously appeared in the public literature. The first result is an improved lower bound for the maximal Gaussian perimeter of convex sets—the first constant improvement since Nazarov’s construction in 2003 [27]. The second result is an improved upper and lower bound on the optimal exponential base for L2 − L1 moment comparison inequalities on the Hamming cube. In particular, the improved lower bound answers a MathOverflow question posed by Noam Lifshitz in 2014 [38]. The third result is an optimal Khintchine-type inequality when the Rademacher random variables are conditioned on the middle slice of the Hamming cube, and we call it balanced Szarek’s inequality. In this case, the random variables are no longer independent, but interestingly, the classical semigroup approach can still obtain the optimal constant. The fourth result is an improved lower bound on an autocorrelation inequality related to the size estimate of g-Sidon sets. This constant is recorded as the C1a constant in a curated collection of optimization constants in mathematics initiated by Tao [34] and maintained by Davis, Ivanisvili, and Tao [6]. This improvement is of interest from the prospective of building automatic reasoning systems with the assistance of LLMs, as will be explained in the next paragraph. These examples, together with recent works obtained in a similar fashion [16, 3, 9, 39, 17], suggest that the current frontier reasoning models may carry out sophisticated analytical arguments, explore the boundary of existing methods, and implement new ideas with a meaningful degree of accuracy. We hope this note helps the community stay informed about the growing power of these new tools in mathematical exploration and discovery. On the other hand, there has been a series of works on building automated reasoning systems with the assistance of LLMs [12, 36, 40]. For each problem, these systems first parameterize it, encode the parametrization in computer code, for example in Python, and then iteratively refine (evolve) the code with the help of LLMs. The idea of building automated reasoning systems is very interesting, but one may hope for a more ambitious use of LLMs. For example, the C1a constant has been extensively studied using computer-assisted approaches, whether based on LLMs or on classical implementations, for searches in explicit and structured discrete spaces; ∗ †

University of California, Irvine, [email protected] University of California, Irvine, [email protected]

1

see [6] for detailed information about this constant. In particular, the best known lower bound was obtained by exhausting certain large but finite sets using about 20,000 CPU hours [7]. In this note, we present a tiny improvement to the lower bound for the C1a constant, obtained within a few minutes of conversation with Grok, suggesting that explicit and structural searching leaves extra room that can be uncovered through LLM-based conversational approaches1 . This example, together with the other examples presented in this note, suggests the potential for building reasoning systems that explore mathematical questions through natural language and mathematical arguments. The rest of the paper is organized as follows. In the remainder of the introduction, we present concise statements of each problem and explain why they are interesting. The proofs are presented in Section 2. In Appendix A, we provide links that fully disclose our corresponding conversations with Grok, which led to the new results for each question.

1.1

Maximal Gaussian perimeter of convex sets

For a convex set K ⊆ Rn , its Gaussian perimeter (or Gaussian surface area) is defined as Z 2 GSA(K) = φn (x) dσ(x), φn (x) := (2π)−n/2 e−∥x∥ /2 . ∂K

The extremal quantity of interest is Γ(n) := sup{GSA(K) : K ⊆ Rn is convex}. The problem is to determine the growth of Γ(n) as n → ∞. Ball[1] proved that 1p log n ≤ Γ(n) ≤ 4n1/4 , e Nazarov[27] later showed that the order n1/4 is sharp by proving Γ(n) Γ(n) ≤ lim sup 1/4 < 0.64. 1/4 n n→∞ n The upper bound for Γn was used to obtain sharper dimension dependence in multivariate Berry–Esseen bound over convex sets [2]. Raič [30] further refined the multivariate Berry–Essen bound and improved the upper bound of Γ(n) to r
2 Γ(n) ≤ + 0.59 n1/4 − 1 , π The problem is also relevant in learning theory. Klivans, O’Donnell, and Servedio showed that Gaussian perimeter controls low-degree Hermite approximation, and hence the complexity of PAC and agnostic learning under Gaussian distributions [18]. More recently, Nadimpalli and Pascale[25] revisited Nazarov’s construction and recovered the Nazarov’s lower-bound constant e−5/4 by an alternative argument based on convex influence. Here we show that the constant in the lower bound can be improved further: e−5/4 < lim inf n→∞

Theorem 1. We have

Γn ≥ 0.31258. n→∞ n1/4 This is about 9% improvement of the previous best known lower bound. lim inf

In [27], the analytical argument is sophisticated, and several intermediate results are stated without proof. In Section 2, we follow the same general strategy as in [27], while giving more detailed proofs and correcting a minor numerical inaccuracy in the original paper. We found that reasoning LLMs were able to work through these arguments, provide proofs for some statements left to the reader in [27], and fine-tune the parameters in Nazarov’s construction, leading to the present improvement. 1

The term “conversational approach” was taken from a post by Tao [35].

2

1.2

L2 -L1 moment comparison inequality

For a function f : {−1, 1}n → R we write 1/p

 X

∥f ∥p := 2−n

|f (x)|p 

,

x∈{−1,1}n

and we denote by deg f its Fourier–Walsh degree, i.e. the degree of its multilinear expansion. We are interested in the optimal exponential base n o d C∗ := inf C > 0 : ∥f ∥2 ≤ C ∥f ∥1 for every d ≥ 1, n ≥ 1, deg f ≤ d . The problem of determining C∗ was posed on √ MathOverflow by Noam Lifshitz in 2014, who asked in particular whether one can take C∗ = 2 [38]. The elementary example d Y

f (x) =

(1 + xj )

j=1

√

√ already shows that C∗ ≥ 2, since ∥f ∥1 = 1 and ∥f ∥2 = 2d/2 . The guess C∗ = 2 is also natural for structural reasons: for√degree 1 it is exactly the sharp Khinchine inequality, due to Szarek [33]; and if the same base 2 were valid already for degree 2, then one would obtain the constant 2 for quadratic Rademacher chaoses, in accordance with the conjecture of Pelczyński; see [15, Remark 6]. On the upper-bound side, the real hypercontractivity yields C∗ ≤ e, see [28, Theorem 9.22]. A a more subtale argument via conformal maps, Hahn–Banach and Riesz representation theorem combined with complex hypercontractivity gives improvement C∗ ≤ 2.69..., see [8]. We should also note that for d-homogeneous chaoses h one has the sharper estimate ∥h∥2 ≤ ed/2 ∥h∥1 by Beckner’s complex hypercontractivity [15, Theorem 2]. A useful observation is that if g is real-valued and f = g 2 , then ∥f ∥2 = ∥f ∥1



∥g∥4 ∥g∥2

2 .

Thus any family exhibiting large L4 /L2 growth immediately yields lower bounds for C∗ . The Gaussian model suggests taking g to be a Hermite polynomial evaluated√on a long linear form. The next theorem shows that this already forces the base to be at least 3. √ Theorem 2. One has 2.408... ≥ C∗ ≥ 3. We should note that the improvement in the upper bound from 2.69 . . . to 2.408 . . . was also known to the authors of [8]. They did not revise the paper to reflect this, since the refinement follows verbatim from the argument in [8]: one only needs to observe that the constant C, defined in equation (191) there as the infimum of an explicit function, is in fact equal √ to 2.408 . . .. On the lower-bound side, we note that the failure of the conjecture C∗ = 2 in this generality was already known to some experts; see, for instance, A. Samorodnitsky (private communication). Thus, Theorem 2 should not be regarded as a new mathematical fact in itself. What is striking, however, is that reasoning models such as Grok were able to recover both bounds, even though these observations do not seem to have been recorded in the published literature.

3

1.3

Optimal balanced Szarek’s inequality

The Khintchine inequality states that for independent Rademacher random variables ε1 , . . . , εn and every p ≥ 1, Ap

n X

a2i

1/2



≤ E

i=1

n X

a i εi

p 1/p

≤ Bp

n X

i=1

a2i

1/2

,

i=1

where the constants Ap , Bp > 0 depend only on p. Determining the optimal constants in this inequality has attracted considerable attention: in particular, the optimal constant when p = 1 is due to Szarek [33], while the optimal constants for all p were determined by Haagerup [13]. A natural analogue is obtained by replacing the discrete cube {−1, 1}n with the middle slice n n o X n Ωn := x ∈ {−1, 1} : xi = 0 , i=1

equipped with the uniform measure (here we Pnneed n be even). Equivalently, one conditions the Rademacher random vector on the event i=1 xi = 0 . In this model the coordinates are no longer independent, so the classical proofs of Khintchine inequality may not directly apply. For p ≥ 2, such balanced Khintchine inequalities were studied by Spektor [32]. Later, Herscovici and Spektor [14] computed the optimal constants when p is even integer. By contrast, the optimal constant for the endpoint p = 1 does not seem to have been documented in the literature. In the result below, we derive such balanced Khintchine-type iequality at the endpoint p = 1 with optimal constant, which may be viewed as a balanced analogue of Szarek’s sharp L1 inequality. Somewhat surprisingly, despite the loss of independence, the semigroup framework due to Kwapień, Latala, and Oleszkiewicz [19, 21, 21] (see the second proof of Theorem 32 in [26] for a modern presentation) still remains effective in obtaining the optimal constant. To deal with the lack of independence, we invoke a recent result of Filmus [10] on orthogonal bases for the slice of the Hamming cube. Theorem 3 (Balanced Szarek’s inequality). Let n ∈ N be even and n ≥ 4. Let x be distributed uniformly on n n o X Ωn := x ∈ {−1, 1}n : xi = 0 , i=1

and let a = (a1 , . . . , an ) ∈ Rn . Then v u n n X u X 2 cn tE ai xi ≤ E ai xi , i=1

where

i=1

s cn =

n−2 2(n − 1)

is optimal. Remark 4. The optimality is witnessed by the following examples, which is also the extremizer for classical Szarek’s inequality. Let a = (1, 1, 0, ..., 0). Then |x1 + x2 | = 2 with probability P n−2 q 2(n/2−2 ) E| n n−2 n−2 i=1 ai xi | √ = = , and |x + x | = 0 otherwise. Thus, P 1 2 n 2 2(n−1) 2(n−1) . E| n (n/2 ) i=1 ai xi |

4

1.4

An autoconvolution inequality and g-Sidon sets

Let an integrable f : R → R≥0 be supported on [−1/4, 1/4]. We are interested in the largest universal constant c > 0 such that !2 Z 1/4 sup f ∗ f (x) ≥ c f (x) dx x∈R

−1/4

holds for every such function f . This constant c corresponds to the asymptotic size estimate for the g-Sidon sets: a set A ⊂ {1, 2, . . . , n} is called g-Sidon if every integer m has at most g representations of the form m = a + b with a, b ∈ A. Denoting the largest size of a g-Sidon set for fixed n and g as βg (n) := max{|A| : A ⊂ {1, 2, . . . , n} is g-Sidon}, [24] showed that the leading asymptotic constant for βg (n) is a universal constant σ when n and g go to infinity, which can be reformulated in terms of the constant c as 2 . σ2 Thus, studying the sharp autoconvolution inequality is equivalent to determining the leading asymptotic constant for the maximal size of g-Sidon sets. There have been many efforts devoted to bounding the constant c from above and below [31, 22, 23, 24, 7, 12, 36, 40], where the best published bounds are 1.28 ≤ c ≤ 1.50286. c=

Recent progress on the upper bound [12, 36, 40] is obtained through automatic reasoning systems assisted with LLMs, which iteratively modify Python codes describing candidate mathematical objects. At a high level, the reasoning LLMs in [12, 36, 40] are used to explore the space formed by certain programming codes. One may naturally ask whether further improvements can still be achieved using existing techniques from the literature, and whether large language models can help uncover and exploit this remaining potential. We provide positive evidence on this specific question. In the following theorem, we provide an improved lower bound by refining an estimate in [7], which was suggested by LLMs. Theorem 5. Let f : R → R≥0 be supported on [−1/4, 1/4]. Then for every such function f , the following holds true !2 Z 1/4 sup f ∗ f (x) ≥ 1.2802 f (x) dx . x∈R

−1/4

The improvement can be obtained from Grok with a single natural prompt, simply by asking it to tighten an inequality in [7]. The improvement is comparable to those obtained for the upper bound by LLM-based reasoning systems; see C1a constant [6] for a neat record of recent progress on this question. From a mathematical perspective, however, this improvement is not particularly substantial, as it represents only a minor refinement of the estimate in [7]. Moreover, we do not believe the authors missed anything essential, as their primary goal was to showcase the methodology. Nevertheless, we find this result impressive, since it shows that a purely conversational approach may extract a further improvement for a problem that has been extensively studied through structured and explicitly parameterized exhaustive search.

2

Proofs

2.1

Proof of Theorem 1

The proof presented in this section is the same as Nazarov’s construction of a random convex set in [27]. We show that a choice of specific parameters improves the previous lower bound. 5

The random convex set is constructed in Rn+1 as in Nazarov [27] to avoid indexing φ, and we write Γn := Γ(n) for brevity. Lemma 6 (Nazarov’s exact expectation identity). Fix n ≥ 1, ρ > 0, and an integer N ≥ 1. Let x1 , . . . , xN be i.i.d. uniform on the unit sphere S n ⊂ Rn+1 and set Q :=

N \

{x ∈ Rn+1 : ⟨x, xj ⟩ ≤ ρ}.

j=1

Then

1 2 E GSA(Q) = N √ e−ρ /2 2π

Z φn (y) 1 − p(|y|)

N −1

dy,

(1)

Rn

where p(r) depends only on r ≥ 0 and is given by !−1 Z √ 2 2 Z √r2 +ρ2  r +ρ  t2  n−2 t2  n−2 2 2 p(r) = 1− 2 dt dt. 1− 2 √ 2 2 r + ρ r + ρ 2 2 − r +ρ ρ

(2)

Remark 7. In [27], the exponent of the integrand in the expression for p(r) was given as n−1 2 instead of n−2 2 without proof, which is a minor inaccuracy that does not affect the final result. Here we include a proof for the reader’s convenience. Proof. Since the laws of xj ’s are continuous and xj are independent, we have P(xi = xj for some i = ̸ j) = 0. Thus we may assume the normals x1 , . . . , xN are pairwise distinct. For each j, let Hj := {x ∈ Rn+1 : ⟨x, xj ⟩ = ρ},

Fj := Q ∩ Hj .

Each Fj is a (possibly empty) facet of Q with outer unit normal xj . Moreover, ∂Q is covered by the union of facets, and intersections Fi ∩ Fj (i ̸= j) have codimension at least 2, hence σ-measure 0. Therefore, Z GSA(Q) =

φn+1 (x) dσ(x) = ∂Q

N Z X j=1

φn+1 (x) dσ(x).

Fj

Taking expectation and using the fact that {xi }i∈[N ] has the same distribution, Z  E GSA(Q) = N · E φn+1 (x) dσ(x) . F1

By rotational invariance of φn+1 , we may assume x1 = en+1 . Therefore Z Z 1 −ρ2 /2 φn+1 (x) dσ(x) = √ e φn (y)1{y:(y,ρ)∈F1 } dy, 2π F1 Rn and by Tonelli, we have that Z  Z
1 −ρ2 /2 E φn+1 (x) dσ(x) = √ e φn (y)P (y, ρ) ∈ F1 dy 2π F1 Rn
It remains to compute P (y, ρ) ∈ F1 . For fixed y ∈ Rn , the point (y, ρ) belongs to F1 iff it is not cut off by any of the other half-spaces, i.e. ⟨(y, ρ), xj ⟩ ≤ ρ,

j = 2, . . . , N.

Let r := |y| and define
p(r) := Px∼Unif(S n ) ⟨(y, ρ), x⟩ > ρ , 6

which depends only on |(y, ρ)| by rotation invariance of Unif(S n ). Thus it only depends on r for fixed ρ. By independence, P (y, ρ) ∈ F1 ) = (1 − p(r))N −1 . Hence

Z E

 φn+1 dσ

F1

1 2 = √ e−ρ /2 2π

Z

φn (y) (1 − p(|y|))N −1 dy,

Rn

and multiplying by N gives (1). It remains to compute p(r). Fix y with |y| = r and put u := (y, ρ) ∈ Rn+1 . For x ∼ Unif(S n ), t u by rotational invariance, the random variable |u| := ⟨ |u| , x⟩ has the same distribution as x1 as u we may assume |u| = e1 . By [4, Eq. (8)], x1 has density proportional to (1 − |x1 |2 )(n−2)/2 . Therefore, (n−2)/2 t2 dt 1− 2 |u| ρ p(r) = Z (n−2)/2 . |u|  t2 dt 1− 2 |u| −|u| Z |u| 

Substituting |u| =

p r2 + ρ2 completes the proof.

Lemma 8 (Radialization). As in Nazarov [27], define Z ∞ −1 n−1 −t2 /2 f (t) := t e , c := f (t) dt .

(3)

0

Then for every n ≥ 1, every measurable g : [0, ∞) → [0, ∞], Z Z ∞ φn (y) g(|y|) dy = c f (r) g(r) dr. Rn

(4)

0

Proof. Let dσ denote the usual surface measure on S n−1 . By polar coordinates and Tonelli, Z Z ∞Z 2 −n/2 φn (y)g(|y|) dy = (2π) e−r /2 g(r)rn−1 dσ(θ) dr Rn 0 S n−1 Z ∞ 2 −n/2 n−1 = (2π) |S | rn−1 e−r /2 g(r) dr. 0

Taking g ≡ 1 gives 1 = (2π)

−n/2

|S

n−1

Z ∞ |

f (r) dr, 0

and hence (2π)−n/2 |S n−1 | =

−1

Z ∞ f (r) dr

= c.

0

Substituting this into the previous display gives the claim. Lemma 9 (Uniform upper bound for p(r)). Define the explicit Nazarov main term  ρ4  2 1 1 exp e−ρ /2 . L(n, ρ) := √ 4n 2π ρ

(5)

Fix constants a > 0 and W > 0. Then there exist √ constants Ca,W > 0 and n0 (a, W ) such that for all n ≥ n0 (a, W ), all ρ = an1/4 , and all r = n − 1 + w with |w| ≤ W , one has  wρ2  C  a,W p(r) ≤ L(n, ρ) exp √ exp √ . (6) n n 7

Proof. Put S := r2 + ρ2 . Then the denominator in (2) equals √ √ √ Z 1 n−2 Γ(n/2) S (1 − u2 ) 2 du = S π . Γ((n + 1)/2) −1 By Wendel’s inequality [37], r

2 , n

Γ(n/2) ≥ Γ((n + 1)/2)

r

Γ(n/2) ≥ Γ((n + 1)/2) and hence

√ √ S π

2πS . n

For the numerator, use log(1 − u) ≤ −u − Thus, for 0 < t <

√

u2 , 2

0 < u < 1.

S, 

  n−2 2 n−2 4 t2  n−2 2 ≤ exp − 1− t − t . S 2S 4S 2

Since t ≥ ρ on the interval of integration, Z √S  ρ

Z ∞    t2  n−2 n−2 4 n−2 2 2 ρ exp − t dt. 1− dt ≤ exp − S 4S 2 2S ρ

Set 1 − Φ(x) m(x) := = φ1 (x)

R ∞ −u2 /2 du x e , 2 /2 −x e

x > 0.

It follows from integration by parts that m(x) ≤ x−1 , i.e., equivalently, Z ∞ 1 2 2 x > 0, e−u /2 du ≤ e−x /2 . x x Applying this with r x=ρ

n−2 , S

we get, for n ≥ 3, r     Z ∞ Z ∞ n−2 2 S S n−2 2 −u2 /2 exp − t dt = e du ≤ exp − ρ . 2S n − 2 ρ√(n−2)/S (n − 2)ρ 2S ρ Therefore

Z √S  ρ

1−

 n−2 t2  n−2 S n − 2 4 2 dt ≤ exp − ρ2 − ρ . S (n − 2)ρ 2S 4S 2

Hence

r  n−2 1 S n n − 2 4 p(r) ≤ √ exp − ρ2 − ρ . 2S 4S 2 2πρ n − 2 S √ Now set ρ = an1/4 and r = n − 1 + w, with |w| ≤ W . Then √ √ S = r2 + ρ2 = ( n − 1 + w)2 + ρ2 = n + (2w + a2 ) n + Oa,W (1),

uniformly for |w| ≤ W . Hence n−2 2w + a2 =1− √ + Oa,W (n−1 ), S n 8

n−2 1 = + Oa,W (n−3/2 ). 2 S n

√ Since ρ2 = a2 n and ρ4 = a4 n, we get −

ρ2 wρ2 n−2 2 ρ4 ρ =− + √ + + Oa,W (n−1/2 ), 2S 2 2n n

and

n−2 4 ρ4 ρ = − + Oa,W (n−1/2 ). 4S 2 4n Adding the last two displays gives −

−

n−2 2 n−2 4 ρ4 ρ2 wρ2 √ + ρ − ρ = − + + Oa,W (n−1/2 ), 2S 4S 2 2 4n n

uniformly for |w| ≤ W . Substituting the exponent expansion together with r
n S = exp Oa,W (n−1/2 ) n−2 S gives C   ρ2  wρ2  1 1 ρ4  a,W exp √ , p(r) ≤ √ exp − + exp √ 2 4n n n 2π ρ which is exactly (6) by the definition of L(n, ρ). Increasing n0 (a, W ) if necessary so that n ≥ 3 completes the proof. √ Lemma 10 (Uniform Laplace form for c f ( n − 1 + w)). Fix W > 0. There exist constants CW > 0 and n1 (W ) such that for all n ≥ n1 (W ) and all |w| ≤ W ,  θ √ 1 2 n,W (w) √ c f ( n − 1 + w) = √ e−w exp , π n √ Proof. Let t0 := n − 1 and u := w/t0 . Then log

|θn,W (w)| ≤ CW .

(7)

f (t0 + w) w2 = (n − 1) log(1 + u) − t0 w − . f (t0 ) 2

For n ≥ 4W 2 + 1 we have |u| ≤ 1/2. Using | log(1 + u) − (u − u2 /2)| ≤ 2|u|3 for |u| ≤ 1/2 and the identities (n − 1)u = t0 w, (n − 1)u2 = w2 , we obtain log

f (t0 + w) = −w2 + OW (n−1/2 ) f (t0 )

uniformly for |w| ≤ W.

R∞ Next, 0 f (t) dt = 2n/2−1 Γ(n/2), so cf (t0 ) can be estimated by Stirling bounds for Γ(n/2), giving cf (t0 ) = √1π exp(O(n−1 )). Combining the two estimates yields (7). Proof of Theorem 1. Fix the rational parameters a :=

6131 = 1.2262, 5000

b :=

2387 = 2.387, 1000

W := 6.

For each n, set ρ := an1/4 ,

N :=

j

b k , L(n, ρ)

where L(n, ρ) is defined in (5). Since L(n, ρ) → 0 exponentially fast in N L(n, ρ) → b

as n → ∞. 9

√

n, we have (8)

Let Q ⊂ Rn+1 be the random polytope from Lemma 6. By definition of Γ(n + 1), Γ(n + 1) ≥ E GSA(Q). Using Lemma 6 and radialization (4), 1 2 E GSA(Q) = N √ e−ρ /2 c 2π Restrict to r =

√

Z ∞ f (r) 1 − p(r)

N −1

dr.

(9)

0

n − 1 + w with |w| ≤ W :

1 2 E GSA(Q) ≥ N √ e−ρ /2 2π

Z W

√ √
N −1 c f ( n − 1 + w) 1 − p( n − 1 + w) dw.

(10)

−W

By Lemma 9, for |w| ≤ W and large n,  wρ2  C  √ a,W exp √ . p( n − 1 + w) ≤ qn (w) := L(n, ρ) exp √ n n Since qn (w) → 0 uniformly on [−W, W ], for large n we have qn (w) ≤ 1/2 there, and log(1 − x) ≥ −x − x2 on [0, 1/2] gives
(1 − qn (w))N −1 ≥ exp − (N − 1)qn (w) − (N − 1)qn (w)2 . √ Using ρ2 / n = a2 and (8), we get pointwise for each fixed w, 2

(N − 1)qn (w) → bea w , hence

(N − 1)qn (w)2 → 0,

√
N −1 2
lim inf 1 − p( n − 1 + w) ≥ exp − bea w . n→∞

(11)

Also Lemma 10 gives √ 1 2 c f ( n − 1 + w) → √ e−w π

as n → ∞.

(12)

Next, by the definition (5) we have the exact identity 1 2 4 √ e−ρ /2 = ρe−ρ /(4n) L(n, ρ). 2π Therefore, using (8) and ρ = an1/4 , N 1 −ρ2 /2 N L(n, ρ) −ρ4 /(4n) 4 √ e = ρe → b a e−a /4 1/4 1/4 n n 2π

as n → ∞.

(13)

Apply Fatou’s lemma to (10) and use (11), (12), (13). We obtain Γ(n + 1) 1 4 ≥ b a e−a /4 · √ 1/4 n→∞ π n

Z W

lim inf

2
exp − w2 − bea w dw.

−W

Certified numerical lower bound (with W = 6). Let 2

F (w) := exp − w − be

a2 w

Z 6 ,

J :=

F (w) dw. −6

10

(14)

We lower bound J by the composite trapezoidal rule with step h := 5 · 10−4 . Let Nh := 24000 and wk := −6 + kh (0 ≤ k ≤ Nh ). Define T := h



1 2 F (w0 ) +

NX h −1

 F (wk ) + 21 F (wNh ) .

k=1

By the standard error estimate for the composite trapezoidal rule [5, Sec. 4.4, Thm. 4.5], if f ∈ C 2 ([A, B]), h = (B − A)/m, and ! m−1 f (A) + f (B) X Th = h + f (A + kh) , 2 k=1

then there exists ξ ∈ (A, B) such that Z B f (x) dx − Th = − A

B − A 2 ′′ h f (ξ). 12

In particular, Z B

B − A 2 ′′ h ∥f ∥L∞ [A,B] . 12

f (x) dx − Th ≤ A

Applying this with A = −6, B = 6, f = F , and m = 24000, gives |J − T | ≤

12 2 h max |F ′′ | = h2 max |F ′′ |. 12 [−6,6] [−6,6] 2

Writing F = eg with g(w) = −w2 −bea w , one has F ′′ = (g ′′ +(g ′ )2 )F . A termwise global estimate −bt = 1/(be), sup 2 −bt = (2/b)2 e−2 , sup 2 −w2 = 1/e, sup −w2 = using t>0 t e w∈R w e w∈R |w|e √ supt>0 te 1/ 2e yields, for all w ∈ R, |F ′′ (w)| ≤ M (a) := 2 +

a4 4 4a2 + + √ + 4a4 e−2 . e e e 2e

For a = 6131/5000 one checks M (a) < 6.476. A direct finite computation gives T ≥ 0.33310717054594 and therefore J ≥ T − M (a)h2 ≥ 0.33310555154594. Moreover, 4

b a e−a /4 ≥ 1.6632596039

1 √ ≥ 0.5641895835. π

Hence the right-hand side of (14) with W = 6 is at least 1.6632596039 × 0.5641895835 × 0.33310555154594 > 0.312584. Finally, since (n/(n + 1))1/4 → 1, this implies lim inf n→∞

Γ(n) ≥ 0.312584, n1/4

which is exactly Theorem 1.

11

2.2

Proof of Theorem 2

Proof of Theorem 2. Let γ be the standard Gaussian measure on R, and let Hem denote the monic probabilists’ Hermite polynomial of degree m, normalized so that Z Hem (x)Hek (x) dγ(x) = m! δmk . R

In particular, ∥Hem ∥2L2 (γ) = m!. For N ≥ 1, define SN (x) :=

x1 + · · · + xN √ , N

x ∈ {−1, 1}N ,

and Fm,N (x) := Hem (SN (x))2 . Since Hem has degree m, the function Hem (SN (x)) is a polynomial of total degree at most m in the coordinates x1 , . . . , xN . After multilinear reduction on the cube (using x2i = 1), its Fourier–Walsh degree is still at most m. Hence deg Fm,N ≤ 2m. If ε1 , . . . , εN are i.i.d. Rademacher signs, then   ε1 + · · · + εN 2 √ , ∥Fm,N ∥1 = E Hem N and  ∥Fm,N ∥2 = Now

E Hem

ε1 + · · · + εN √ N

ε1 + · · · + εN √ =⇒ G N

4 !1/2 .

(N → ∞),

where G ∼ N (0, 1), and in fact all moments converge. Since He2m and He4m are fixed polynomials, it follows that ∥Hem ∥2L4 (γ) ∥Fm,N ∥2 lim = . N →∞ ∥Fm,N ∥1 ∥Hem ∥2L2 (γ) We now use the asymptotics of Larsson-Cohn. By [20, Theorem 2.1(b)], for every fixed p > 2, √ ∥Hem ∥Lp (γ) = c(p) m−1/4 m! (p − 1)m/2 (1 + o(1)) (m → ∞), where c(p) > 0 is an explicit constant. Taking p = 4, we obtain √ ∥Hem ∥L4 (γ) = c(4) m−1/4 m! 3m/2 (1 + o(1)), and therefore

∥Hem ∥2L4 (γ) ∥Hem ∥2L2 (γ)

= c(4)2 m−1/2 3m (1 + o(1)).

Consequently, lim

m→∞

∥Hem ∥2L4 (γ) ∥Hem ∥2L2 (γ) 12

!1/(2m) =

√ 3.

Let C <

√ 3. By the previous limit, one can choose m so large that ∥Hem ∥2L4 (γ) ∥Hem ∥2L2 (γ)

> C 2m .

Fix such an m. Then, by the convergence in N , for all sufficiently large N , ∥Fm,N ∥2 > C 2m . ∥Fm,N ∥1 But deg Fm,N ≤ 2m, so the inequality ∥f ∥2 ≤ C d ∥f ∥1 √ √ fails for d = 2m. Since this happens for every C < 3, we conclude that C∗ ≥ 3.

2.3

Proof of Theorem 3

Let n ∈ N be even. In this section, we show that the balanced P Szarek’s inequality, i.e., (x1 , ..., xn ) is uniformly distributed on Ωn := {(x1 , ..., xn ) ∈ {−1, 1}n : ni=1 xi = 0}, holds with the optimal constant. The derivation of the optimal constant follows the semigroup proof framework for Szarek’s inequality (also called the L1 Khintchine inequality) due to Kwapień, Latala, and Oleszkiewicz [19, 21, 21] (see the second proof of Theorem 32 in [26] for a modern presentation). In their setting, the random variables are independent, whereas in our setting they are slightly dependent. Their framework, however, still works. To deal with the lack of independence, we invoke a recent result of Filmus [10] on orthogonal bases for the slice of the Hamming cube. This may suggest that a certain symmetry of the distribution can be a suitable replacement for the independence condition in some Khintchine-type inequalities. 2.3.1

Harmonic Multilinear Polynomial

Let f : Rn → R be a multilinear polynomial. We call f a harmonic multilinear polynomial if n X ∂f i=1

∂xi

(x) = 0

for all x ∈ Rn .

One can check that the harmonic multilinear polynomials on Rn form a vector space, which
n we denote by Hn . Note that Hn has dimension ⌊ n ⌋ and every element in Hn is a multilinear 2 polynomial of degree at most n2 ; see [10, Lemma 2] for a proof. A probability distribution µ on Rn is called exchangeable if it is invariant under permutations of the coordinates. Given an exchangeable distribution µ with mild regular conditions, we define the inner product on Hn by ⟨f, g⟩ = Ex∼µ [f (x)g(x)], and set ∥f ∥22 = ⟨f, f ⟩. Filmus [10, Section 3] constructed a finite collection of functions {χB }B∈Bn that forms an orthogonal basis of Hn with respect to any exchangeable probability distribution µ, and showed that every f ∈ Hn admits the following Young-Fourier expansion (whenever ∥χB ∥22 > 0 for all B ∈ Bn ): X f= fb(B)χB , B∈Bn B⟩ where fb(B) = ⟨f,χ . We will use the following facts from [10, Section 3]: every χB in the basis ∥χB ∥22

is a homogeneous multilinear polynomial indexed by B, where B is a list of increasing numbers in [n]; the length of B, denoted by |B|, equals the degree of χB . 13

Filmus [10, Section 6] further defined a Laplacian operator L by X 1 f (i,j) , Lf = f − n
2

1≤i<j≤n

where f (i,j) = f (x(i,j) ) and x(i,j) is obtained from x by swapping its ith and jth coordinates. He also showed that L admits the following spectral decomposition: X 2|B|(n + 1 − |B|) Lf = fb(B)χB . n(n − 1) B∈Bn

2.3.2

Balanced Szarek’s inequality with best constant

Theorem 11. Let n ∈ N be even. For any function f : Ωn → R, there exists a unique f˜ ∈ Hn such that f˜(x) = f (x) for all x ∈ Ωn .
Proof. Since the dimension of functions on Ωn is nn , which matches the cardinality of {χB }B∈Bn , 2 it suffices to show that {χB }B∈Bn is a set of linearly independent vectors when restricted to Ωn . Equip Rn with the uniform probability measure on Ωn . Thus it suffices to show that {χB }B∈Bn forms an orthogonal basis with respect to the uniform probability measure on Ωn . Since the uniform probability measure on Ωn is exchangeable, by Theorem 9 in [10], we know that ⟨χA , χB ⟩ = 0 for any A, B ∈ Bn with A = ̸ B. It remains to show that ∥χB ∥22 > 0 for all B ∈ Bn . Theorem 10 in [10] shows that, for B ∈ Bn of length d, ∥χB ∥22 = cB ∥χd ∥22 Q Q where χd = di=1 (x2i−1 −x2i ) and cB = di=1 (bi −2(i−1))(b2i −2(i−1)−1) . By the definition of B ∈ Bn (see the definition of the top set under the Definition 4 in [10]) , one can deduce that bi ≥ 2i for all i ∈ [d]. Hence cB > 0 and it suffices to show that ∥χd ∥22 > 0. Let a ∈ Ωn such that ai = (−1)i for all i ∈ [n]. Since χd (a) ̸= 0 and Pr{x = a} > 0 under the uniform probability measure on Ωn , we conclude that ∥χd ∥22 > 0, completing the proof. For the rest of the section, we equip Rn with the uniform probability measure on Ωn , which is exchangeable. Hence, many of the results in [10] apply. Let n ∈ N be even, and let (a1 , ..., an ) ∈ Rn . By Theorem 11, there exists f ∈ Hn such that f (x) = |

n X

ai xi |

for all x ∈ Ωn .

i=1

Theorem 12 (Spectrum is supported on even indices). The spectrum of f defined above is supported on the even indices, i.e., X f (x) = fb(B)χB (x). B∈Bn |B| even

Proof. Note that f (x) = f (−x) for all x ∈ Ωn . Looking at this fact from the Young-Fourier expansion of f , we have that X X fb(B)χB (x) = fb(B)(−1)|B| χB (x), for all x ∈ Ωn B∈Bn

B∈Bn

due to the fact that χB is a homogeneous multilinear polynomial of degree |B|. After rearranging the terms, we have that X fb(B)χB (x) = 0, for all x ∈ Ωn . B∈Bn |B| is odd

14

Since the functions {χB }B∈Bn are linearly independent by Theorem 11, we know that fb(B) = 0 for all B ∈ Bn with |B| odd, completing the proof. A fact used in the Kwapień-Latala-Oleszkiewicz’s proof of the Szarek’s inequality is a tightened Poincaré inequality for f discussed above. We now derive an analogue on the middle slice. Theorem P13 (A tightened Poincaré inequality). Let n ∈ N be even. Let f ∈ Hn such that f (x) = | ni=1 ai xi | for all x ∈ Ωn . Var(f ) := Eunif{Ωn } (f − Eunif{Ωn } f )2 . Then we have the tightened Poincaré inequality 4 Var(f ) ≤ ⟨f, Lf ⟩. n Proof. ⟨f, Lf ⟩ =

X 2|B|(n + 1 − |B|) B∈Bn

n(n − 1)

fb(B)2 ∥χB ∥22 =

X even |B|≥2

2|B|(n + 1 − |B|) b 2 f (B) ∥χB ∥22 , n(n − 1)

since fb(B) = 0 for B with |B| odd by Theorem 12. Moreover, Var(f ) = ⟨f − Eunif{Ωn } f, f − Eunif{Ωn } f ⟩ = ⟨f − fb(∅), f − fb(∅)⟩ =

X

fb(B)2 ∥χB ∥22 .

even |B|≥2

By [10, Lemma 2], |B| ≤ n2 . Hence we have 2|B|(n+1−|B|) ≥ n4 for even |B| ≥ 2 by direct n(n−1) calculation, completing the proof. Another fact used in the Kwapień-Latala-Oleszkiewicz’s proof of the Szarek’s inequality is a pointwise bound for Lf . We now derive an analogue on the middle slice. Lemma 14. Let n ∈ N be even and n ≥ 4. Let f be the function discussed above. Then Lf (x) ≤

2 f (x), n−1

for all x ∈ Ωn .

P Proof. Denote by Sn (x) = ni=1 ai xi . Note that for x ∈ Ωn , X X
Sn(i,j) (x) = Sn (x) + (ai − aj )(xj − xi ) i<j

i<j

  n 1X (ai xj − ai xi − aj xj + aj xi ) = Sn (x) + 2 2 i̸=j   n 1X 1X 1X 1X ai xj − ai xi − aj xj + aj xi = Sn (x) + 2 2 2 2 2 i̸=j i̸=j i̸=j i̸=j   n n X n 1X 1 = Sn (x) + (ai ( xj ) − ai xi ) − (n − 1)Sn (x) 2 2 2 i=1

j=1

n n X 1 1X − (n − 1)Sn (x) + (aj ( xi ) − aj xj ) 2 2 j=1 i=1   n 1 1 1 1 = Sn (x) − Sn (x) − (n − 1)Sn (x) − (n − 1)Sn (x) − Sn (x), 2 2 2 2 2   n =( − n)Sn (x). 2

15

since

n X i=1

xi = 0

Hence, for x ∈ Ωn Lf (x) = |Sn (x)| −

1 X (i,j)
|Sn (x)| n 2

i<j

1 X (i,j) ≤ |Sn (x)| − n
| Sn (x)| 2

i<j

  1 n = |Sn (x)| − n
|( − n)Sn (x)| 2 2 2 = f (x). n−1

Proof of Theorem 3. The upper bound follows from Jensen’s inequality. For the lower bound, we proceed asPfollows. Let f (x) be the unique multilinear harmonic polynomial over Rn such that f (x) = | ni=1 ai xi | for all x ∈ Ωn . Then by Theorem 13 and Lemma 14, we have 4 2 Var(f ) ≤ ⟨f, Lf ⟩ ≤ ∥f ∥22 . n n−1 Since Var(f ) = ∥f ∥22 − (Ef )2 , we obtain s Ef ≥

n−2 ∥f ∥2 , 2(n − 1)

completing the proof. Remark 15. It would be interesting to explore further which symmetry condition of the distribution of (x1 , . . . , xn ) leads to Khintchine-type inequalities, without assuming that the xi are independent.

2.4

Proof of Theorem 5

The lower bound c ≥ 1.28 was proved in [7]. The key estimate, established in [7, Lemma 3], is that for all m, n ∈ N, 2 1 c ≥ bn,m − − 2, m m where bn,m is defined in terms of a discrete set Bn,m ; see [7, Lemma 3] for the definition of Bn,m and further details. Using a large-scale computation, Cloninger and Steinerberger [7, Section 3.3] showed that for n = 24 and m = 50, bn,m ≥ 1.28 +

2 1 + 2, m m

which yields c ≥ 1.28. Their computation required about 20000 CPU hours on a high-performance server. Our improvement comes from a small refinement of the error term. Namely, we replace the estimate above by 2 1 c ≥ bn,m − − . m 2m2 Using the same computed value of b24,50 , this gives c ≥ 1.2802.

16

The argument is very simple: the only change is in the bound for ε ∗ ε. In [7, Lemma 3], one uses 1 ∥ε ∗ ε∥L∞ (R) ≤ 2 . m   Since ε is supported on − 14 , 41 , which has length 12 , we have ∥ε∥L1 (R) ≤

1 1 ∥ε∥L∞ (R) ≤ . 2 2m

Therefore, ∥ε ∗ ε∥L∞ (R) ≤ ∥ε∥L∞ (R) ∥ε∥L1 (R) ≤

1 1 1 · = . m 2m 2m2

Substituting this into the proof of [7, Lemma 3] yields c ≥ bn,m −

2 1 , − m 2m2

as claimed.

Acknowledgments P.I. acknowledges partial support from the NSF CAREER grant DMS-2152401, a Simons Fellowship, and a Humboldt Research Fellowship for Experienced Researchers.

A

Chat Links • Theorem 1. Chat link with Grok 4.20(Beta). https://grok.com/share/c2hhcmQtNA_452b1841-e7b1-4695-a6aa-cf3210365900?rid= f9834ecf-f716-4a48-81c6-15aefa0dcba7 • Upper bound in Theorem 2. Chat link with Grok 4 Expert. https://grok.com/share/c2hhcmQtNA_84e20fd0-d81a-4809-9728-83d37e88cd5f?rid= f6483ca7-f772-4e51-a8e2-f62926b6dfab • Lower bound in Theorem 2. Chat link with Grok 4 Heavy. https://grok.com/share/c2hhcmQtNA_826731f3-afbc-42a5-aa61-acca3ec1324a?rid= de4a202d-357d-4ece-9954-16c0dc5951d5 • Theorem 3. Chat link with Grok 4.30(Beta). https://grok.com/share/c2hhcmQtNA_de1dfee6-b59f-4fa8-a439-aa176d2308c6 • Theorem 5. Chat link with Grok 4 Heavy. https://grok.com/share/c2hhcmQtNQ_f4d17f80-4582-4679-b931-06277fd4cfd4?rid= a60436ae-eaba-4638-a0fd-47b231f19cd0

References [1] K. Ball. The Reverse Isoperimetric Problem for Gaussian Measure. Discrete & Computational Geometry, 10:411–420, 1993.

17

[2] V. Bentkus. On the dependence of the Berry–Esseen bound on dimension. J. Statist. Plann. Inference, Volume 113, 2003, pp. 385–402. [3] S. Bubeck, C. Coester, R. Eldan, T. Gowers, Y. T. Lee, A. Lupsasca, M. Sawhney, R. Scherrer, M. Sellke, B. K. Spears, D. Unutmaz, K. Weil, S. Yin, N. Zhivotovskiy. Early science acceleration experiments with GPT-5. arXiv preprint arXiv:2511.16072, 2025. [4] S. Bubeck, J. Ding, R. Eldan, M. Z. Rácz. Testing for high-dimensional geometry in random graphs. Random Structures & Algorithms, Volume 49, Issue 3, 2016, pp. 503–532. [5] R. L. Burden, J. D. Faires, and A. M. Burden, Numerical Analysis, 10th ed., Cengage Learning, Boston, MA, 2016. [6] C1a Sidon set autocorrelation constant. Optimization Constants in Mathematics. https://teorth.github.io/optimizationproblems/constants/1a.html, accessed April 16, 2026. [7] A. Cloninger, S. Steinerberger. On suprema of autoconvolutions with an application to Sidon sets. Proceedings of the American Mathematical Society, Volume 145, Issue 8, 2017, pp. 3191–3200. [8] A. Eskenazis and P. Ivanisvili, Polynomial inequalities on the Hamming cube, Probability Theory and Related Fields 178 (2020), 235–287. [9] T. Feng, T. Trinh, G. Bingham, J. Kang, S. Zhang, S.-h. Kim, K. Barreto, C. Schildkraut, J. Jung, J. Seo, C. Pagano, Y. Chervonyi, D. Hwang, K. Hou, S. Gukov, C.-C. Tsai, H. Choi, Y. Jin, W.-Y. Li, H.-A. Wu, R.-A. Shiu, Y.-S. Shih, Q. V. Le, T. Luong. Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems. arXiv preprint arXiv:2601.22401, 2026. [10] Y. Filmus. An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube. The Electronic Journal of Combinatorics, Volume 23, Issue 1, 2016. [11] Y. Filmus, H. Hatami, S. Heilman, E. Mossel, R. O’Donnell, S. Sachdeva, A. Wan, K. Wimmer. Real Analysis in Computer Science: A collection of Open Problems. Simons Institute (2014). https://simons.berkeley.edu/sites/default/files/openprobsmerged.pdf. [12] B. Georgiev, J. Gómez-Serrano, T. Tao, A. Z. Wagner. Mathematical exploration and discovery at scale. arXiv preprint arXiv:2511.02864, 2025. [13] U. Haagerup. The best constants in the Khintchine inequality. Studia Mathematica, Volume 70, 1981, pp. 231–283. [14] O. Herscovici, S. Spektor. The best constant in the Khinchine inequality for slightly dependent random variables. arXiv preprint arXiv:1806.03562, 2020. [15] P. Ivanisvili and T. Tkocz, Comparison of moments of Rademacher chaoses, Ark. Mat. 57 (2019), no. 1, 121–128. [16] P. Ivanisvili, X. Xie. Counterexample to majority optimality in NICD with erasures. arXiv preprint arXiv:2510.20013, 2025. [17] H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, B. Dong. Automated Conjecture Resolution with Formal Verification. arXiv preprint arXiv:2604.03789, 2026.

18

[18] A. Klivans, R. O’Donnell, and R. A. Servedio. Learning geometric concepts via Gaussian surface area. In Proc. 49th IEEE Symposium on Foundations of Computer Science (FOCS), pages 541–550, 2008. [19] S. Kwapień, R. Latala, K. Oleszkiewicz. Comparison of moments of sums of independent random variables and differential inequalities. J. Funct. Anal., Volume 136, Issue 1, 1996, pp. 258–268. [20] L. Larsson-Cohn, Lp -norms of Hermite polynomials and an extremal problem on Wiener chaos, Ark. Mat. 40 (2002), 133–144. [21] R. Latala, K. Oleszkiewicz. On the best constant in the Khinchin-Kahane inequality. Studia Math., Volume 109, Issue 1, 1994, pp. 101–104. [22] G. Martin, K. O’Bryant. The symmetric subset problem in continuous Ramsey theory. Experimental Mathematics, Volume 16, Issue 2, 2007, pp. 145–165. [23] G. Martin, K. O’Bryant. The supremum of autoconvolutions, with applications to additive number theory. Illinois Journal of Mathematics, Volume 53, Issue 1, 2009, pp. 219–235. [24] M. Matolcsi, C. Vinuesa. Improved bounds on the supremum of autoconvolutions. Journal of Mathematical Analysis and Applications, Volume 372, Issue 2, 2010, pp. 439–447. [25] S. Nadimpalli and C. Pascale. On the Maximal Gaussian Perimeter of Convex Sets, Revisited. Preprint, 2025. arXiv:2508.20079. [26] P. Nayar and T. Tkocz. Extremal sections and projections of certain convex bodies: a survey. Harmonic Analysis and Convexity, 343-390, 2023. [27] F. Nazarov. On the maximal perimeter of a convex set in Rn with respect to a Gaussian measure. In Geometric Aspects of Functional Analysis (2001–2002), pages 169–187. Lecture Notes in Mathematics, Vol. 1807, Springer, 2003. [28] R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, Cambridge, 2014. [29] K. Oleszkiewicz. Comparison of moments via Poincaré-type inequality. In Advances in stochastic inequalities (Atlanta, GA, 1997), pp. 135–148, Contemp. Math., Volume 234, Amer. Math. Soc., Providence, RI, 1999. [30] M. Raič. A multivariate Berry–Esseen theorem with explicit constants. 25(4A):2824–2853, 2019.

Bernoulli,

[31] A. Schinzel, W. M. Schmidt. Comparison of L1 - and L∞ -norms of squares of polynomials. Acta Arithmetica, Volume 104, Issue 3, 2002, pp. 283–296. [32] S. Spektor. Restricted Khinchine inequality. Canadian Mathematical Bulletin, Volume 59, 2016, pp. 204–210. [33] S. J. Szarek. On the best constants in the Khinchin inequality. Studia Mathematica, Volume 58, 1976, pp. 197–208. [34] T. Tao. A crowdsourced repository for optimization constants? What’s new, 22 January 2026. https://terrytao.wordpress.com/2026/01/22/ a-crowdsourced-repository-for-optimization-constants/.

19

[35] T. Tao. I encountered no issues with hallucinations or other AI-generated nonsense. Mathstodon post, 2 October 2025. Status 115306550084415351. https://mathstodon.xyz/ @tao/115306550084415351, accessed April 20, 2026. [36] Y. Wang, S.-R. Su, Z. Zeng, E. Xu, L. Ren, X. Yang, Z. Huang, X. He, L. Ma, B. Peng, H. Cheng, P. He, W. Chen, S. Wang, S. S. Du, Y. Shen. ThetaEvolve: Test-time Learning on Open Problems. arXiv preprint arXiv:2511.23473, 2025. [37] J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), no. 9, 563–564. doi:10.2307/2304460. [38] What is the minimal Ck , such that every f : {−1, 1}n → R of degree at most k satisfies ∥f ∥2 ≤ Ck ∥f ∥1 ? MathOverflow, Question 184286, 2014. Available at https: //mathoverflow.net/questions/184286/ (accessed April 3, 2026). [39] D. P. Woodruff, V. Cohen-Addad, L. Jain, J. Mao, S. Zuo, M. Bateni, S. Branzei, M. P. Brenner, L. Chen, Y. Feng, L. Fortnow, G. Fu, Z. Guan, Z. Hadizadeh, M. T. Hajiaghayi, M. JafariRaviz, A. Javanmard, K. C. S., K.-i. Kawarabayashi, R. Kumar, S. Lattanzi, E. Lee, Y. Li, I. Panageas, D. Paparas, B. Przybocki, B. Subercaseaux, O. Svensson, S. Taherijam, X. Wu, E. Yogev, M. Zadimoghaddam, S. Zhou, Y. Matias, J. Manyika, V. Mirrokni. Accelerating Scientific Research with Gemini: Case Studies and Common Techniques. arXiv preprint arXiv:2602.03837, 2026. [40] M. Yuksekgonul, D. Koceja, X. Li, F. Bianchi, J. McCaleb, X. Wang, J. Kautz, Y. Choi, J. Zou, C. Guestrin, Y. Sun. Learning to Discover at Test Time. arXiv preprint arXiv:2601.16175, 2026.

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