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REVIEW 2 major objections 3 minor 1 cited by

Sums and differences of sets (improvement over AlphaEvolve)

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper constructs an explicit integer set with more than 10^43546 elements and proves the lower bound θ ≥ 1.173050 for the sums and differences of sets problem.

desk verdict A simple coordinate-bounded twist on the V(m,L) simplex yields a genuine record lower bound, but the paper's proof rests on a 15-hour computation you cannot audit from the PDF. read the letter →

arxiv 2505.16105 v1 pith:PYYVPZZO submitted 2025-05-22 math.NT

Robert Gerbicz This is my paper
classification math.NT MSC 11B75
keywords SumsetDifferencesetAlphaEvolvelowerboundexplicitconstructiondigitencodingboundedcoordinatesintegersets
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

Let $K>1$ be fixed. The sums and differences of sets problem asks for the largest exponent $\theta$ such that two integer sets $A,B$ with $|A|$ arbitrarily large can satisfy $|A+B|\le K|A|$ while $|A-B|$ grows at least like a constant times $|A+B|^\theta$. The paper claims to raise the known lower bound for $\theta$ from $1.1584$ to $\theta\ge 1.173050$, using an explicit integer set $U$ with more than $10^{43546}$ elements. The improvement comes from a new encoding: restrict the coordinates of the underlying vectors to a small bound $B$, then encode with base $2B+1$ instead of the much larger base used in earlier constructions. This makes $\max(U)$ far smaller while keeping $|U+U|$ and $|U-U|$ comparably large, which is exactly what the inequality $\theta\ge 1+\log(|U-U|/|U+U|)/\log(2\max(U)+1)$ rewards. The bound is verified by exact integer arithmetic at the parameter choice $m=81411$, $L=65536$, $B=5$.

What carries the argument

The load-bearing object is the bounded-coordinate digit set. Earlier constructions encoded every vector of the simplex $x_1+\cdots+x_m\le L$ using digits roughly the size of $2L$, so the largest encoded integer grows like $(2L)^m$. The new set $W(m,L,B)$ additionally requires each coordinate $x_k\le B$, and encodes with base $2B+1$; the coordinate bound does little harm because typical coordinates are near $L/m$, but it shrinks $\max(U)$ from roughly $(2L)^m$ to roughly $(2B+1)^m$. The four formulas (2.3)--(2.6) convert the problem into exact binomial sums: inclusion-exclusion for $|W|$, the identity $|U+U|=|W(m,2L,2B)|$, a signed sum over the number of positive coordinates for $|U-U|$, and a greedy expression for $\max(U)$. These formulas let the paper search candidates in floating point and then verify the winner in exact integer arithmetic.

What would settle it

Recompute the four exact quantities $|U|$, $|U+U|$, $|U-U|$, and $2\max(U)+1$ for $m=81411$, $L=65536$, $B=5$ from formulas (2.3)--(2.6) with arbitrary-precision integers, then evaluate $\theta = 1+\log(|U-U|/|U+U|)/\log(2\max(U)+1)$; if the result is less than $1.173050$, Theorem 2.2 is false. A convenient internal check is the exact identity $|U+U|=|W(81411,131072,10)|$ from (2.4).

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.2: for the sums and differences of sets problem, $\theta \ge 1.173050$. The theorem is established by exhibiting one explicit set $U = U(m,L,B)$, defined as the image under the base-$(2B+1)$ digit map $g(x_1,\dots,x_m)=\sum_{k=0}^{m-1}x_k(2B+1)^k$ of the set $W(m,L,B)$ of all $m$-tuples of nonnegative integers with coordinate sum at most $L$ and each coordinate at most $B$. Because $g$ is injective on $W+W$ and $W-W$, the counts $|U+U|$, $|U-U|$ and $\max(U)$ are given exactly by the binomial formulas (2.4), (2.5) and (2.6). Inserting those counts into the cited log-ratio inequality gives the numerical lower bound. The chosen parameters $m=81411$, $L=65536$, $B=5$ produce $|U| > 10^{43546}$, and a 15-hour exact integer computation at those parameters confirms the final value $\theta=1.173050$.

Load-bearing premise

The weakest load-bearing point is that the 15-hour exact integer computation for $(m,L,B)=(81411,65536,5)$ is correct: the paper prints only rounded approximations for $|U|$, $|U+U|$, $|U-U|$, and $2\max(U)+1$, and points to a link without a checksum or full transcript, so the value $\theta=1.173050$ cannot be fully audited from the PDF alone.

Editorial extensions

If this is right

  • The lower bound for $\theta$ in the sums and differences of sets problem becomes $1.173050$, improving on the previous accepted values $1.14465$ and $1.1584$.
  • The construction gives an explicit, algorithmically describable set $U$ with more than $10^{43546}$ elements that realizes the bound.
  • The exact counting formulas (2.3)--(2.6) are valid for every $(m,L,B)$, so the same machinery can produce bounds at other scales without new mathematics.
  • The paper's own final estimate says further increases of $L$ at $B=5$ would add less than $0.0001$ to $\theta$, so this particular family of constructions is close to its own limit.

Reading between the lines

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

  • The observed heuristic $m \approx \frac54 L$ at $B=5$ is used only as a search guide; proving or disproving such a scaling law for this family would turn the single verified point into a systematic optimization method.
  • The same digit-encoding trick -- bounding each coordinate to shrink the maximum encoded integer -- could be applied to other additive-combinatorics constructions where a large $\max(U)$ inflates the denominator of a log-ratio bound.
  • Because the exact verification is distributed through a link rather than included in full in the archive record, an independent reimplementation of formulas (2.3)--(2.6) at $(81411,65536,5)$ would settle Theorem 2.2 without relying on that link.
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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 / 3 minor

Summary. The paper presents a new lower bound θ ≥ 1.173050 for the sums and differences of sets problem, improving the previous bounds 1.14465 and AlphaEvolve's 1.1584. The construction generalizes [3] by restricting the coordinates of the integer vectors to x_k ≤ B and encoding them in base 2B+1. Section 2 derives closed formulas: (2.3) for |W(m,L,B)| by inclusion-exclusion, (2.4) for |U+U| = |W(m,2L,2B)|, (2.5) for |U−U| by separating positive and non-positive coordinates, and (2.6) for 2·max(U)+1 from the greedy maximal digit string. Theorem 2.2 reports that an exact integer computation for (m,L,B) = (81411, 65536, 5) gives θ = 1.173050, with only rounded decimal approximations and a bit.ly link to the code and exact values.

Significance. If the reported computation is correct, the result is a genuine improvement over the best known lower bound, and the paper demonstrates that the simple coordinate-bound idea of §1 is effective. The analytical content is transparent: formulas (2.3), (2.5), and (2.6) are explicitly stated, the construction is explicit, and the resulting set has more than 10^43546 elements. However, the central numerical verification is not auditable from the manuscript alone, since the theorem rests on a 15-hour computation whose exact output is not included and whose archive link lacks a checksum or version identifier. The mathematical derivation is sound as far as it goes, but the computer-assisted step needs to be made reproducible for the theorem to be fully verified.

major comments (2)
  1. [Theorem 2.2, proof] The proof of Theorem 2.2 depends entirely on the claimed 15-hour exact integer computation for (m,L,B) = (81411, 65536, 5). The manuscript prints only ten-significant-digit rounded approximations and refers to the bit.ly link [1] with no commit hash, checksum, or exact transcript. A reader cannot verify from the PDF alone that the exact values of s(U), d(U), and q(U) satisfy log(d/s)/log(q) ≥ 0.173050, and a last-digit error in any typed value or a mismatch in the linked artifact could shift θ below the claimed bound. Please include the exact integers (or a stable, versioned archive with checksums) and a short verification script, or give a certificate in an appendix.
  2. [Equation (2.4)] The displayed equality |U+U| = |W(k,2L,2B)| contains a typo (k should be m). More importantly, the equality is asserted with only the easy inclusion '⊆' explained; the reverse inclusion, i.e., that every vector in W(m,2L,2B) splits into two vectors in W(m,L,B), requires a short argument. Since this identity is used in the computation of s(U), it should be justified explicitly, for example by a splitting lemma for integer boxes with sum constraints.
minor comments (3)
  1. [Before (2.6)] The statement 'Since each coordinate is at most B, we may assume L ≤ mB' is correct but deserves a sentence: if L > mB, replacing L by mB does not change W(m,L,B) nor U. In formula (2.6), the case L = mB gives t = m and L%B = 0, making the middle term 2(L%B)(2B+1)^{m-t-1} formally contain a negative exponent; state explicitly that this term is zero in that boundary case.
  2. [References [1], [4]] The references to [1] and [4] are bit.ly short links with no version information. Please provide permanent archive links (for example DOI or arXiv) and, for [1], include a commit hash or checksum so that the exact computational data can be independently verified.
  3. [Equation (2.5)] In the paragraph preceding (2.5), 'the absolute value of the sum of the remaining non-positive coordinates' is phrased slightly confusingly; it means the sum of the absolute values of the non-positive coordinates, which is at most L. The formula itself is correct, but the wording could be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bound is an explicit computational construction whose only external input is the prior inequality from [3].

full rationale

The derivation chain is self-contained and non-circular. The paper defines the box-restricted set W(m,L,B), counts it via inclusion-exclusion in (2.3), counts the sumset in (2.4) as W(m,2L,2B), counts the difference set in (2.5) by independently choosing positive and non-positive coordinates, and computes q(U)=2 max(U)+1 in (2.6) by the greedy base-(2B+1) representation. These are combinatorial counting identities for the explicit construction U=U(m,L,B), not definitions of the target quantity. Theorem 2.2 uses the external inequality theta >= 1 + log(|U-U|/|U+U|)/log(2 max(U)+1) from Gyarmati--Hennecart--Ruzsa [3], then reports the result of a 15-hour exact integer computation for the single triple (m,L,B)=(81411,65536,5). The floating-point search for good parameters is a heuristic selection method, not a fitted prediction: no claim is made that the reported value is the optimal global value, and the lower bound only needs the explicitly computed U to satisfy the displayed inequality. There is no self-citation chain, no renaming of a known result, and no ansatz smuggled in via citation. The only legitimate concern is reproducibility: the exact integer values are not printed and the linked bit.ly artifact lacks a commit hash or checksum, so the numerical content of Theorem 2.2 cannot be independently audited from the PDF alone. That is a verification gap, not circularity, and does not raise the circularity score. The typo in (2.4) where k should be m is harmless because the surrounding text and example confirm the intended count is W(m,2L,2B).

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The construction has three explicit integer parameters selected by search rather than derived from constraints. The proof also leans on the prior lower-bound formula from [3], an unstated splitting lemma behind (2.4), and the correctness of large GMP computations.

free parameters (3)
  • m (number of coordinates) = 81411
    Chosen by floating-point search near 5L/4 for L = 65536 to maximize theta; no proof of global optimality.
  • L (coordinate sum bound) = 65536
    Fixed at 2^16; small searches with L = 64 and the observed m about 5L/4 heuristic motivated the choice.
  • B (coordinate bound) = 5
    Selected from the restricted search 1 <= B <= 7 where B = 5 gave the best theta at L = 64 and was then used for L = 65536.
assumptions (4)
  • domain assumption theta >= 1 + log(|U-U|/|U+U|)/log(2 max(U)+1) for any nonnegative set U containing 0
    Used in the proof of Theorem 2.2; cited from [3], not proved in this paper.
  • standard math Base (2B+1) representation with digits in the admissible ranges is unique, so g is injective on W+W and W-W
    Invoked before equations (2.4)-(2.6); a standard but unstated property of positional notation.
  • domain assumption Every vector in W(m,2L,2B) is the sum of two vectors in W(m,L,B)
    Needed for equality (2.4); true by an interval-sum argument but not shown in the text.
  • domain assumption The GMP program in [1] computes the enormous integer values exactly
    The numerical conclusion of Theorem 2.2 rests on this; no formal verification or certificate is provided.

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Cite this review

Pith. "Pith review of Sums and differences of sets (improvement over AlphaEvolve)." pith.science (2026). https://pith.science/paper/PYYVPZZO

@misc{pith2026250516105,
  author       = {Pith},
  title        = {Pith review of: Sums and differences of sets (improvement over AlphaEvolve)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PYYVPZZO}},
  note         = {Machine review of arXiv:2505.16105}
}
abstract

On May 14, 2025, DeepMind announced that AlphaEvolve, a large language model applied to a set of mathematical problems, had matched or exceeded the best known bounds on several problems. In the case of the sum and difference of sets problem, AlphaEvolve, using a set of $54265$ integers, improved the known lower bound of $\theta=1.14465$ to $\theta=1.1584$. In this paper, we present an improved bound $\theta=1.173050$ using an explicit construction of a U set that contains more than $10^{43546}$ elements. For fast integer and floating-point arithmetic, we used the (free) GMP library.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sums and differences of sets: a further improvement over AlphaEvolve

    math.CO 2025-06 conditional novelty 5.0 of 10

    A new family of U sets gives the best known lower bound theta >= 1.173077 for the sums and differences of finite integer sets.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [3]

    Gyarmati, F

    K. Gyarmati, F. Hennecart, and I. Z. Ruzsa (2007). Sums and d ifferences of fi- nite sets. Functiones et Approximatio Commentarii Mathematici, 37 (1):175–186, 2007. https://gyarmatikati.web.elte.hu/publ/sumdiffv.pdf

    work page 2007

  2. [4]

    Alexander Novikov, Ngˆ an Vu, Marvin Eisenberger, Emilien Dupont , Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehra- bian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Hollan d, Alex Davies, Se- bastian Nowozin, Pushmeet Kohli and Matej Balog (2025). AlphaEvo lve: A coding agent for scient...

    work page 2025

  3. [1]

    https://bit.ly/43mWCVC

    Sums and differences of sets. https://bit.ly/43mWCVC

    work page

  4. [2]

    https://gmplib.org/

    The GNU Multiple Precision Arithmetic Library. https://gmplib.org/

    work page

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