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arxiv: 2607.01458 · v1 · pith:C3U4JMJRnew · submitted 2026-07-01 · 🧮 math.CO · math.CA· math.MG

Sharp Lower Bounds for Sumsets in Hypercubes

Felipe Gon\c{c}alves , Danylo Radchenko This is my paper

Pith reviewed 2026-07-03 19:52 UTC · model grok-4.3

classification 🧮 math.CO math.CAmath.MG
keywords sumsetshypercubesadditive combinatoricslower boundsmixed volumeslattice pathssup-convolution
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The pith

Sumsets of n subsets inside an m-hypercube obey a sharp lower bound |A1+⋯+An| ≥ (|A1|⋯|An|)^{1/p} with p = n log(m+1)/log(nm+1).

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

The paper proves a lower bound on the size of the sumset formed by n subsets of the integer lattice each confined to the hypercube with coordinates running from 0 to m. The bound states that the cardinality of the sum is at least the product of the individual cardinalities raised to the power 1/p, where p is defined as n times log of (m+1) divided by log of (nm+1), and this exponent cannot be improved. The result resolves a conjecture that had been circulating in additive combinatorics and extends the only previously known sharp cases (binary hypercubes for any n, and ternary for n=2). The proof proceeds by establishing a stronger inequality for the sup-convolution of functions on Z^d and then specializing to indicator functions.

Core claim

For any sets A_j ⊆ {0,1,…,m}^d the inequality |A1+⋯+An| ≥ (|A1|⋯|An|)^{1/p} holds with p = n log(m+1)/log(nm+1), and the exponent is best possible. The same conclusion holds when the upper limits m_j are allowed to differ across the sets. This follows from a stronger functional inequality on sup-convolutions that is proved via a mixed-volume representation of the lattice-path norm together with a sharp one-dimensional inequality.

What carries the argument

Mixed-volume representation of the lattice-path norm together with a sharp one-dimensional functional inequality on Z.

If this is right

  • The bound remains valid when each set A_j is confined to its own hypercube {0,…,m_j}^d.
  • The exponent p cannot be replaced by any smaller number while keeping the inequality true for all choices of sets.
  • The inequality specializes to the known sharp results when m=1 for any n and when m=2 and n=2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mixed-volume approach may extend to other discrete Brunn-Minkowski-type statements on product sets.
  • Small explicit examples in low dimension and small m can be checked computationally to verify the bound saturates.
  • The functional form may yield new inequalities for the support size of convolutions of arbitrary nonnegative functions rather than just indicators.

Load-bearing premise

A sharp one-dimensional functional inequality holds for the relevant functions on the integers.

What would settle it

Explicit sets A_j inside {0,…,m}^d whose sumset has cardinality strictly smaller than the right-hand side for the stated p would falsify the claim.

Figures

Figures reproduced from arXiv: 2607.01458 by Danylo Radchenko, Felipe Gon\c{c}alves.

Figure 1
Figure 1. Figure 1: A geometric interpretation of Theorem 3 for 𝑛 = 2. The yellow and blue polygons are 𝑄2 ( 𝑓𝑖). The upper left green region is a component of the complement of 𝑄2( 𝑓1) ∪ 𝑄2 ( 𝑓2) in the Minkowski sum 𝑄2( 𝑓1) + 𝑄2 ( 𝑓2) and its area is the mixed volume 𝑉(𝑄2( 𝑓1), 𝑄2( 𝑓2)). The green figure on the bottom right shows the path of maximal area from the lower left to the upper right rectangle. The theorem asserts … view at source ↗
Figure 2
Figure 2. Figure 2: Decomposition of 𝑄2( 𝑓1) into 𝑄2( 𝑓 ′ 1 ) ∪ 𝑄2 ( 𝑓 ′′ 1 ). Let 𝑗in ≤ 𝑗out be the 𝑗-coordinates where 𝛾 enters and leaves the column 𝑖 = 𝑖0. We can construct two new valid lattice paths: 𝛾 ′ for 𝑓 ′ 1 ⊗ 𝑓2 which goes from (0, 0) to (𝑖0, 𝑀2) by following 𝛾 and then moving straight up from (𝑖0, 𝑗out), and 𝛾 ′′ for 𝑓 ′′ 1 ⊗ 𝑓2 which goes from (𝑖0, 0) to (𝑀1, 𝑀2) by moving straight up to (𝑖0, 𝑗in) and then foll… view at source ↗
Figure 3
Figure 3. Figure 3: Splitting the path 𝛾 into 𝛾 ′ and 𝛾 ′′ at the vertical line 𝑥 = 𝑖0. By symmetry, this property holds for every direction 𝑒𝑙 , meaning that 𝛾 only changes direction at the points of the subgrid 𝐼 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

We prove a sharp lower bound for the cardinality of sumsets of subsets of $\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets $A_j\subseteq \{0,1,2,\dots,m\}^d$ we show that \[|A_1+\dots+A_n|\;\geq\; (|A_1|\cdots|A_n|)^{1/p},\qquad p=\frac{n\log(m+1)}{\log(nm+1)},\] with the exponent best possible. The only previously known sharp cases were $A_j\subseteq \{0,1\}^d$, for all $n\ge1$, and $A_j\subseteq \{0,1,2\}^d$ for $n=2$. We also prove a sharp inequality in the case when $A_j\subseteq\{0,1,\dots,m_j\}^d$ for different $m_j$. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on $\mathbb{Z}^d$, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves sharp lower bounds on sumset cardinalities for subsets A_j of the hypercube {0,1,...,m}^d in Z^d: |A1+⋯+An| ≥ (|A1|⋯|An|)^{1/p} with p = n log(m+1)/log(nm+1), and shows the exponent is optimal. It also treats the case of distinct m_j. The result is obtained as a corollary of a stronger sup-convolution inequality for functions on Z^d, derived from a novel mixed-volume representation of the lattice-path norm combined with a sharp one-dimensional functional inequality on Z.

Significance. If correct, the result resolves a folklore conjecture in additive combinatorics with the first sharp bounds for general m beyond the cases m=1 (all n) and m=2 (n=2). The mixed-volume approach to the lattice-path norm constitutes a methodological advance with potential for further applications in discrete geometry and inequalities.

major comments (2)
  1. [Proof of the sup-convolution inequality (via mixed-volume representation and 1D inequality)] The central claim rests on transferring the mixed-volume representation of the lattice-path norm to a strict lower bound on discrete sumset cardinalities via the 1D functional inequality. The manuscript must supply an explicit discretization or approximation argument (with error term shown to vanish in the relevant regime) to confirm that the inequality passes to the indicator functions of the sets A_j without loss of sharpness; this step is load-bearing for both the inequality and the optimality statement.
  2. [Sharpness / equality cases] Equality cases are asserted to arise exactly from the 1D construction. The interaction between the mixed-volume representation and the support constraints {0,…,m}^d must be verified to ensure that the extremal examples in the hypercube achieve equality without additional discretization error; otherwise the sharpness claim requires adjustment.
minor comments (2)
  1. Clarify the precise statement of the one-dimensional functional inequality (including the class of functions to which it applies) and its proof, as this is invoked as a black box for the higher-dimensional result.
  2. Add a short comparison table or paragraph contrasting the new exponent with the previously known sharp cases (m=1 and m=2, n=2) to highlight the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of the results. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Proof of the sup-convolution inequality (via mixed-volume representation and 1D inequality)] The central claim rests on transferring the mixed-volume representation of the lattice-path norm to a strict lower bound on discrete sumset cardinalities via the 1D functional inequality. The manuscript must supply an explicit discretization or approximation argument (with error term shown to vanish in the relevant regime) to confirm that the inequality passes to the indicator functions of the sets A_j without loss of sharpness; this step is load-bearing for both the inequality and the optimality statement.

    Authors: The mixed-volume representation is developed directly for the lattice-path norm on the discrete group Z^d, and the sup-convolution inequality is proved for arbitrary nonnegative functions on Z^d. The one-dimensional functional inequality holds exactly on Z. Indicator functions of subsets A_j ⊆ {0,…,m}^d are admissible inputs, so the passage to the sumset cardinality bound is immediate and exact, with no approximation or discretization step required. We will insert a short clarifying paragraph after the statement of the functional inequality to make this direct applicability explicit. revision: partial

  2. Referee: [Sharpness / equality cases] Equality cases are asserted to arise exactly from the 1D construction. The interaction between the mixed-volume representation and the support constraints {0,…,m}^d must be verified to ensure that the extremal examples in the hypercube achieve equality without additional discretization error; otherwise the sharpness claim requires adjustment.

    Authors: The extremal configurations are obtained by taking identical one-dimensional extremal sets in each coordinate; these sets lie inside {0,…,m} and therefore inside the hypercube. Because both the mixed-volume identity and the one-dimensional inequality are exact equalities on Z, the resulting functions on Z^d attain equality in the sup-convolution inequality with no error term. We will add an explicit verification of equality attainment (including the coordinate-wise construction) in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent mixed-volume representation and 1D inequality

full rationale

The paper derives the sumset bound as a corollary of a sup-convolution inequality obtained from a novel mixed-volume representation of the lattice-path norm plus a sharp 1D functional inequality on Z. These steps are presented as external to the target cardinality result, with no reduction of the final inequality to fitted parameters, self-definitions, or self-citation chains. The exponent is shown best possible via explicit equality cases, and the argument is self-contained against the stated conjecture without importing uniqueness theorems or ansatzes from prior self-work. No load-bearing step reduces by construction to the input data or target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract indicates reliance on standard convex geometry and analysis; no explicit free parameters, ad-hoc axioms, or new postulated entities are described.

pith-pipeline@v0.9.1-grok · 5769 in / 1067 out tokens · 23539 ms · 2026-07-03T19:52:55.429046+00:00 · methodology

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Reference graph

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