{"id":"9323dd15-c1bb-4902-97c3-3a50e4285b42","arxiv_id":"2507.01359","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For functions supported on the binary cube, the full range of exponents in the sharp dimension-free Hausdorff-Young and diagonal Young inequalities is characterized exactly.","lead":"This paper finds the exact set of exponents for which the Hausdorff-Young and diagonal Young inequalities hold with sharp constant 1 when the functions live on binary cubes, and derives sharp entropy inequalities from the boundary cases. A generalist reader should care because it completes a program begun by Beckner 50 years ago and connects Fourier analysis to additive combinatorics and information theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 13's anti-diagonal proof swaps the coefficient 2q in (9.4) for 2^q, so the claimed Theorem 6 range is false for q>2; the d=1 test f=g=1_{0,1} already fails at q=3.","rationale":"The reader's concern about unshipped computer algebra is a legitimate reproducibility issue, but it is not the load-bearing problem: even granting every Mathematica output, Theorem 6 is false as stated. The precise failure is an internal algebraic mismatch in Lemma 13, where the desired anti-diagonal bound requires the coefficient 2^q on the cosh term, while inequality (9.4) can only supply the coefficient 2q. These are equal only at q=2. For q=3, the purported anti-diagonal bound is already false at the center point, giving G_3(1/2,1/2)=5/4>1. This invalidates Lemma 8, and therefore Theorem 6 and Corollary 7. The counterexample is one-dimensional and uses only f=g=1_{\\{0,1\\}}, so no computational certificate can repair the theorem. The correct verdict is REJECT for the central Young-inquality claim, even though Theorem 1 and the entropic corollaries near q=2 may be salvageable.","tokens_in":23838,"tokens_out":26240,"duration_ms":260933,"concrete_test":"Do the hand arithmetic for q=3, p=2 (the endpoint allowed by (1.17)) and f=g=1_{\\{0,1\\}} on Z: ||f*g||_3=(1+2^3+1)^{1/3}=10^{1/3}approx 2.154, while ||f||_2||g||_2=2, so (1.16) fails. For Lemma 8's open domain, repeat with q=3, p=1.9, obtaining LHS 10^{1/3}approx 2.154 and RHS 2^{2/1.9}approx 2.074. Also verify the anti-diagonal inequality at t=0: it becomes 2^{2q/p}=2q+2 >= 2^q+2, i.e. 8 >= 10 for q=3, which is false. No CAS is needed.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing error is in the proof of Lemma 13 (anti-diagonal case, Section 9). After substituting x=1/(1+e^{pt/q}), proving G_q(x,1-x)<=1 is equivalent to (2 cosh(pt/(2q)))^{2q/p} >= (2 cosh(t/q))^q + 2. The authors claim this follows from Lemma 12's first inequality (9.4), namely 2^{2q/p}[(cosh(pt/(2q)))^{2q/p}-1] >= 2q[(cosh(t/q))^q -1], using 2^{2q/p}-2q=2. Rearranging (9.4) gives 2^{2q/p}(cosh(pt/(2q)))^{2q/p} >= 2q(cosh(t/q))^q + 2, whose right-hand coefficient is 2q, not the 2^q needed in the desired inequality. For q=2 these coincide, but for q>2 they do not. Consequently the desired inequality is false at t=0 for q=3: it reads 2^{2q/p} >= 2^q+2, i.e. 2q+2 >= 2^q+2, which fails since 8 >= 10 is false. Equivalently, G_3(1/2,1/2)=2^{1-3}+2^{0}=5/4>1, contradicting the assertion at the start of Section 9 that (9.1) makes G_q(1/2,1/2)=1. A direct one-dimensional check kills Theorem 6: take q=3, p=2 (or p=1.9, which lies in Lemma 8's domain) and f=g=1_{\\{0,1\\}}. Then ||f*g||_3=10^{1/3}approx 2.154 while ||f||_p||g||_p=2^{2/p}approx 2.074 for p=1.9, so (1.16) and (1.19) fail. Corollary 7 accordingly fails for kappa=3: ~E_3({0,1})=2^3+1+1=10 > 2^3=8.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":24306,"tokens_out":7864,"duration_ms":66450,"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":[{"comment":"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.","section":"Section 9, Lemma 13 and Eq. (9.4)"},{"comment":"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.","section":"Section 4 (Lemma 10) and Section 9 (Lemma 12)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"The sentence 'random variables ... taking value s in the binary cube' contains a typo; it should read 'taking values'.","section":"Section 1.2"}],"recommendation":"reject","confidential_remarks":"To the editor: the counterexample in Section 9 is directly fatal to Theorem 6 and Corollary 7, so I cannot recommend revision. Even if Theorem 1 is sound, the paper's central claim about both classical inequalities is false. The unshipped Mathematica verification for Theorem 1 would also need to be made available in any future submission. I saw no indication of misconduct; the error is a straightforward algebraic mismatch in Lemma 13."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, the short version: don't let the Young convolution half of this paper take up your time. Theorem 6 is false. The bug is in Lemma 13. After the anti-diagonal substitution, the paper needs to prove (2 cosh(pt/2q))^{2q/p} >= (2 cosh(t/q))^q + 2. Lemma 12's (9.4) gives the same left side but with 2q (cosh(t/q))^q + 2 on the right. For q=2 the two coincide, but for q>2 they do not, and the desired inequality already fails at t=0: it would say 2q+2 >= 2^q+2. Direct d=1 counterexample: q=3, p=2, f=g=1_{0,1}. Then ||f*g||_3 = 10^{1/3} ≈ 2.154, while ||f||_2||g||_2 = 2. So Lemma 8, Theorem 6, Corollary 7, and Corollary 9 all fall. That is a load-bearing flaw, not a gap.\n\nCredit where due: the Hausdorff-Young half (Theorem 1) is a different matter. The continuous range q in (2,4) genuinely extends the integer-exponent results, the induction via the two-point inequality is clean, and the necessity arguments check out. The additive-energy and entropic corollaries would be attractive if the proof rested on shipped code. It does not: Lemma 10's ODE verification is Mathematica-only, with no code or certificates, so an independent reader cannot audit it. Corollary 5's equality computation also looks garbled in places.\n\nProportionately, the Hausdorff-Young material may be salvageable as a separate paper. As submitted, the paper makes false claims in the Young section and cannot be accepted. If it is resubmitted after removing or fixing that section, it deserves a serious referee. For now, I would not cite it and would not bring it to reading group.","headline":"Theorem 6 is false — the Young convolution half of the paper collapses on a coefficient error, though the Hausdorff-Young half may be sound.","tokens_in":24873,"tokens_out":7352,"would_cite":false,"duration_ms":69790,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42A05","05D05","94A17","42B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["sharp estimate","Fourier transform","Fourier restriction","additive energy","entropy","Hausdorff–Young inequality","Young's convolution inequality","binary cube"],"falsifier":"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.","tokens_in":23593,"feed_emoji":"📐","tokens_out":11121,"duration_ms":118190,"temperature":0.7,"pith_summary":"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.","feed_headline":"Binary-cube Fourier inequalities get exact exponent range","feed_subtitle":"On {0,1}^d, Hausdorff–Young and Young hold with constant 1 in two enlarged, sharply characterized regions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the sharp constants and the entropic-uncertainty template that the binary-cube results refine.","marker":"[2]"},{"why":"Proves the q=4 case of the binary Hausdorff–Young estimate, the finite-sum starting point generalized here.","marker":"[8]"},{"why":"Treats the even-integer q cases up to 10 and introduces the additive-energy formulation.","marker":"[5]"},{"why":"Completes all even integer q=2k via the binomial inequality that Lemma 4 extends to all real q>2.","marker":"[9]"},{"why":"Provides the Legendre function integral representation, the Legendre ODE, and the gamma-function identities used in Lemma 10 and in the necessity arguments.","marker":"[11]"},{"why":"Supplies the entropy uncertainty principle whose binary-cube sharpening is Corollary 5.","marker":"[7]"},{"why":"Provides the computer-algebra system used for the symbolic ODE verification and the exact grid evaluations on which Lemmas 10 and 12 rely.","marker":"[17]"}],"fun_headline_variants":["Sharp Fourier inequalities on binary cubes pinned down","Exact exponent ranges for binary-cube Fourier estimates","Optimal Hausdorff-Young and Young on {0,1}^d","Dimension-free Fourier bounds on binary cubes characterized","Binary cube Fourier inequalities: exact thresholds found"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Fourier inequalities on binary cubes pinned down","Exact exponent ranges for binary-cube Fourier estimates","Optimal Hausdorff-Young and Young on {0,1}^d","Dimension-free Fourier bounds on binary cubes characterized","Binary cube Fourier inequalities: exact thresholds found"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1383,"prompt_tokens":1009,"completion_tokens":374,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":298}},"tokens_in":625,"tokens_out":374,"duration_ms":4063,"temperature":1.0,"reasoning_tokens":298,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:54:31.225249+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Inequalities in Fourier analysis","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp constants and the entropic-uncertainty template that the binary-cube results refine."},{"cited_title":"A bound on partitioning clus ters","cited_arxiv_id":null,"evidence_quote":"Proves the q=4 case of the binary Hausdorff–Young estimate, the finite-sum starting point generalized here."},{"cited_title":"Additive energies on discrete cubes","cited_arxiv_id":null,"evidence_quote":"Treats the even-integer q cases up to 10 and introduces the additive-energy formulation."},{"cited_title":"On binomial sums, additive energies, and lazy random walks","cited_arxiv_id":null,"evidence_quote":"Completes all even integer q=2k via the binomial inequality that Lemma 4 extends to all real q>2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Legendre function integral representation, the Legendre ODE, and the gamma-function identities used in Lemma 10 and in the necessity arguments."},{"cited_title":"A note on entropy","cited_arxiv_id":null,"evidence_quote":"Supplies the entropy uncertainty principle whose binary-cube sharpening is Corollary 5."},{"cited_title":"Mathematica, Version 14.2, 202 5","cited_arxiv_id":null,"evidence_quote":"Provides the computer-algebra system used for the symbolic ODE verification and the exact grid evaluations on which Lemmas 10 and 12 rely."}],"review_version":1}