=== PAGE 1 ===
A Sharp Small-Coefficient Variant of Khintchine’s
Inequality and the Sharp π/2 Theorem
Lei Yu∗
September 1, 2026
Abstract
We prove a refined quadratic normal approximation for the first abso-
lute moment of normalized weighted Rademacher sums with bounded max-
imal coefficients.
For any weight vector w ∈Rn satisfying ∥w∥2 = 1 and
∥w∥∞≤β with sufficiently small β > 0, we establish the uniform error bound
E |Pn
i=1 wiXi| −
p
2/π
 = O(β2) over all admissible weight configurations.
Our proof combines zero-bias Stein’s method and refined small-ball probabil-
ity estimates to exploit symmetry cancellation and control the non-smooth
residual of the absolute-value test function. An explicit extremal construc-
tion further verifies the optimality of this quadratic convergence rate. As an
application, we establish an asymptotically sharp refinement of the Friedgut–
Kalai–Naor (FKN) theorem for Boolean functions, also known as the sharp
π/2 theorem, characterizing the level-1 Fourier energy for functions deviating
far from dictatorships.
1
Introduction
Let X1, X2, . . . , Xn be independent identically distributed Rademacher random vari-
ables, i.e., P(Xi = 1) = P(Xi = −1) = 1/2. Given a weight vector w = (w1, . . . , wn) ∈
∗L. Yu is with the School of Statistics and Data Science, LPMC, KLMDASR, and LEBPS,
Nankai University, Tianjin 300071, China (e-mail: leiyu@nankai.edu.cn). This work was supported
by the National Key Research and Development Program of China under grant 2023YFA1009604
and the NSFC under grant 62101286.
1
arXiv:2608.29703v1  [math.PR]  30 Aug 2026

=== PAGE 2 ===
Rn, define the normalized weighted Rademacher sum
W =
n
X
i=1
wiXi,
subject to the normalization condition ∥w∥2
2 = Pn
i=1 w2
i = 1 and maximal coeffi-
cient bound ∥w∥∞= max1≤i≤n |wi| ≤β. By the classical central limit theorem, W
converges in distribution to a standard Gaussian random variable Z ∼N(0, 1). A
fundamental Gaussian approximation for symmetric distributions is the first abso-
lute moment matching: E |W| →E |Z| =
p
2/π as β →0. K¨onig, Sch¨utt, and
Tomczak-Jaegermann [1] proved the following variant of Khintchine’s inequality
E|W| −
p
2/π
 ≤

1 −
r
2
π

β,
(1)
which can be also recovered with an implicit coefficient C by applying the Berry–
Esseen Theorem with proper truncation argument [2, Theorem 5.16].
In this work, we rigorously prove that the absolute moment approximation error
decays quadratically in β, establishing a sharp improvement over the bound above.
We define
G(β) = sup
(
E

n
X
i=1
wiXi
 : n ≥1, ∥w∥2 = 1, ∥w∥∞≤β
)
,
which quantifies the worst-case Gaussian approximation error over all admissible
weighted Rademacher sums. Our main theorem establishes the optimal quadratic
expansion of G(β), uniform over all valid weight configurations.
Theorem 1 (Sharp Small-Coefficient Variant of Khintchine’s Inequality). There
exist universal constants C, β0 > 0 such that for all 0 < β ≤β0, all integers n ≥1,
and all weight vectors w ∈Rn with ∥w∥2 = 1 and ∥w∥∞≤β,
E |W| −
r
2
π
 ≤Cβ2.
(2)
Consequently, G(β) =
p
2/π +O(β2), and the quadratic rate is optimal in the order.
To confirm the quadratic rate is sharp, construct the uniform weight vector wi =
1/√n with n = ⌈β−2⌉. This satisfies ∥w∥2 = 1 and ∥w∥∞= 1/√n ≤β. Applying
Stirling’s formula yields
E

1
√n
n
X
i=1
Xi
 ≥
r
2
π −Cβ2 + O(β4),
2

=== PAGE 3 ===
with C =
1
2√π for even n and C =
3
2
√
2π for odd n, providing a matching upper
bound. Thus G(β) =
p
2/π + O(β2), and the O(β2) rate is optimal.
We next prove (2) by combining Stein’s method and the classical CLT.
2
Preliminaries
We recap core tools from Stein’s method and zero-bias transformation required for
the main proof, including regularity properties of the Stein equation solution for the
absolute value function.
2.1
Zero-Bias Transformation
Stein’s characterization for standard normality states that a random variable Z ∼
N(0, 1) if and only if
E[Zf(Z)] = E[f ′(Z)]
for all absolutely continuous functions f with integrable derivatives. For any mean-
zero, unit-variance random variable W, its zero-bias transform W ∗is the unique
random variable satisfying
E[Wf(W)] = E[f ′(W ∗)]
for all admissible f. For weighted Rademacher sums W = Pn
i=1 wiXi with ∥w∥2 = 1,
we construct W ∗explicitly. Let U1, . . . , Un be independent uniform random variables
on [−1, 1], independent of {Xi}. The zero-bias transform of each summand wiXi is
wiUi, verified via integration by parts. Let I be an independent random index with
P(I = i) = w2
i , and define leave-one-out sums
Yi =
X
j̸=i
wjXj.
The zero-bias transform of W is then
W ∗= YI + wIUI.
This explicit coupling is a standard tool for refined normal approximation of sym-
metric sums [3].
3

=== PAGE 4 ===
2.2
Stein Equation for the Absolute Value Function
Let a = E |Z| =
p
2/π denote the standard Gaussian first absolute moment. We
solve the Gaussian Stein equation for the test function h(x) = |x|:
f ′(x) −xf(x) = |x| −a.
(3)
The unique bounded solution to (3) is
f(x) = ex2/2
Z x
−∞
(|t| −a) e−t2/2dt.
(4)
The solution possesses uniform derivative regularity away from the origin, critical for
our local error analysis.
Lemma 1. There exists a universal constant C0 > 0 such that the Stein solution f
in (4) satisfies
sup
x̸=0
|f ′′(x)| ≤C0,
sup
x̸=0
|f ′′′(x)| ≤C0.
Proof. For x > 0, using
R ∞
−∞(|t| −a)e−t2/2 dt = 0, we may rewrite (4) as
f(x) = −ex2/2
Z ∞
x
(t −a)e−t2/2 dt = −1 + aR(x),
(5)
where
R(x) := ex2/2
Z ∞
x
e−t2/2 dt.
Substituting (5) into (3) yields
f ′(x) = x (−1 + aR(x)) + x −a = axR(x) −a.
(6)
By the Mills ratio inequality, 0 ≤xR(x) ≤1 for x > 0, so |f ′(x)| ≤a for x >
0. The case x < 0 follows by symmetry since f is odd, hence f ′ is even. Thus
supx̸=0 |f ′′(x)| ≤a < ∞.
We now prove the bound on f ′′. Differentiating (3) for x̸ = 0 gives
f ′′(x) −f(x) −xf ′(x) = sgn(x).
(7)
Using (5) and (6), we obtain for x > 0,
f ′′(x) = a(1 + x2)R(x) −ax.
4

=== PAGE 5 ===
Using the standard Mills ratio expansion R(x) = x−1 −x−3 + O(x−5), we obtain
f ′′(x) = a(1 + x2)
 x−1 −x−3 + O(x−5)

−ax = −ax−3 + O(x−3) = O(x−3).
Thus f ′′ is bounded on (0, ∞), and by symmetry on (−∞, 0). Hence ∥f ′′∥∞≤C0.
Finally, we establish the bound on f ′′′ away from 0. Differentiating (7) for x̸ = 0
gives
f ′′′(x) = xf ′′(x) + 2f ′(x).
Since f ′′(x) = O(|x|−3), we have
|f ′′′(x)| ≤|x| |f ′′(x)| + 2|f ′(x)| ≤C′ + 2 ∥f ′∥∞< ∞,
uniformly for x̸ = 0. Hence supx̸=0 |f ′′′(x)| ≤C1.
2.3
Taylor Estimate
The only non-smoothness of f ′ occurs at x = 0, which dominates the residual approx-
imation error. To quantify this singularity, define the truncated second derivative
d(y) =
(
f ′′(y)
y̸ = 0,
0
y = 0.
Lemma 2. There exists a universal constant C1 > 0 such that for all y, u ∈R,
|f ′(y + u) −f ′(y) −u d(y)| ≤C1
 u2 + |u| 1{|y|≤|u|}

.
(8)
Proof. Case 1: The line segment connecting y and y+u does not contain 0. Then f ′′′
is uniformly bounded on the segment by Lemma 1, and standard Taylor expansion
yields a remainder of order O(u2).
Case 2: The segment crosses 0. This forces |y| ≤|u|. Since f ′ is absolutely
continuous and f ′′ are bounded a.e., as a standard consequence of the fundamen-
tal theorem of calculus for absolutely continuous functions, f ′ is Lipschitz contin-
uous with Lipschitz constant at most K = supx̸=0 |f ′′(x)| < ∞.
We obtain the
bound |f ′(y + u) −f ′(y)| + |ud(y)| ≤2K |u|, which is absorbed by the indicator
term |u| 1{|y|≤|u|}.
Combining both cases establishes (8).
5

=== PAGE 6 ===
3
Proof of Theorem 1
We decompose the approximation error via the zero-bias Stein identity, bound incre-
mental differences between Rademacher and uniform variables, and control small-ball
probabilities via the Berry–Esseen theorem to yield quadratic error decay.
For fixed index i and state y ∈R, define the increment discrepancy
Ai(y) = EX [f ′(y + wiX)] −EU [f ′(y + wiU)] ,
where X is Rademacher and U is uniform on [−1, 1].
Both variables have zero
mean, so linear Taylor terms vanish under expectation. Applying Lemma 2 with
y + u = wiX and y = wiU yields
|Ai(y)| ≤C2w2
i + C2 |wi| 1{|y|≤|wi|}.
(9)
By the Stein identity and zero-bias characterization,
E |W| −a = E [f ′(W) −Wf(W)] = Ef ′(W) −Ef ′(W ∗).
Substituting the zero-bias construction of W ∗, we expand the error:
|E |W| −a| ≤
n
X
i=1
w2
i E |Ai(Yi)|
≤C2
n
X
i=1
w4
i + C2
n
X
i=1
|wi|3 P (|Yi| ≤|wi|) .
(10)
We bound the small-ball probabilities P(|Yi| ≤|wi|) using the classical Berry–
Esseen theorem. The variance of leave-one-out sums is
σ2
i = Var(Yi) = 1 −w2
i .
For β ≤1/2, we have σ2
i ≥3/4. The third absolute moment of the summands of Yi
satisfies
ρi =
X
j̸=i
|wj|3 ≤∥w∥∞
X
j̸=i
w2
j ≤β.
The Berry–Esseen inequality gives a uniform distributional error bound:
sup
x∈R
|P(Yi ≤x) −Φ(x/σi)| ≤CBE
ρi
σ3
i
≤C3β.
(11)
6

=== PAGE 7 ===
Since σi ≥
p
3/4 and |wi| ≤β, there exists a universal constant C4 > 0 such that
P (|Z| ≤|wi|/σi) ≤C4|wi| ≤C4β. Combining with the Berry–Esseen error term, we
obtain
P (|Yi| ≤|wi|) ≤P

|Z| ≤|wi|
σi

+ 2C3β
≤C4β + 2C3β ≤C5β.
(12)
Using the normalization and maximal weight bound:
n
X
i=1
w4
i ≤∥w∥2
∞
n
X
i=1
w2
i ≤β2,
n
X
i=1
|wi|3 ≤∥w∥∞
n
X
i=1
w2
i ≤β.
(13)
Substituting (12) and (13) into (10) yields
|E |W| −a| ≤C2β2 + C2C5β · β ≤Cβ2.
This establishes the upper bound of Theorem 1.
4
Application to Strengthening FKN Theorem
As a direct and significant application of our quadratic Gaussian approximation
theorem for weighted Rademacher sums, we derive a refined level-1 Fourier weight
bound for Boolean functions, which strengthens the classical Friedgut–Kalai–Naor
(FKN) theorem in the regime where the maximal absolute value of single-coordinate
Fourier coefficient tends to zero, namely, functions deviate far from dictatorships.
4.1
The Sharp π
2 Theorem
Let f : {±1}n →{±1} be a Boolean function. Denote its level-1 Fourier weight and
maximal level-1 Fourier coefficient by
W1(f) :=
n
X
i=1
bf 2
{i},
θ := max
1≤i≤n
 bf{i}
,
where bfS = EX[f(X)χS(X)] are the standard Boolean Fourier coefficients and χS(X) =
Q
i∈S Xi are Rademacher characters.
We state the refined FKN-type bound as a corollary of our main theorem.
7

=== PAGE 8 ===
Theorem 2 (Sharp π
2 Theorem). There exists a universal constant C > 0 such that
for every Boolean function f : {±1}n →{±1} with sufficiently small θ = maxi | bf{i}|,
W1(f) =
n
X
i=1
bf 2
{i} ≤2
π + Cθ2.
Moreover, the quadratic rate is attained by majority functions in the sense of order,
and thus, is optimal in the order.
This theorem improves the existing π
2 theorem [2] where the latter stating W1(f) ≤
2
π + Cθ. Our upper bound is asymptotically optimal as θ →0. For odd n, consider
the majority function g(x) = sgn(Pn
i=1 Xi)

. For this function, using the refined
Stirling expansion, as n →∞,
θ =
bg{i}
 =
1
2n−1
n −1
n−1
2

=
s
2
π(n −1)

1 −
1
4(n −1) + O(n−2)

,
and
W1(g) =
n
4n−1
n −1
n−1
2
2
= 2
π

1 +
1
2(n −1) + O(n−2)

.
Therefore, as θ →0,
W1(g) = 2
π + θ2
2 + O(θ4),
confirming the optimality of the quadratic order.
Denote a = Ef. By applying Chang’s bound, the present author [4] derived a
bound on W1 that is sharp for a = 1
2, θ = 1
4 and asymptotically sharp for a = 1
2, θ ↑1
2.
By applying the variant of Khintchine’s inequality of K¨onig, Sch¨utt, and Tomczak-
Jaegermann in (1), the present author [4] derived another bound: W1(f) ≤2
π + Cθ
with explicit C. Further improvements of FKN-type bound can be also found in [5].
4.2
Proof of Theorem 2
Associate to the linear Fourier spectrum of f the weighted Rademacher sum
Z =
n
X
i=1
bf{i}Xi,
where Xi are independent Rademacher variables. By orthogonality of characters,
E[Z2] = W1(f),
E[fZ] =
X
i
bf{i}E[fXi] =
X
i
bf 2
{i} = W1(f).
8

=== PAGE 9 ===
Since f(x) ∈{±1}, we have the pointwise inequality fZ ≤|Z|. Taking expectation
yields the key deterministic spectral bound:
W1(f) = E[fZ] ≤E|Z|.
(14)
Let s =
p
W1(f) and define normalized weights
wi =
bf{i}
s .
These weights satisfy the normalization ∥w∥2 = 1 and maximal coefficient bound
∥w∥∞= θ
s =: β.
By our Theorem 1, for all sufficiently small β > 0,
E

n
X
i=1
wiXi
 =
r
2
π + O(β2).
Rescaling by s recovers the expectation for the unnormalized sum Z:
E|Z| = s ·
 r
2
π + O
θ2
s2
!
=
r
2
π s + O
θ2
s

.
(15)
Combining (14) and (15), we obtain
s2 ≤
r
2
π s + O
θ2
s

.
Multiplying both sides by s eliminates the denominator:
s3 ≤
r
2
π s2 + O(θ2).
(16)
Rewrite (16) as
s2
 
s −
r
2
π
!
≤Cθ2,
for some universal constant C > 0. For small θ, the right-hand side is negligible,
which forces s to be close to
p
2/π. Precisely, suppose s =
p
2/π + δ for δ > 0.
Then 2
π δ ≤Cθ2, i.e., δ = O(θ2). This yields the tight bound
s ≤
r
2
π + O(θ2).
Squaring both sides gives
s2 = W1(f) ≤2
π + O(θ2).
(17)
9

=== PAGE 10 ===
Acknowledgments
Generative-AI use disclosure: During the preparation of this manuscript, the author
used ChatGPT (GPT-5.5-mini model) as an auxiliary tool for exploring proof ideas
and improving exposition. The authors take full responsibility for all mathematical
claims.
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[2] R. O’Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
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[4] L. Yu. On the Φ-stability and related conjectures. Probability Theory and Related
Fields, 186:1045–1080, 2023.
[5] L. Yu. On average distance, level-1 fourier weight, and chang’s lemma. arXiv
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