Boolean Fourier Analysis

• Boolean Fourier analysis is the study of functions on the discrete hypercube using Walsh expansions to capture spectral coefficients, degree, and influence.
• It translates combinatorial and probabilistic queries into spectral properties, forming a central toolkit in theoretical computer science and learning theory.
• Recent work emphasizes exact recovery from low-degree data and extends analysis to biased, non-binary, geometric, and quantum settings.

Boolean Fourier analysis studies functions on the discrete hypercube through their expansion in the Walsh, or parity, basis. In the $\{-1,1\}^n$ convention, it turns combinatorial, probabilistic, and algorithmic questions into statements about spectral coefficients, degree, influence, and noise stability, and it has become a central toolkit across theoretical computer science, combinatorics, learning theory, social choice, hardness of approximation, and related parts of analysis (O'Donnell, 2021). Contemporary work also emphasizes exact reconstruction from low-degree data, extensions to biased and non-binary domains, operator-valued and diagrammatic analogues, and quantum or support-theoretic variants of the classical spectrum (Chen, 15 Apr 2026, Chakraborty et al., 2024, Jones et al., 2024, 0810.2435, Carlet et al., 2022).

1. Walsh basis, Fourier expansion, and degree

For functions $f : \{-1,1\}^n \to \mathbb{R}$, the Walsh characters are

$$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$

and they form an orthonormal basis under the uniform inner product

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$

Every $f$ therefore admits a unique Fourier–Walsh expansion

$$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$

with Parseval and Plancherel identities

$$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$

For Boolean-valued $f : \{-1,1\}^n\to\{-1,1\}$, Parseval reduces to $\sum_S \widehat f(S)^2=1$, so the squared coefficients form a probability distribution on subsets of $[n]$; this is often viewed as the spectral sample of $f : \{-1,1\}^n \to \mathbb{R}$0 (O'Donnell, 2021, Chen, 15 Apr 2026).

The Fourier degree is

$f : \{-1,1\}^n \to \mathbb{R}$1

and the degree-$f : \{-1,1\}^n \to \mathbb{R}$2 truncation retains only coefficients with $f : \{-1,1\}^n \to \mathbb{R}$3. For low-degree identifiability questions it is convenient to write

$f : \{-1,1\}^n \to \mathbb{R}$4

where $f : \{-1,1\}^n \to \mathbb{R}$5 counts the retained coefficients and $f : \{-1,1\}^n \to \mathbb{R}$6 the discarded tail (Chen, 15 Apr 2026).

The same formalism admits several equivalent encodings. On $f : \{-1,1\}^n \to \mathbb{R}$7, one may index characters by subsets or linear forms over $f : \{-1,1\}^n \to \mathbb{R}$8, writing $f : \{-1,1\}^n \to \mathbb{R}$9. This viewpoint is particularly useful for Fourier sparsity and Fourier dimension, where the support $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$0 is regarded as a subset of $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$1, and

$$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$2

measures the number of independent parity directions needed to generate the spectrum (Sanyal, 2014).

Boolean Fourier analysis also extends beyond the uniform cube. On the $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$3-biased cube $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$4, the orthonormal basis is built from

$$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$5

while on finite Abelian groups $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$6 one expands against the character group $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$7,

$$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$8

These variants preserve the same Parseval paradigm while changing the ambient geometry and arithmetic of the spectrum (O'Donnell, 2021, Chakraborty et al., 2024).

2. Influence, noise, and hypercontractive structure

A central theme of the subject is that local perturbations of the input correspond to simple spectral operators. For $$\chi_S(x) = \prod_{i\in S} x_i,\qquad \chi_\emptyset \equiv 1,$$9, the discrete derivative in coordinate $\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$0 is

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$1

and for Boolean $\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$2 the influence of coordinate $\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$3 is

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$4

The total influence, or average sensitivity, is

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$5

and the Poincaré inequality on the cube gives $\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$6 (O'Donnell, 2021).

Noise acts diagonally in the Fourier basis. For $\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$7, the noise operator is

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$8

and the noise stability is

$\langle f,g\rangle = \mathbb{E}[f(x)g(x)].$9

For Boolean functions, the corresponding noise sensitivity at flip rate $f$0 is

$f$1

The level weights $f$2 therefore govern robustness under random perturbations, with higher levels attenuated faster (O'Donnell, 2021, Tan, 2012).

The key analytic inequality is Bonami–Beckner hypercontractivity. In one standard form, if $f$3 and $f$4, then

$f$5

For degree-$f$6 functions this yields moment comparisons and strong control of high-degree mass. In the biased setting, the corresponding operator

$f$7

acts on the $f$8-biased Fourier basis, and the influence identity becomes

$f$9

This biased formalism is essential in sharp-threshold theory and in recent work on Boolean PCSP polymorphisms (Banakh et al., 24 Apr 2026).

These operators underwrite much of the subject’s global geometry. They connect expectation derivatives to influences, yield quantitative threshold phenomena, and force low-degree or low-influence functions to behave approximately like juntas or Gaussian halfspaces under suitable regularity assumptions (O'Donnell, 2021, Tan, 2012).

3. Structural theorems and spectral complexity

Several of the field’s foundational theorems can be phrased as rigidity statements about the Fourier spectrum. The Friedgut–Kalai–Naor theorem asserts that if a Boolean function has almost all Fourier weight on levels $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$0 and $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$1, then it is close to a dictator or affine function. Friedgut’s Junta theorem states that bounded total influence forces approximation by a junta depending on few coordinates. The Kahn–Kalai–Linial theorem gives

$$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$2

together with the more quantitative lower bound

$$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$3

for a universal constant $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$4 (O'Donnell, 2021).

A different form of rigidity concerns coefficient norms rather than level concentration. For every degree-$$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$5 Boolean function $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$6, there is a universal constant $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$7 such that

$$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$8

The exponent $$f(x) = \sum_{S\subseteq [n]} \widehat f(S)\chi_S(x),\qquad \widehat f(S)=\mathbb{E}[f(x)\chi_S(x)],$$9 is best possible, and the Bohnenblust–Hille constant has subexponential growth: $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$0 This inequality implies nontrivial control on the density of large Fourier coefficients and has recently been used to obtain $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$1 sample complexity for learning bounded low-degree Boolean functions (Defant et al., 2017, Zhang, 2024).

Spectral support itself leads to two further complexity measures. The Fourier sparsity is $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$2, and the Fourier dimension

$$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$3

equals the non-adaptive parity decision tree complexity: $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$4 For a Boolean function with Fourier sparsity $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$5, one has the unconditional bound

$$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$6

and, assuming a conjecture of Tsang et al. on constant restrictions to low-codimension affine subspaces, the improved bound

$$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$7

follows. The address function shows that $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$8 would be essentially tight up to polylogarithmic factors (Sanyal, 2014).

Parity decision tree complexity also admits an additive-combinatorial formulation. If $$\sum_{S\subseteq[n]}\widehat f(S)^2=\mathbb{E}[f(x)^2],\qquad \langle f,g\rangle=\sum_S \widehat f(S)\widehat g(S).$$9, then there exists a parity decision tree of depth $f : \{-1,1\}^n\to\{-1,1\}$0 computing $f : \{-1,1\}^n\to\{-1,1\}$1, and a probabilistic construction shows that naive random sampling of parities from the Fourier support already suffices to obtain this optimal upper bound. The proof is organized around “bucket complexity” and the folding structure of the support under sums $f : \{-1,1\}^n\to\{-1,1\}$2 in $f : \{-1,1\}^n\to\{-1,1\}$3 (Mande et al., 2020).

4. Exact recovery from low-degree data

A recent identifiability problem asks when the truncated spectrum $f : \{-1,1\}^n\to\{-1,1\}$4 determines the entire Boolean function. For a uniformly random $f : \{-1,1\}^n\to\{-1,1\}$5, the threshold occurs at half the dimension, up to an $f : \{-1,1\}^n\to\{-1,1\}$6 window. If

$f : \{-1,1\}^n\to\{-1,1\}$7

then with probability $f : \{-1,1\}^n\to\{-1,1\}$8 there exists another Boolean function $f : \{-1,1\}^n\to\{-1,1\}$9 with the same degree-$\sum_S \widehat f(S)^2=1$0 Fourier coefficients: $\sum_S \widehat f(S)^2=1$1 Conversely, for every fixed $\sum_S \widehat f(S)^2=1$2, if

$\sum_S \widehat f(S)^2=1$3

then with probability at least $\sum_S \widehat f(S)^2=1$4, $\sum_S \widehat f(S)^2=1$5 is uniquely determined by $\sum_S \widehat f(S)^2=1$6 even among all bounded $\sum_S \widehat f(S)^2=1$7 (Chen, 15 Apr 2026).

The lower bound is a counting argument. Each low-degree coefficient is a normalized sum of $\sum_S \widehat f(S)^2=1$8 signs and therefore has at most $\sum_S \widehat f(S)^2=1$9 possible values, so the total number of degree-$[n]$0 data vectors is at most $[n]$1. When

$[n]$2

this image is negligible compared with the $[n]$3 Boolean functions on the cube, forcing collisions. The natural location of the threshold is $[n]$4 because

$[n]$5

and the relevant binomial tails are Gaussian on the $[n]$6 scale (Chen, 15 Apr 2026).

The upper bound is constructive. Let $[n]$7 be the span of Walsh characters of degree at most $[n]$8, let $[n]$9 be orthogonal projection onto $f : \{-1,1\}^n \to \mathbb{R}$00, and define

$f : \{-1,1\}^n \to \mathbb{R}$01

If $f : \{-1,1\}^n \to \mathbb{R}$02 lies in the uniqueness regime, then with probability at least $f : \{-1,1\}^n \to \mathbb{R}$03,

$f : \{-1,1\}^n \to \mathbb{R}$04

so $f : \{-1,1\}^n \to \mathbb{R}$05 for every $f : \{-1,1\}^n \to \mathbb{R}$06. This yields a decoder: $f : \{-1,1\}^n \to \mathbb{R}$07 The resulting full truth-table reconstruction costs $f : \{-1,1\}^n \to \mathbb{R}$08, while membership queries at a single $f : \{-1,1\}^n \to \mathbb{R}$09 cost $f : \{-1,1\}^n \to \mathbb{R}$10. The same argument also gives a simple robustness margin: if the low-degree coefficients are perturbed by $f : \{-1,1\}^n \to \mathbb{R}$11 and

$f : \{-1,1\}^n \to \mathbb{R}$12

then the reconstructed sign pattern is unchanged (Chen, 15 Apr 2026).

This threshold theorem sharply contrasts with structured classes. Linear threshold functions are determined by their degree-$f : \{-1,1\}^n \to \mathbb{R}$13 and degree-$f : \{-1,1\}^n \to \mathbb{R}$14 coefficients, via Chow parameters, and polynomial threshold functions admit robust low-degree identifiability results. Random Boolean functions, by contrast, require almost half the Fourier levels for exact determination, showing that exact recovery from low degrees is atypical in the absence of structure (Chen, 15 Apr 2026).

5. Beyond the uniform cube: biased, non-binary, geometric, and quantum variants

Boolean Fourier analysis extends naturally to product-biased measures and to finite Abelian groups. For

$f : \{-1,1\}^n \to \mathbb{R}$15

characters decompose coordinatewise, Parseval still holds, and sparsity may be defined as $f : \{-1,1\}^n \to \mathbb{R}$16 in the dual group $f : \{-1,1\}^n \to \mathbb{R}$17. For $f : \{-1,1\}^n \to \mathbb{R}$18-sparse Boolean functions over such groups, the analogue of “granularity” becomes cyclotomic rather than dyadic: large coefficients are close to values of the form

$f : \{-1,1\}^n \to \mathbb{R}$19

for integer polynomials $f : \{-1,1\}^n \to \mathbb{R}$20. The smallest nonzero coefficient obeys

$f : \{-1,1\}^n \to \mathbb{R}$21

which is qualitatively weaker than the $f : \{-1,1\}^n \to \mathbb{R}$22 behavior available over $f : \{-1,1\}^n \to \mathbb{R}$23. These structural results support a sparsity tester using

$f : \{-1,1\}^n \to \mathbb{R}$24

queries, together with an $f : \{-1,1\}^n \to \mathbb{R}$25 lower bound for adaptive sparsity testing (Chakraborty et al., 2024).

The same spectral viewpoint has been transported to sparse analogues of the cube. On high-dimensional expanders and measured posets, the role of Fourier levels is played by a decomposition into approximate eigenspaces of random-walk operators $f : \{-1,1\}^n \to \mathbb{R}$26 and $f : \{-1,1\}^n \to \mathbb{R}$27. For a $f : \{-1,1\}^n \to \mathbb{R}$28-high-dimensional expander, the subspaces

$f : \{-1,1\}^n \to \mathbb{R}$29

form an approximate orthogonal decomposition of $f : \{-1,1\}^n \to \mathbb{R}$30, and each $f : \{-1,1\}^n \to \mathbb{R}$31 is an approximate eigenvector of $f : \{-1,1\}^n \to \mathbb{R}$32 with eigenvalue $f : \{-1,1\}^n \to \mathbb{R}$33. This supports an HDX analogue of the Friedgut–Kalai–Naor theorem and yields a sparse model of the slice or cube with only $f : \{-1,1\}^n \to \mathbb{R}$34 points at a given level, rather than $f : \{-1,1\}^n \to \mathbb{R}$35 (Dikstein et al., 2018).

A different extension replaces linear Fourier polynomials by trigonometric ones. The periodic Fourier representation writes

$f : \{-1,1\}^n \to \mathbb{R}$36

and the minimum number of nonzero $f : \{-1,1\}^n \to \mathbb{R}$37 is the periodic Fourier sparsity. This quantity equals the number of qubits required for exact computation in $f : \{-1,1\}^n \to \mathbb{R}$38, and it satisfies the lower bound

$f : \{-1,1\}^n \to \mathbb{R}$39

Although one-layer $f : \{-1,1\}^n \to \mathbb{R}$40 therefore needs exponentially many qubits for functions such as $f : \{-1,1\}^n \to \mathbb{R}$41, $f : \{-1,1\}^n \to \mathbb{R}$42, and $f : \{-1,1\}^n \to \mathbb{R}$43, every symmetric Boolean function can be computed exactly by depth-2 $f : \{-1,1\}^n \to \mathbb{R}$44 with polynomially many qubits (Mori, 2018).

Quantum Boolean analysis replaces Walsh characters by Pauli strings. A quantum Boolean function is a Hermitian unitary $f : \{-1,1\}^n \to \mathbb{R}$45 with $f : \{-1,1\}^n \to \mathbb{R}$46, and every operator on $f : \{-1,1\}^n \to \mathbb{R}$47 qubits expands as

$f : \{-1,1\}^n \to \mathbb{R}$48

The corresponding depolarizing-noise operator

$f : \{-1,1\}^n \to \mathbb{R}$49

satisfies a quantum Bonami–Gross–Beckner inequality in normalized Schatten norms, and quantum versions of the Goldreich–Levin algorithm, FKN theorems, and property testing can be developed in this basis (0810.2435).

Support geometry itself has also been studied directly. For Boolean or pseudo-Boolean functions on $f : \{-1,1\}^n \to \mathbb{R}$50, balancing sets determine the vanishing pattern of Walsh and Fourier–Hadamard coefficients. In this language, nonempty sets are “fully balanced” exactly when they are affine spaces, and the indicator of a fully balanced set is a minimum-weight codeword in an appropriate Reed–Muller code (Carlet et al., 2022).

6. Algorithmic uses and current research fronts

In learning theory, Fourier methods remain foundational. The Goldreich–Levin algorithm identifies all coefficients above threshold $f : \{-1,1\}^n \to \mathbb{R}$51 in $f : \{-1,1\}^n \to \mathbb{R}$52 time, and recent use of the Boolean Bohnenblust–Hille inequality gives optimal logarithmic sample complexity for learning bounded degree-$f : \{-1,1\}^n \to \mathbb{R}$53 Boolean functions. Specifically, if

$f : \{-1,1\}^n \to \mathbb{R}$54

then one can construct $f : \{-1,1\}^n \to \mathbb{R}$55 from $f : \{-1,1\}^n \to \mathbb{R}$56 random examples with

$f : \{-1,1\}^n \to \mathbb{R}$57

with probability at least $f : \{-1,1\}^n \to \mathbb{R}$58, yielding $f : \{-1,1\}^n \to \mathbb{R}$59 samples. The same analysis implies that every bounded degree-$f : \{-1,1\}^n \to \mathbb{R}$60 function is $f : \{-1,1\}^n \to \mathbb{R}$61-close in $f : \{-1,1\}^n \to \mathbb{R}$62 to a $f : \{-1,1\}^n \to \mathbb{R}$63-junta with

$f : \{-1,1\}^n \to \mathbb{R}$64

(O'Donnell, 2021, Zhang, 2024).

The Fourier formalism has also been used to design optimization-based solvers for hybrid Boolean constraints. In FourierSAT, each clause type—CNF, XOR, cardinality, or NAE—is converted to a multilinear polynomial $f : \{-1,1\}^n \to \mathbb{R}$65 on $f : \{-1,1\}^n \to \mathbb{R}$66, and the global objective

$f : \{-1,1\}^n \to \mathbb{R}$67

satisfies

$f : \{-1,1\}^n \to \mathbb{R}$68

where $f : \{-1,1\}^n \to \mathbb{R}$69. Projected gradient descent then exploits the algebraic gradient of the Fourier expansion to search for good assignments, and randomized rounding satisfies

$f : \{-1,1\}^n \to \mathbb{R}$70

(Kyrillidis et al., 2019).

A more recent analytical use arises in random-matrix algorithms. For a broad class of nonlinear iterative methods, including power iteration, belief propagation, approximate message passing, and many gradient-like updates, symmetrized Boolean/Walsh characters of the matrix entries organize the iterate into “Fourier diagrams.” Cyclic diagrams are asymptotically negligible, tree diagrams form a basis of asymptotically independent Gaussian vectors, and restricting to the tree sector yields an exact state evolution in the large-$f : \{-1,1\}^n \to \mathbb{R}$71 limit (Jones et al., 2024).

Influence theory has also entered constraint satisfaction. In Boolean PCSPs, two phenomena are singled out as complexity indicators: preservation of coordinate influence under random $f : \{-1,1\}^n \to \mathbb{R}$72-to-$f : \{-1,1\}^n \to \mathbb{R}$73 minors and sharp-threshold behavior. These mechanisms produce hardness criteria for minions of unate functions and polynomial threshold functions, and tractability criteria when sharp thresholds force the presence of polymorphisms such as $f : \{-1,1\}^n \to \mathbb{R}$74, $f : \{-1,1\}^n \to \mathbb{R}$75, $f : \{-1,1\}^n \to \mathbb{R}$76, or alternating threshold operations (Banakh et al., 24 Apr 2026).

Outside classical TCS, Boolean Fourier analysis has been used to study many-body quantum sign structures. Representing the sign function on spin configurations as a polynomial on $f : \{-1,1\}^n \to \mathbb{R}$77 yields complexity measures such as the participation ratio and reconstruction complexity of the spectrum, and motivates explicit “Fourier learning” and “Hadamard learning” algorithms. On $f : \{-1,1\}^n \to \mathbb{R}$78-spin kagome instances, Fourier learning achieves sign overlap $f : \{-1,1\}^n \to \mathbb{R}$79 with training fraction $f : \{-1,1\}^n \to \mathbb{R}$80, whereas the dense neural-network baseline remains poor even for $f : \{-1,1\}^n \to \mathbb{R}$81. Augmenting a neural architecture with a small number of parity features also markedly improves generalization on frustrated regimes (Schurov et al., 13 Aug 2025).

A current analytic frontier concerns spectral entropy. On the biased cube, one now has the lower bound

$f : \{-1,1\}^n \to \mathbb{R}$82

complementing the Fourier-Entropy-Influence program (Chang, 11 Nov 2025). In the uniform case, a recent upper bound shows

$f : \{-1,1\}^n \to \mathbb{R}$83

refining the classical expectation that spectral entropy should be linearly controlled by total influence (Han, 2023). These results suggest that, even in its most classical form, Boolean Fourier analysis remains an active domain in which identifiability, entropy, geometry, and algorithmic complexity continue to be reorganized around the spectrum.