{"id":"59ed42a7-0b41-4812-8c48-7cebe2b4e4c4","arxiv_id":"2507.06115","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp Young's convolution inequality and its reverse on {0,1}^d are proved, with optimal diagonal exponent p_r = 2r/log_2(2+2^r).","lead":"This paper finds the exact best possible exponents in Young's convolution inequality for functions supported on the discrete cube, and proves a matching reverse inequality. The sharp constants yield bounds for additive energies and sumsets, recovering and unifying several known results.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Lemma 2.1 reduction and the scalar inequality proof in §3 hold up under scrutiny.","rationale":"The reader's accepted verdict is supported. The weakest assumption identified by the reader, Lemma 2.1, is indeed the load-bearing hinge of the paper, and I examined it closely. The reduction is mathematically sound: the scalar inequality is equivalent to the one-dimensional convolution inequality for positive boundary values, the zero cases follow from the norm embeddings under exactly the conditions p,q≤r (or p,q≥r in the reverse case), and the induction step is a correct application of Minkowski's inequality. The subsequent Sections 3 and 4 contain the real substance, and I traced the sign bookkeeping in Lemma 3.1 and Proposition 3.2. In particular, the reductions involving the function φ(t), the use of φ(t)=φ(1/t), and the inequalities p<2, 1≤p≤2 for r>1, and p<1 for r<1 are all consistent; no step reverses an inequality without appropriate compensation. The off-diagonal necessary conditions and the r=2 sufficiency proof also check out, including the endpoint identity for f_y(y^3) after the substitution y↦1/y. Thus no genuine correctness risk to the central theorem emerged. Since the proof is long and computational in places, a symbolic or high-precision numerical verification of the main reduction chain remains a worthwhile sanity check, but I do not see a reason to change the verdict.","tokens_in":13904,"tokens_out":42474,"duration_ms":387477,"concrete_test":"Symbolically re-derive the chain (3.9)→(3.13) in Proposition 3.2 for generic r, keeping track of the direction of every inequality after raising to the power p/(r−p) and after using φ(t)=φ(1/t); confirm that the reductions to (3.16), (3.17), and (3.18) preserve the required signs for both r>1 and r<1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the argument from Lemma 2.1 through Lemma 3.1 and Proposition 3.2 and found no load-bearing flaw. The reduction is valid: the d=1 base case is exactly the scalar inequality after rescaling by f1,g1, with the degenerate cases handled by the embeddings ℓ^q⊆ℓ^r and ℓ^p⊆ℓ^r when p,q≤r (and the reverse embeddings when p,q≥r for r<1); the induction uses Minkowski for r>1 and reverse Minkowski for r<1, and the final application of the d=1 case to the coefficient arrays a_{u_d}=‖f_{u_d}‖_p, b_{v_d}=‖g_{v_d}‖_q is correct. In Lemma 3.1 the reduction to w=x^r and the monotonicity/sign analysis of f′ are consistent, including the key fact that f(0)=f(1)=0 forces the required sign of f once f′ has at most one zero. In Proposition 3.2 the long chain from (3.5) to (3.18) was checked step by step: the factorization of the quadratic, the reduction to φ(t), the symmetry φ(t)=φ(1/t), the bounds using p<2, and the final sign arguments for k′ are all directionally correct in both regimes r>1 and r<1. The sharpness argument via x=y=1 in the d=1 scalar inequality is also sound. I therefore do not see a concrete failure mode for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes sharp Young-type convolution inequalities for functions on Z^d supported on the hypercube {0,1}^d. The main result (Theorem 1.1) proves for each r ≥ 1 the diagonal inequality ||f*g||_{ℓ^r} ≤ ||f||_{ℓ^{p_r}}||g||_{ℓ^{p_r}} with p_r = 2r/log_2(2+2^r), and shows that no larger exponent is possible. Theorem 1.8 gives the analogous reverse inequality for 0 < r < 1 with the same exponent p_r, and sharpness is proved. The paper also derives necessary off-diagonal conditions (Propositions 1.3 and 1.9), a complete characterization for r = 2 (Theorem 1.4), and applications to additive energies and sumset bounds (Corollaries 1.6, 1.7, 1.10, 1.11). The proof strategy is to reduce the convolution inequality to a scalar inequality via an induction over coordinates (Lemma 2.1), then verify the scalar inequality by detailed calculus in Lemma 3.1 and Proposition 3.2.","tokens_in":14201,"tokens_out":49606,"duration_ms":469115,"significance":"The result is significant: it determines the sharp exponent for Young's convolution inequality on the hypercube in the diagonal case and provides a unified treatment of the forward and reverse inequalities, yielding known Brunn-Minkowski-type and additive-energy estimates as corollaries. The proof is self-contained and purely analytical, with a clean reduction to a one-variable inequality; this is a positive feature given that independent concurrent work uses computer-assisted verification. I checked the key steps in Lemma 2.1, Lemma 3.1, and Proposition 3.2 and found no gap in the central argument. The off-diagonal classification for r = 2 is a useful additional contribution.","major_comments":[],"minor_comments":[{"comment":"The sentence after Corollary 1.2 stating that f = 1_{0,1}^d shows failure when 1/p+1/q < log_2(2^r+2)/2 appears incorrect: for f = g = 1_{0,1}^d the actual threshold is log_2(1+2^r)/r. The sharpness of the line follows from the one-dimensional example f = g = 1_{0,1}, which gives the threshold log_2(2+2^r)/r. Please correct the example and the displayed threshold.","section":"Corollary 1.2"},{"comment":"In the proof of Theorem 1.4, the statement that 'the remaining bounds follow by interpolation' is made in one sentence. Since the theorem is an if-and-only-if statement over a continuum of exponents, please provide the precise bilinear Riesz-Thorin interpolation step or cite a standard reference so the sufficiency part is fully self-contained.","section":"Theorem 1.4"},{"comment":"In several displayed equations, the notation '2r' is used where the context requires 2^r (for example in the expressions for f'(w) and g(w) in Lemma 3.1). Please ensure superscripts are unambiguous in the final typeset version.","section":"Section 3"},{"comment":"The proof of Proposition 1.9 is dismissed as 'entirely analogous' to Proposition 1.3. Because the reverse inequality reverses the required sign of h'(1), a brief sentence recording the sign condition would remove any ambiguity for the reader.","section":"Proposition 1.9"},{"comment":"The Remark after the proofs of Propositions 1.3 and 1.9 says that combined with Corollary 1.2 'would imply false estimates'; please spell out which false estimates would be implied, as the current phrasing is cryptic.","section":"Section 4.1, Remark"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript overlaps with the independent work [4], which already proved Theorem 1.1 using a computer-assisted approach. The present paper's contribution remains substantial because its proof is analytic and it includes the reverse inequality, the r = 2 off-diagonal classification, and the applications. The authors properly disclose the overlap in a remark. No other concerns about fit or attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nYou should know this paper proves the sharp Young convolution inequality on the discrete hypercube in the diagonal case, with exponent p_r = 2r/log2(2+2^r), and the matching reverse inequality for 0<r<1. The diagonal theorem was found simultaneously by Crmarič, Kovač, and Shiraki via a computer-assisted proof; this paper gives a purely analytic proof, which is a genuinely different route, and the reverse inequality and the sharp off-diagonal classification at r=2 are new and go beyond that concurrent work. The applications to additive energies and sumsets are clean corollaries and recover known sharp results.\n\nThe core argument is a reduction (Lemma 2.1) to a scalar inequality in two variables, followed by a proof of that inequality for p=q=p_r. I traced the reduction and the scalar proof; they hold up. Lemma 3.1 handles x=y by one-variable calculus, and Proposition 3.2 reduces the general case to the geometric mean via a derivative inequality. The derivative chain from (3.5) to (3.18) is long and easy to get lost in, but I found no sign error and no circular step. The sharpness argument with f=g=1_{cube} is correct.\n\nSoft spots are minor. The proof of Proposition 3.2 is technical and would benefit from an informal explanation of why the diagonal case should be extremal. There is a typo in Corollary 1.2: it mentions both f and g, but the inequality only involves f. Interpolation details for Corollary 1.5 are omitted but standard. The off-diagonal necessary conditions in Proposition 1.3 are not shown to be sufficient except at r=2, so the off-diagonal picture is incomplete, but the paper is upfront about it. The reverse inequality for r<1 is a nice addition that ties into discrete Prékopa–Leindler.\n\nThe citation pattern is honest; the concurrent work is cited, and the authors' own prior results appear only as applications. The paper is self-contained.\n\nMy take: this deserves a serious referee. The central theorem is now in the literature twice, but the analytic proof and the extra results justify publication. I would send it to a good harmonic analysis referee and expect acceptance after minor revision.\n\nBest,","headline":"Sharp diagonal Young on the hypercube, proved analytically, with a genuine reverse inequality and a clean r=2 off-diagonal classification; the proof is long but sound.","tokens_in":14746,"tokens_out":2794,"would_cite":true,"duration_ms":26519,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["39A12","26D15","11B30","11B13"],"pacs":[],"model":"deepseek-v4-flash","headline":"Functions on the discrete cube obey a sharp Young convolution inequality whose diagonal exponent $p_r=2r/\\log_2(2+2^r)$ is best possible, and the reverse inequality uses the same exponent.","keywords":["Young's convolution inequality","hypercube","sharp constants","reverse Young inequality","additive energies","sumset bounds","higher additive energy","diagonal exponent"],"falsifier":"Maximize the ratio $[1+(x+y)^r+(xy)^r]^{1/r} / [(1+x^{p_r})^{1/p_r}(1+y^{p_r})^{1/p_r}]$ over $x,y\\ge 0$ for a fixed $r>1$; any value above $1$ disproves Theorem 1.1. For $0<r<1$, any value below $1$ in the corresponding ratio for the reverse inequality disproves Theorem 1.8. The $d=1$ case is already decisive by the reduction lemma.","tokens_in":13728,"feed_emoji":"📐","tokens_out":18189,"duration_ms":181432,"temperature":0.7,"pith_summary":"Young's convolution inequality controls the $\\ell^r$ norm of a convolution by products of $\\ell^p$ norms. When the functions are supported on the discrete cube $\\{0,1\\}^d$, the paper shows that in the diagonal case $p=q$ the exponent can be raised from the classical value $2r/(r+1)$ to $p_r = 2r/\\log_2(2+2^r)$, and that no larger exponent is possible. The same $p_r$ appears in a reverse inequality for $0<r<1$, where convolution is bounded below by the product of $\\ell^{p_r}$ norms for nonnegative functions. These two results imply sharp bounds for additive energies and for the size of sumsets in the cube, recovering and extending earlier results. The proof is a chain of reductions: the $d$-dimensional inequality becomes a two-variable scalar inequality, and that scalar inequality is controlled by a one-variable calculus computation.","feed_headline":"Exact exponent found for Young convolution on the hypercube","feed_subtitle":"The exact exponent 2r/log2(2+2^r) also controls a reverse inequality that bounds sumsets.","key_machinery":"The load-bearing mechanism is Lemma 2.1, an induction over coordinates that reduces the $d$-dimensional convolution inequality to the two-variable scalar inequality $$[1+(x+y)^r+(xy)^r]^{1/r} \\le (1+x^p)^{1/p}(1+y^q)^{1/q}$$ for all $x,y\\ge 0$, with the inequality reversed for $r<1$. The induction uses the triangle inequality for the $\\ell^r$ norm along the last coordinate and works separately for $r>1$ and $r<1$, so checking the scalar inequality is equivalent to checking the whole theorem. The proof of the scalar inequality in the diagonal case then splits: Proposition 3.2 shows that the function $$H_r(x,y)=1+(x+y)^r+(xy)^r-(1+$x^{{p_r}}$)^{r/p_r}(1+$y^{{p_r}}$)^{r/p_r}$$ is maximized for $r>1$ and minimized for $r<1$ on the diagonal $x=y$, using the sign of the differential expression $x\\partial_x H-y\\partial_y H$; Lemma 3.1 verifies the resulting one-variable inequality by analyzing the derivative of a function with $w=x^r$. Sharpness comes from evaluating the scalar inequality at $x=y=1$.","core_discovery":"The paper's central discovery is that Young's convolution inequality on functions supported on $\\{0,1\\}^d$ is governed, in the diagonal case $p=q$, by the exponent $p_r = 2r/\\log_2(2+2^r)$. For $r\\ge 1$ it proves $\\|f*g\\|_{\\ell^r(\\mathbb{Z}^d)} \\le \\|f\\|_{\\ell^{p_r}(\\mathbb{Z}^d)}\\|g\\|_{\\ell^{p_r}(\\mathbb{Z}^d)}$ for all real-valued $f,g$, and the indicator of the whole hypercube shows the exponent cannot be increased. For $0<r<1$ it proves the reverse inequality with the same exponent for nonnegative functions, and the exponent cannot be decreased. In the off-diagonal range $p\\ne q$ the paper gives necessary restrictions on $p$ and $q$ along the line $1/p+1/q = \\log_2(2+2^r)/r$, and it fully characterizes the valid range when $r=2$. The sharp inequality for $f=g$ follows by a standard interpolation step, and the reverse inequality has a limiting $r\\to 0$ form that is a sharp sumset bound.","pith_inferences":["A natural conjecture, suggested by the $r=2$ characterization and by the necessary conditions of Proposition 1.3, is that the interval for $p$ and $q$ in Proposition 1.3 is sufficient for every $r>1$; the paper proves sufficiency only for $r=2$.","The same coordinatewise reduction should transfer to functions supported on wider boxes such as $\\{0,1,\\dots,m-1\\}^d$, with exponents depending on $m$ through an analogous scalar inequality; the paper does not pursue this.","Because the reverse inequality has a meaningful $r\\to0$ limit, the sumset estimates sit at the endpoint of a scale of Young-type inequalities; interpolating along that scale could yield intermediate bounds for partial sumsets, a direction not addressed here."],"forward_implications":["For $f=g$, a standard interpolation step transfers the diagonal bound to all pairs with $1/p+1/q=\\log_2(2+2^r)/r$, giving sharp off-diagonal control on the same efficiency line.","The $k$-higher additive energy of two sets $A,B\\subset\\{-1,1\\}^d$ satisfies $\\tilde E_k(A,B)\\le |A|^{q_k/2}|B|^{q_k/2}$ with $q_k=\\log_2(2+2^k)$, and the exponent is optimal; for $k=2$ this extends previous single-set bounds to pairs.","The sumset of subsets $A,B\\subset\\{0,1\\}^d$ obeys $|A+B|\\ge |A|^{(\\log_2 3)/2}|B|^{(\\log_2 3)/2}$, matching the sharp sumset bound on the cube.","For $r=2$, the inequality $\\|f*g\\|_2\\le\\|f\\|_p\\|g\\|_q$ holds on the line $1/p+1/q=(\\log_2 6)/2$ exactly for $4/3\\le p,q\\le 1/((\\log_2 3)/2-1/4)$."],"supporting_citations":[{"why":"Supplies the classical converse Young inequality whose sharp hypercube version is Theorem 1.8.","marker":"[8]"},{"why":"The sharp sumset bound recovered as the r to 0 limit of the reverse inequality.","marker":"[1]"},{"why":"One of the independent proofs of the sharp sumset bound recovered by Corollary 1.11.","marker":"[11]"},{"why":"The other independent proof of the same sharp sumset bound, also recovered by Corollary 1.11.","marker":"[6]"},{"why":"Previous additive-energy bound for the case k=2 and A=B, extended to pairs by Corollary 1.6.","marker":"[7]"},{"why":"Previous additive-energy bounds for general integer k with A=B, extended to distinct sets by Corollary 1.6.","marker":"[5]"},{"why":"Supplies the definition of higher moments of convolutions used in the additive-energy formulation.","marker":"[9]"},{"why":"Supplies the definition of k-higher additive energies used in the applications.","marker":"[10]"}],"fun_headline_variants":["Sharp exponent for Young convolution on hypercube","Forward and reverse Young inequalities sharp on hypercube","Hypercube convolution inequality made sharp","Exact exponent rules hypercube convolutions","Young convolution sharpened on discrete hypercube"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the claim that the scalar inequality $[1+(x+y)^r+(xy)^r]^{1/r} \\le (1+x^{p_r})^{1/p_r}(1+y^{p_r})^{1/p_r}$ holds for every $x,y\\ge 0$; if any pair of nonnegative numbers violates it, the $d$-dimensional theorem fails, because Lemma 2.1 shows the two statements are equivalent.","fun_headline_variants_meta":{"raw":{"variants":["Sharp exponent for Young convolution on hypercube","Forward and reverse Young inequalities sharp on hypercube","Hypercube convolution inequality made sharp","Exact exponent rules hypercube convolutions","Young convolution sharpened on discrete hypercube"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001086,"raw_usage":{"total_tokens":4495,"prompt_tokens":860,"completion_tokens":3635,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":3570}},"tokens_in":476,"tokens_out":3635,"duration_ms":30981,"temperature":1.0,"reasoning_tokens":3570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:13:13.377294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Maximize the ratio $[1+(x+y)^r+(xy)^r]^{1/r} / [(1+x^{p_r})^{1/p_r}(1+y^{p_r})^{1/p_r}]$ over $x,y\\ge 0$ for a fixed $r>1$; any value above $1$ disproves Theorem 1.1. For $0<r<1$, any value below $1$ in the corresponding ratio for the reverse inequality disproves Theorem 1.8. The $d=1$ case is already decisive by the reduction lemma.","supporting_citations":[{"cited_title":"Leindler","cited_arxiv_id":null,"evidence_quote":"Supplies the classical converse Young inequality whose sharp hypercube version is Theorem 1.8."},{"cited_title":"Discrete Brunn-Minkowski Inequality for subsets of the cube","cited_arxiv_id":"2404.04486","evidence_quote":"The sharp sumset bound recovered as the r to 0 limit of the reverse inequality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"One of the independent proofs of the sharp sumset bound recovered by Corollary 1.11."},{"cited_title":"Hajela and P","cited_arxiv_id":null,"evidence_quote":"The other independent proof of the same sharp sumset bound, also recovered by Corollary 1.11."},{"cited_title":"A bound on partitioning clusters","cited_arxiv_id":null,"evidence_quote":"Previous additive-energy bound for the case k=2 and A=B, extended to pairs by Corollary 1.6."},{"cited_title":"Additive energies on discrete cubes","cited_arxiv_id":null,"evidence_quote":"Previous additive-energy bounds for general integer k with A=B, extended to distinct sets by Corollary 1.6."},{"cited_title":"Shkredov","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of higher moments of convolutions used in the additive-energy formulation."},{"cited_title":"Energies and structure of additive sets","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of k-higher additive energies used in the applications."}],"review_version":1}