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Inequalities in Fourier analysis on binary cubes

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Pith's one-line read The paper exactly characterizes the exponent ranges in which sharp Hausdorff–Young and Young inequalities hold on binary cubes with constant 1 for every dimension.

desk verdict Theorem 6 is false — the Young convolution half of the paper collapses on a coefficient error, though the Hausdorff-Young half may be sound. read the letter →

arxiv 2507.01359 v1 pith:ZT5MK4W2 submitted 2025-07-02 math.CA cs.ITmath.COmath.IT

Tonći Crmarić , Vjekoslav Kovač , Shobu Shiraki This is my paper · ORCID
classification math.CAcs.ITmath.COmath.IT MSC 42A0505D0594A1742B05
keywords sharpestimateFouriertransformrestrictionadditiveenergyentropyHausdorff–YounginequalityYoung'sconvolutionbinarycube
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper settles the exact exponent ranges for which the classical Hausdorff–Young inequality $\|\hat f\|_{L^q(\mathbb T^d)}\le \|f\|_{\ell^p(\{0,1\}^d)}$ and the equal-exponent Young convolution inequality hold with constant $1$ for functions supported on the binary cube $\{0,1\}^d$ in every dimension $d$. The answer is a curved boundary: for $q\ge 2$ the Hausdorff–Young range requires $1/p\ge (1/q)\log_2 \binom{q}{q/2}$, with $1/p\ge 1/2$ for $q\in[1,2)$ and $p\le 1$ at $q=\infty$; the Young range requires $1/p\ge (1/(2q))\log_2(2q+2)$ for $q<\infty$ and $1/p\ge 1/2$ at $q=\infty$. These characterizations imply sharp bounds on generalized additive energies of subsets of the cube, a sharp entropic uncertainty principle on $\mathbb T^d\times\mathbb Z^d$, a sharp lower bound on the entropy of sums of independent binary-cube random variables, and the exact range of dimension-free Fourier restriction estimates. The proof reduces both inequalities to one- and two-variable pointwise inequalities that are established by ordinary differential equations, monotonicity arguments, and two computer-assisted checks.

What carries the argument

The load-bearing objects are the two-point inequality (Lemma 4) and the four-point inequality (Lemma 8). Lemma 4 is the $d=1$ case of Theorem 1: for $\alpha,\beta\ge 0$ and $p,q$ in the sharp range, $(\int_0^1 |\alpha e^{2\pi i t}+\beta|^q\,dt)^{1/q}\le (\alpha^p+\beta^p)^{1/p}$. Lemma 8 plays the same role for Theorem 6. Lemma 4 is proved by defining $F_q(x)=\int_0^1 |x^{1/p}e^{2\pi i t}+(1-x)^{1/p}|^q\,dt$ and showing $F_q(x)\le 1$ on $[0,1]$ via a second-order ordinary differential equation with coefficients $a_q,b_q,c_q$; the ODE is verified symbolically, and the boundary points $x=0$ and $x=1/2$ are strict local maxima, so any interior maximum would force two zeros in a function that can have only one. Lemma 8 is proved by a two-variable analogue, $G_q(x,y)\le 1$ on the square, with the domain dissected along the curves $y(1-x)/(x(1-y))=a$; along these curves repeated differentiation reduces the count of stationary points to an impossible sign pattern.

What would settle it

Evaluate the residual of the differential equation for $F_q$ at a single non-exceptional point, say $q=3$, $x=1/4$, to high precision; exactly zero is required, so any nonzero residual would falsify Lemma 10 and with it Theorem 1. For Lemma 12, re-evaluate $\partial_q\tilde\Phi(q,u)$ on the claimed $2101\times 901$ grid using exact arithmetic and the stated Lipschitz bounds; any grid value at or below $1/50$ after accounting for the bounds would falsify Lemma 8 and Theorem 6.

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Extended reading notes

Core claim

The central discovery is that the classical Fourier inequalities, restricted to functions supported on $\{0,1\}^d$, remain valid with constant $1$ in strictly larger exponent regions than on all of $\mathbb Z^d$, and that the enlarged regions are exactly characterized by the displayed formulas. The Hausdorff–Young inequality holds for all $d$ exactly when $1/p\ge 1/q\log_2\binom{q}{q/2}$ for $q\ge 2$, with the two endpoint regimes $1/p\ge 1/2$ for $q<2$ and $p\le 1$ at $q=\infty$. The equal-exponent Young inequality holds exactly when $1/p\ge (1/(2q))\log_2(2q+2)$ for $q\in[1,\infty)$ and $1/p\ge 1/2$ at $q=\infty$. Both regions are optimal, and because the constants are exactly $1$ in every dimension, the estimates are automatically dimension-free; the dual form is a Fourier restriction estimate to the binary cube with the same exponent range. These characterizations are proved by reducing each inequality to a one- or two-variable pointwise inequality, then proving those inequalities by ODE analysis, monotonicity, and computer-assisted verification.

Load-bearing premise

The whole characterization rests on two computer-assisted checks being exactly right: the symbolic verification that the function $F_q$ satisfies the displayed second-order differential equation, and the grid-plus-Lipschitz verification that a certain derivative is positive on $[1,4]\times[0,3]$; neither set of commands, outputs, or code is included.

Editorial extensions

If this is right

  • For every $\kappa\ge 1$ and every $A\subseteq\{0,1\}^d$, the generalized additive energy satisfies $E_\kappa(A)\le |A|^{\log_2\binom{2\kappa}{\kappa}}$, and $A=\{0,1\}^d$ attains equality, so the exponent is optimal.
  • The reflected additive energy satisfies $\tilde E_\kappa(A)\le |A|^{\log_2(2\kappa+2)}$, again with equality for the full cube.
  • Every $f$ supported on $\{0,1\}^d$ with $\|f\|_2=1$ obeys $H_{\mathbb T^d}(|\hat f|^2)+(\frac{1}{\ln 2}-1)H_{\mathbb Z^d}(|f|^2)\ge 0$, with equality at nonzero entropy for constant $f$ on the cube.
  • For independent random variables $X,Y$ valued in $\{0,1\}^d$, $H(X+Y)\ge \frac{3}{4}(H(X)+H(Y))$, and the constant $3/4$ is sharp.
  • The Fourier restriction estimate to the binary cube has a dimension-free constant exactly in the region (1.6); outside that region, no dimension-free constant exists.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension the authors do not pursue: the same two-point reduction might characterize which finite subsets of $\mathbb Z^d$ admit dimension-free Hausdorff–Young with constant 1, with the binary cube being the case where the two-point inequality has a closed-form boundary.
  • The sharp entropy constants can be stress-tested by exhaustive enumeration of probability distributions on $\{0,1\}^d$ for small $d$; the proof is analytic, so such a census would be a verification rather than a substitute.
  • Remark 15 indicates that a full three-exponent Young characterization on binary cubes will require new extremal analysis: for $r=2$, the full-cube test function is not always extremal when one exponent is below $4/3$.
  • Because the tensor-power argument forces any dimension-free constant to be exactly 1, a numerical violation of either pointwise inequality at any exponent would immediately destroy the restriction corollary as well, making the two computer-assisted checks the natural targets for independent verification.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. This paper studies sharp dimension-free versions of the Hausdorff–Young and equal-exponent Young inequalities for functions supported on the binary cube {0,1}^d. Theorem 1 characterizes the exponent range for ||\hat f||_{L^q} ≤ ||f||_{\ell^p} across all dimensions d, and Theorem 6 gives an analogous characterization for the diagonal Young inequality ||f*g||_{\ell^q} ≤ ||f||_{\ell^p}||g||_{\ell^p}. The proofs reduce to two- and four-point inequalities (Lemmas 4 and 8), which are established via Legendre-function differential equations, one-dimensional monotonicity arguments, and computer-assisted verification. Consequences include generalized additive energy bounds (Corollaries 2 and 7), a binary Beckner–Hirschman entropic uncertainty principle (Corollary 5), an entropy bound for sums of independent binary-cube variables (Corollary 9), and dimension-free Fourier restriction estimates (Corollary 3).

Significance. If correct, Theorem 1 would be a substantial result: it would settle, for all real q, the exact enlarged Hausdorff–Young range on binary cubes, unify and extend the integer-q results of Kane–Tao and de Dios Pont–Greenfeld–Ivanisvili–Madrid, and yield sharp dimension-free restriction estimates with constant 1 through the tensor-power trick. The entropic corollaries with explicit sharp constants (1/ln2 - 1) and 3/4 are attractive, and the equality computations in Corollaries 5 and 9 are genuine. However, Theorem 6 is false as stated, and the proof of the four-point inequality (Lemma 8) contains a concrete algebraic error. Moreover, the computer-assisted verification for Theorem 1 is not shipped, so even the remaining main theorem is not independently auditable from the manuscript.

major comments (2)
  1. [Section 9, Lemma 13 and Eq. (9.4)] The claimed reduction is algebraically incorrect. The anti-diagonal estimate to be proved is (2 cosh(pt/(2q)))^{2q/p} ≥ (2 cosh(t/q))^q + 2. Rearranging (9.4) gives 2^{2q/p}(cosh(pt/(2q)))^{2q/p} ≥ 2q(cosh(t/q))^q + 2, because (9.1) implies 2^{2q/p} - 2q = 2. The right-hand side has coefficient 2q, not 2^q, so (9.4) cannot imply the displayed estimate. At t=0 the displayed estimate would require 2^{2q/p} ≥ 2^q + 2, i.e. 2q+2 ≥ 2^q+2, which fails for every q>2. Consequently Lemma 13 is false for q>2. A concrete one-dimensional counterexample is f=g=1_{\{0,1\}} with q=3, p=2 (which satisfies (1.17)): ||f*g||_{\ell^3(Z)} = (1+2^3+1)^{1/3} = 10^{1/3} > 2 = ||f||_{\ell^2(Z)}||g||_{\ell^2(Z)}. The same choice with p=1.9, q=3 falsifies Lemma 8 as stated, since p=1.9 lies in its domain. Thus the sufficiency direction of Theorem 6 and Corollary 7 (κ=3 gives \tilde E_3({0,1})=10>8) fail.
  2. [Section 4 (Lemma 10) and Section 9 (Lemma 12)] The two computer-assisted verifications that are load-bearing for the remaining main theorem are not shipped. Lemma 10's proof consists of the statement that Mathematica's Simplify/FullSimplify verifies the ODE; Lemma 12's final step replaces the desired positivity by a check on a 2101 by 901 grid with 'exact expressions' plus explicit Lipschitz bounds, but no code, commands, or output files are included. The text itself calls Lemma 10 'practically unverifiable without an assistance of a computer.' As a consequence, the endpoint exponent range in Theorem 1 cannot be independently audited from the manuscript. This is a support gap, not a stylistic issue; it would need to be fixed by supplying a complete, machine-readable verification before the remaining claims could be accepted.
minor comments (2)
  1. [Throughout] Several displayed formulas contain OCR artifacts such as '\bracehtipupleft' and '\bracehtipdownright' (for example, in Corollary 5 and in the induction step of Section 3.2); these should be cleaned up.
  2. [Section 1.2] The sentence 'random variables ... taking value s in the binary cube' contains a typo; it should read 'taking values'.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: endpoint exponents are derived functions, constants emerge from limits, and self-citations are historical rather than load-bearing.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1's endpoint function p(q) = q / log_2 binom(q, q/2) is obtained from the d=1 necessity test with f = 1_{0,1}, and the sufficiency direction proves the corresponding two-point inequality from the Legendre ODE in Lemma 10 plus the sign analysis of c_q. Lemma 10 is described as a generalization of [9, Lemma 5], but it is proved here independently via the Legendre integral representation and symbolic differentiation; the self-cited paper [9] is used only for integer-case history and context, not as a load-bearing premise. The constants that drive Corollaries 5 and 9, namely (1/ln 2 - 1) and 3/4, are limits of the derived function p(q) as q -> 2+ and q -> 1+, respectively; they are not fitted parameters, and the sharpness examples verify equality independently. Lemma 12's inequalities are established by explicit monotonicity arguments, with one computer-assisted grid check; the lack of shipped code and commands is a reproducibility or rigor concern, not a circular reduction. The possible coefficient slip in Lemma 13's anti-diagonal reduction raised by the skeptic would, if correct, be a correctness defect rather than a self-referential one: the proof would fail to imply the claimed inequality, but it would not be defining the conclusion into the premise. No imported uniqueness theorem, ansatz-by-self-citation, or renaming of a known result carries the argument. Hence no circular step is present; the score reflects only the minor presence of non-load-bearing self-citations and the unshipped computational verification. Correctness risk from the computational checks and the alleged Lemma 13 algebra is real but outside the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numbers are fitted to data or chosen ad hoc. The endpoint exponents p(q) in (2.1) and (9.1) are derived functions, and the constants (1/ln2 - 1) and 3/4 emerge from limits of the sharp inequalities. No new physical entities are postulated; the curves Theta_a and the functions F_q and G_q are proof devices, not entities with independent empirical handles.

assumptions (4)
  • standard math Legendre function integral representation (DLMF 14.12.7) and Legendre ODE (DLMF 14.2.1) hold for the ranges used.
    Used to derive formula (4.6) and the ODE verification in Lemma 10.
  • ad hoc to paper Mathematica's Simplify, FullSimplify, and exact-arithmetic evaluations over the stated subdivisions correctly verify the algebraic claims in Lemmas 10 and 12.
    No code or certificates are shipped, so the central two-point and four-point inequalities depend on these unarchived computations.
  • standard math The gamma and trigamma duplication identities, and the strict decrease of psi' on (0, infinity), hold as stated.
    Used in Section 2 to prove monotonicity of the endpoint function p(q).
  • standard math Rolle's theorem, the intermediate value theorem, and Minkowski's inequality apply in the claimed regimes.
    Used throughout Lemmas 4, 8, 10, and 12 and in the induction steps.

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Cite this review

Pith. "Pith review of Inequalities in Fourier analysis on binary cubes." pith.science (2026). https://pith.science/paper/ZT5MK4W2

@misc{pith2026250701359,
  author       = {Pith},
  title        = {Pith review of: Inequalities in Fourier analysis on binary cubes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZT5MK4W2}},
  note         = {Machine review of arXiv:2507.01359}
}
abstract

This paper studies two classical inequalities, namely the Hausdorff-Young inequality and equal-exponent Young's convolution inequality, for discrete functions supported in the binary cube $\{0,1\}^d\subset\mathbb{Z}^d$. We characterize the exact ranges of Lebesgue exponents in which sharp versions of these two inequalities hold, and present several immediate consequences. First, if the functions are specialized to be the indicator of some set $A\subseteq\{0,1\}^d$, then we obtain sharp upper bounds on two types of generalized additive energies of $A$, extending the works of Kane-Tao, de Dios Pont-Greenfeld-Ivanisvili-Madrid, and one of the present authors. Second, we obtain a sharp binary variant of the Beckner-Hirschman entropic uncertainty principle, as well as a sharp lower estimate on the entropy of a sum of two independent random variables with values in $\{0,1\}^d$. Finally, the sharp binary Hausdorff-Young inequality also reveals the exact range of dimension-free estimates for the Fourier restriction to the binary cube.

Figures

Figures reproduced from arXiv: 2507.01359 by the authors.

Figure 1
Figure 1. Regions determined by points (1/p, 1/q) ∈ [0, 1]2 from the exponent ranges (1.2) (left) and (1.6) (right). see the left half of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Regions determined by points (1/p, 1/q) ∈ [0, 1]2 from the exponent ranges (1.15) (left) and (1.17) (right). It is interesting to notice that the constant ln(e/2) also appears in the sharp entropic uncer￾tainty principle on the real line: HR(|fb| 2 ) + HR(|f| 2 ) > ln e 2 . Here we do not define the entropy HR of probability densities on R, as these are not of our current interest, and we rather just refer to [7, 2]… view at source ↗
Figure 3
Figure 3. Graph of F4. The maximum 1 is attained at 0, 1/2, and 1. Proof. The Legendre function Pν of degree ν has the integral representation Pν (z) = Z 1 0 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: A sketch of the function Fq. While the behavior near the points x = 0, 1/2 is well understood, the assumption that Fq also attains its maximum at some point xmax ∈ (0, 1/2) leads to a contradiction. Recall that the point (1/p, 1/q) is at the lower boundary of the range…
Figure 2
Figure 2. Figure 2: Note 1 < p < min{q, 2}. (9.2) Substituting aj = α p j , bj = β p j for j = 0, 1, inequality (1.19) turns into a q/p 0 b q/p 0 + [PITH_FULL_IMAGE:figures/full_fig_p019_2.png]
Figure 5
Figure 5. Figure 5: Graph of G2. The maximum 1 is attained at (0, 0), (0, 1), (1, 0), (1, 1), and (1/2, 1/2). see [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 2
Figure 2. Figure 2: Note that the assignment q 7→ p(q) actually makes sense for all q ∈ (0,∞) and we can compute p ′ (q) = p q  1 − p2 q 2(2q + 2)  , p′ (1) = 3 4 . (9.8) Substitute u = t/q, transforming (9.5) into [PITH_FULL_IMAGE:figures/full_fig_p021_2.png]
Figure 6
Figure 6. Figure 6: The curve Θa passes through the hypothetical maxima of Gq in ∆ (left). The graph of Gq evaluated along the curve Θa (right). All this and the symmetry of Gq about y = 1−x imply that Gq|Θa is decreasing6 near (0, 0), increasing near (1, 1), and has yet another local max…

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