Optimal Young's convolutions inequality and its reverse form on the hypercube

David Beltran; Jos\'e Madrid; Lekha Patil; Paata Ivanisvili

Optimal Young's convolutions inequality and its reverse form on the hypercube

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.06115 v1 pith:OGXQ2KKF submitted 2025-07-08 math.CA math.CO

David Beltran , Paata Ivanisvili , José Madrid , Lekha Patil This is my paper

classification math.CAmath.CO MSC 39A1226D1511B3011B13

keywords Young'sconvolutioninequalityhypercubesharpconstantsreverseYoungadditiveenergiessumsetboundshigherenergydiagonalexponent

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

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

What carries the argument

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 $x=y=1$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $4/3\le p,q\le 1/((\log_2 3)/2-1/4)$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

You 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

The 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.

Soft 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.

The citation pattern is honest; the concurrent work is cited, and the authors' own prior results appear only as applications. The paper is self-contained.

My 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.

Best,

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[Corollary 1.2] 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.
[Theorem 1.4] 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 3] 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.
[Proposition 1.9] 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 4.1, Remark] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sharp exponent is forced by an extremizer and the sufficiency proof is an independent scalar inequality.

full rationale

Trace of the derivation: Theorem 1.1 is proved by Lemma 2.1, which reduces the d-dimensional convolution inequality to the scalar inequality (2.1), followed by Lemma 3.1 for the x = y case and Proposition 3.2, which reduces general (x, y) to x = y via the sign analysis of R'(s). This yields (2.1) for p = q = p_r. Sharpness is then shown independently by evaluating the d = 1 scalar inequality at x = y = 1, forcing p <= p_r; this is not a fitted parameter but the threshold determined by the test function f = g = 1_{cube}. The reverse inequality in Theorem 1.8 follows the same chain with reversed inequalities and reverse Minkowski. Prior work by the authors, such as references [1] and [5], is used only for applications, context, or related known results, and is not load-bearing for the proof of Theorems 1.1 or 1.8. The central claim is therefore self-contained: no equation is equivalent to its own input by construction, no fitted quantity is renamed as a prediction, and no self-citation is used to force the conclusions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests only on standard mathematical background; no new constants are fitted and no new objects are postulated. The optimal exponent is derived from an extremal example, making the argument self-contained.

assumptions (3)

standard math Minkowski's inequality and its reverse in ℓ_r for 0<r<1
Used in Lemma 2.1 to reduce the d-dimensional convolution inequality to a scalar inequality by induction over coordinates.
standard math Hölder's inequality
Used in Corollary 1.2 and in off-diagonal interpolation arguments.
standard math Descartes' rule of signs
Used in the proof of Theorem 1.4 to count sign changes of R'(y) and establish the r=2 case.

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Pith. "Pith review of Optimal Young's convolutions inequality and its reverse form on the hypercube." pith.science (2026). https://pith.science/paper/OGXQ2KKF

@misc{pith2026250706115,
  author       = {Pith},
  title        = {Pith review of: Optimal Young's convolutions inequality and its reverse form on the hypercube},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGXQ2KKF}},
  note         = {Machine review of arXiv:2507.06115}
}

abstract

We establish sharp forms of Young's convolution inequality and its reverse on the discrete hypercube $\{0,1\}^d$ in the diagonal case $p=q$. As applications, we derive bounds for additive energies and sumsets. We also investigate the non-diagonal regime $p\neq q$, providing necessary conditions for the inequality to hold, along with partial results in the case $r = 2$.

Figures

Figures reproduced from arXiv: 2507.06115 by the authors.

**Figure 1.** The case r “ 2 in Corollary 1.5. The blue points represent the exponents p1{p, 1{qq “ p3{4, log2 3 2 ´ 1 4 q and p1{p, 1{qq “ p log2 3 2 ´ 1 4 , 3{4q. The convolution inequality }f ˚ g}2 ď }f}p}g}q for f, g : Z d Ñ R supported in t0, 1u d holds for p1{p, 1{qq in the gray region Ω, and fails in the non-colored region. The validity of the inequality in the interior of the yellow triangles and the boundary segments 1{… view at source ↗

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Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 1 Pith paper

[1]

Discrete Brunn-Minkowski Inequality for subsets of the cube

Lars Becker, Paata Ivanisvili, Dmitry Krachun, and Jóse Madrid. Discrete B runn- M inkowski inequality for subsets of the cube. Preprint: +arXiv:2404.04486+

work page Pith review arXiv
[2]

Sharp estimates for Gowers norms on discrete cubes

Adrian Beker, Tonći Crmarić, and Vjekoslav Kovač. Sharp estimates for G owers norms on discrete cubes. Preprint: +arXiv:2409.12579+

work page Pith review arXiv
[3]

Explicit constructions of RIP matrices and related problems

Jean Bourgain, Stephen Dilworth, Kevin Ford, Sergei Konyagin, and Denka Kutzarova. Explicit constructions of RIP matrices and related problems. Duke Math. J. , 159(1):145--185, 2011

work page 2011
[4]

Inequalities in Fourier analysis on binary cubes

Ton\'ci Crmari c , Vjekoslav Kova c , and Shobu Shiraki. Inequalities in F ourier analysis on binary cubes. Preprint: +arXiv:2507.01359+

work page Pith review arXiv
[5]

Additive energies on discrete cubes

Jaume de Dios Pont, Rachel Greenfeld, Paata Ivanisvili, and Jos\'e Madrid. Additive energies on discrete cubes. Discrete Anal. , pages Paper No. 13, 16, 2023

work page 2023
[6]

Hajela and P

D. Hajela and P. Seymour. Counting points in hypercubes and convolution measure algebras. Combinatorica , 5(3):205--214, 1985

work page 1985
[7]

A bound on partitioning clusters

Daniel Kane and Terence Tao. A bound on partitioning clusters. Electron. J. Combin. , 24(2):Paper No. 2.31, 13, 2017

work page 2017
[8]

Leindler

L. Leindler. On a certain converse of H \"older's inequality. In Linear operators and approximation ( P roc. C onf., M ath. R es. I nst., O berwolfach, 1971) , volume Vol. 20 of Internat. Ser. Numer. Math. , pages 182--184. Birkh\"auser Verlag, Basel-Stuttgart, 1972

work page 1971

Show all 23 references

[9]

Shkredov

Tomasz Schoen and Ilya D. Shkredov. Higher moments of convolutions. J. Number Theory , 133(5):1693--1737, 2013

2013
[10]

Energies and structure of additive sets

Ilya Shkredov. Energies and structure of additive sets. Electron. J. Combin. , 21(3):Paper 3.44, 53, 2014

2014
[11]

D. R. Woodall. A theorem on cubes. Mathematika , 24(1):60--62, 1977

1977
[12]

Bourgain, S

J. Bourgain, S. J. Dilworth, K. Ford, S. Konyagin, and D. Kutzarova, Explicit constructions of RIP matrices and related problems, Duke Math. Journal 159(1): 145--185 (2011)

2011
[13]

de Dios, R

J. de Dios, R. Greenfeld, P. Ivasnisvili and J. Madrid, Additive energies on discrete cubes, Preprint to appear in Discrete Analysis
[14]

Fish, Ben Lund, and A

S. Fish, Ben Lund, and A. Sheffer, A Construction for Difference Sets with Local Properties, European Journal of Combinatorics, 79 (2019), 237--243

2019
[15]

Gyarmati, M

K. Gyarmati, M. Matolcsi and I. Z. Ruzsa, Pl\"unnecke’s Inequality for Different Summands, Bolyai Society Mathematical Studies book series (BSMS,volume 19), Building Bridges, Between Mathematics and Computer Science, pages 309--320
[16]

Green, D

B. Green, D. Matolcsi, I. Z. Ruzsa, G. Shakan and D. Zhelezov, A weighted Prekopa-Leindler inequality and sumsets with quasicubes, preprint
[17]

B. J. Green and T. C. Tao, Compressions, convex geometry and the Freiman-Bilu theorem, Q. J. Math. 57 (2006), no. 4, 495--504

2006
[18]

Hanson and G

B. Hanson and G. Petridis, A Question of Bukh on Sums of Dilates, Discrete Analysis, 2021: Paper No. 13, 21 pp

2021
[19]

Ivanisvili, Convolution estimates and the number of disjoint partitions, The Electronic Journal of Combinatorics, Volume 24, Issue 2 (2017), Paper P2.43

P. Ivanisvili, Convolution estimates and the number of disjoint partitions, The Electronic Journal of Combinatorics, Volume 24, Issue 2 (2017), Paper P2.43

2017
[20]

Kovac, On binomial sums, additive energies, and lazy random walks, Preprint at arxiv.org/abs/2206.01591

V. Kovac, On binomial sums, additive energies, and lazy random walks, Preprint at arxiv.org/abs/2206.01591

arXiv
[21]

Kane and T

D. Kane and T. Tao, A bound on Partitioning Clusters, The Electronic Journal of Combinatorics, Volume 24, Issue 2 (2017), Paper P2.31

2017
[22]

Matolcsi, I

D. Matolcsi, I. Z. Ruzsa, G. Shakan and D. Zhelezov, An analytic approach to cardinalities of sumsets, Combinatorica (2022). https://doi.org/10.1007/s00493-021-4547-0

2022 doi
[23]

T. Tao, V. Vu, Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006

2006

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Discrete Brunn-Minkowski Inequality for subsets of the cube

Lars Becker, Paata Ivanisvili, Dmitry Krachun, and Jóse Madrid. Discrete B runn- M inkowski inequality for subsets of the cube. Preprint: +arXiv:2404.04486+

work page Pith review arXiv

[2] [2]

Sharp estimates for Gowers norms on discrete cubes

Adrian Beker, Tonći Crmarić, and Vjekoslav Kovač. Sharp estimates for G owers norms on discrete cubes. Preprint: +arXiv:2409.12579+

work page Pith review arXiv

[3] [3]

Explicit constructions of RIP matrices and related problems

Jean Bourgain, Stephen Dilworth, Kevin Ford, Sergei Konyagin, and Denka Kutzarova. Explicit constructions of RIP matrices and related problems. Duke Math. J. , 159(1):145--185, 2011

work page 2011

[4] [4]

Inequalities in Fourier analysis on binary cubes

Ton\'ci Crmari c , Vjekoslav Kova c , and Shobu Shiraki. Inequalities in F ourier analysis on binary cubes. Preprint: +arXiv:2507.01359+

work page Pith review arXiv

[5] [5]

Additive energies on discrete cubes

Jaume de Dios Pont, Rachel Greenfeld, Paata Ivanisvili, and Jos\'e Madrid. Additive energies on discrete cubes. Discrete Anal. , pages Paper No. 13, 16, 2023

work page 2023

[6] [6]

Hajela and P

D. Hajela and P. Seymour. Counting points in hypercubes and convolution measure algebras. Combinatorica , 5(3):205--214, 1985

work page 1985

[7] [7]

A bound on partitioning clusters

Daniel Kane and Terence Tao. A bound on partitioning clusters. Electron. J. Combin. , 24(2):Paper No. 2.31, 13, 2017

work page 2017

[8] [8]

Leindler

L. Leindler. On a certain converse of H \"older's inequality. In Linear operators and approximation ( P roc. C onf., M ath. R es. I nst., O berwolfach, 1971) , volume Vol. 20 of Internat. Ser. Numer. Math. , pages 182--184. Birkh\"auser Verlag, Basel-Stuttgart, 1972

work page 1971