On commuting integer matrices

Akshat Mudgal; Jonathan Chapman

REVIEW 1 major objections 7 minor 20 references

On commuting integer matrices

T0 review · 1 major / 7 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that the count of commuting pairs of 3x3 integer matrices in [-N,N]^{3x3} has order N^10, and gives an asymptotic with explicit constant for 2x2 matrices.

desk verdict Sharp count for commuting 3x3 integer matrices and first asymptotic for 2x2; elementary, correct, and worth a serious referee. read the letter →

arxiv 2504.15839 v1 pith:JBXCHY7V submitted 2025-04-22 math.NT math.CO

Jonathan Chapman , Akshat Mudgal This is my paper · ORCID

classification math.NTmath.CO MSC 11D4515A2715B36

keywords commutingintegermatricesrestricteddivisorcorrelationsmatrixcommutatororderofmagnitude2x23x3p-adicdensityzetavalues

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

The paper proves that the number $C_3(N)$ of pairs of $3\times 3$ integer matrices with entries in $[-N,N]$ that commute is of exact order $N^{10}$, matching the trivial lower bound obtained by taking one matrix to be a scalar multiple of the identity. It also establishes the asymptotic $C_2(N)=K(2N)^5+O(N^4\log N)$ for $2\times 2$ matrices, with $K=10\zeta(2)/(3\zeta(3))$, an explicit constant. These results replace earlier upper bounds that carried extra powers of $N$ or logarithmic factors. The proofs are elementary, resting on new pointwise and moment estimates for restricted divisor correlations.

What carries the argument

The carrying object is the restricted divisor correlation $r_N(h)$, the number of $a_1,a_2,a_3,a_4\in[-N,N]$ with $a_1a_2-a_3a_4=h$, together with its moments $I_k(N)=\sum_h r_N(h)^k$. Lemma 1.5 gives the pointwise bound $r_N(h)\ll N^2\sum_{d\mid h,\,d\le N}1/d$ for $0<|h|\le 2N^2$ and the moment bound $I_k(N)\ll_k N^{2k+2}$. These bounds fix the number of admissible off-diagonal entry choices in the rank-$4$ case, while the rank stratification of the matrix $M$ in (4.3) controls how many diagonal choices remain.

What would settle it

Compute $I_3(N)=\sum_{|h|\le 2N^2} r_N(h)^3$ for increasing $N$ and check whether it stays within a constant of $N^8$; a single $N$ with $I_3(N)>C N^8$ for large $C$ would refute Lemma 1.5. A more refined check is to evaluate the pointwise bound (1.5) for $h$ of size roughly $2N^2$ by direct enumeration of the $O(N^4)$ quadruples at moderately large $N$.

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Extended reading notes

Core claim

The central claim is that the naive lower bound is sharp for $3\times 3$ commuting integer matrices: $N^{10}\ll C_3(N)\ll N^{10}$ for every positive integer $N$. For $2\times 2$ matrices the paper goes further and gives the first-order asymptotic $C_2(N)=K(2N)^5+O(N^4\log N)$ with $K=10\zeta(2)/(3\zeta(3))$. The mechanism is to count solutions to $AB=BA$ by first using three quadratic equations to fix the off-diagonal entries and then classifying by the rank of a $6\times 4$ matrix $M$ whose entries are those off-diagonal entries; the rank-$4$ case, which dominates, is controlled by the third moment estimate $I_3(N)=O(N^8)$. A separate local computation over $\mathbb{Z}/p^n\mathbb{Z}$ shows that the local densities multiply to a divergent product, which explains why the circle-method heuristic would predict spurious logarithmic factors.

Load-bearing premise

The load-bearing premise is the moment estimate $I_3(N)=O(N^8)$ for the correlation function $r_N(h)$; if that count of coincident $2\times 2$ determinants were larger, the rank-4 case in the proof of Theorem 1.1 would not fit in $N^{10}$.

Editorial extensions

If this is right

For $d=3$ the commutator variety's point count has the same order as its trivial subfamily, so the generic matrix pair contributes at most a bounded factor.
The $2\times 2$ asymptotic removes the implicit $N^\epsilon$ factor and supplies a numerically explicit leading constant $K\approx 4.56144$.
The $p$-adic density computation shows the local factors do not multiply to a finite constant, so the usual singular series heuristic fails for this problem; the main term instead comes from pairs where one matrix is a scalar multiple of the identity.
For arbitrary finite sets $A\subset\mathbb{R}$ with small doubling, $C_3(A)$ has order $|A|^{10}$ up to a factor depending only on the doubling constant $K$.

Reading between the lines

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

For $d\ge 4$ the same rank stratification cannot suffice on its own: the consistency example in the paper's final remarks shows a full-rank $M$ with empty solution set, so counting $C_d(N)$ will require an additional constraint that has no analogue for $d=3$.
A weighted analogue of $r_N(h)$, using smooth bump functions instead of sharp truncation, should satisfy an identical moment bound and would likely give the same $N^{10}$ order for smooth box-constrained matrices.
The explicit constant in the $2\times 2$ asymptotic suggests that higher-dimensional analogues, when they become available, will express the leading coefficient in terms of zeta values and totient sums.

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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

Tim, quick take on arXiv:2504.15839. The paper proves C_3(N) is within a constant factor of N^10, confirming the Browning–Sawin–Wang speculation for d=3, and gives C_2(N) = K(2N)^5 + O(N^4 log N) with K=10 ζ(2)/(3 ζ(3)). That second result is the first asymptotic for commuting 2x2 integer matrices. The main novelty is Lemma 1.5, a pointwise bound on the determinant correlation function r(h) and its moments, proved by elementary divisor sums. The rest of the paper is a careful rank-based decomposition of the commuting equations for 3x3 matrices, which is the right technical heart. I checked the key reductions: equation (2.6) follows from Lemmas 2.1–2.2, the M X = Y setup in Section 4 is sound, and the rank 2–4 cases are counted correctly using I_3(N) and the rank geometry. The constant in Theorem 1.2 rederives cleanly. The p-adic count (Theorem 1.3) is a nice extra and is consistent with the stated local density.

Soft spots are minor. There are typos: the statement of Lemma 2.1 has an unused v in max{|u|, |v|}; the organisation paragraph says Section 6 proves Lemma 1.5 when it means Lemma 1.4; and the proof of Lemma 1.5 has a stray N^{1+1/2} that should be N^{1+ε} or N^2. None of these affect the arguments. The 'completely elementary' claim in the abstract covers Theorems 1.1–1.3; Theorem 1.6 invokes Solymosi, so a reader might want a footnote to dodge the overstatement, but it is not a substantive issue. The d≥4 discussion is honest: the authors show with a concrete 4x4 example why the naive rank-based approach fails, which is useful context rather than a defect.

The citation pattern is healthy: they engage with [4], [17], and the divisor correlation literature, and they give credit where due. I found no circularity. The paper is sharp, self-contained in its main claims, and a genuine advance within analytic number theory. I would send it to a serious refereed venue. If you are in the area, you should cite it.

Referee Report

1 major / 7 minor

Summary. The paper studies the number C_d(N) of pairs of d×d integer matrices with entries in [-N,N] that commute. Theorem 1.1 proves the sharp order N^{10} for d=3, confirming a conjecture of Browning–Sawin–Wang. Theorem 1.2 gives an asymptotic formula C_2(N) = K(2N)^5 + O(N^4 log N) with an explicit constant K = 10ζ(2)/(3ζ(3)). Theorem 1.3 provides the analogous local count over Z/p^nZ, and Theorem 1.6 extends the d=3 upper bound to arbitrary finite sets A ⊂ R with small doubling. The proofs are elementary, based on new restricted divisor-correlation estimates (Lemma 1.5).

Significance. The main results are significant. The d=3 order-of-magnitude problem was open, and the d=2 asymptotic with explicit constant is new. The proof method is self-contained and avoids harmonic analysis; the pointwise and moment bounds for r(h) are of independent interest. The paper also gives a clean p-adic treatment and a nice application of Solymosi's sum-product bound. However, the lower-bound half of Theorem 1.6 is not proved in the manuscript.

major comments (1)

[Section 6.2 (proof of Theorem 1.6)] The proof of Theorem 1.6 as written establishes only the upper bound C_3(A) ≪ K^6 |A|^{10} (log(2|A|))^3. After Lemma 6.1, the argument splits into rank cases and bounds each case from above; there is no lower-bound construction for arbitrary A. The lower bound K^{-3}|A|^{10} ≪ C_3(A) is asserted in the theorem statement but is not obtained from the scalar-multiple construction used in the integer case, which requires 0 ∈ A. The authors should either supply a proof of this lower bound or modify the statement of Theorem 1.6 to include only the upper bound (and any lower bound that does follow from the given arguments).

minor comments (7)

[Lemma 2.1] The statement reads 'let u, w be coprime, non-zero integers' but should read 'let u, v be coprime, non-zero integers'.
[Lemma 2.2 proof] In the proof, 'b − b′ = |vz|' should read 'b − b′ = vz'.
[Proof of Lemma 1.5] The displayed line for r(0) ends with '16 ζ(2)N^2 log N'; the preceding calculation gives 16/ζ(2) as the coefficient. Since the lemma only needs the order N^2 log N, this typo does not affect the results.
[Proof of Lemma 1.5] The term 'N^{1+1/2}' appears to be a typographical artifact for N^{1+ε}; the subsequent N^2 bound is sufficient regardless.
[Section 4, after (4.1)] The text says the matrices lie in 'Mat2(Z,N)', but they are 3×3 matrices, so this should be 'Mat3(Z,N)'.
[Lemma 4.2] In the final display, the bound is written as '|S3| ≪ N^8 · N^2' but the lemma estimates |S4|, so the subscript should be 4.
[Section 6.3] The phrase 'the the first six rows' contains a duplicated article.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central estimates are proven in-paper and independent of the claims they support.

full rationale

The derivation chain is self-contained. Lemma 1.5, the load-bearing estimate, is proved elementarily in Section 3 from Lemma 3.1, an independent convolution bound that is itself proved from first principles; the proof of the pointwise bound r(h) << N^2 sum_{d|h,d<=N} 1/d fixes a1,a3 and counts a2,a4 by a congruence, and the moment bound I_k(N) << N^{2k+2} is deduced from this pointwise bound and Lemma 3.1, not assumed. Section 4 then uses only these estimates: the rank-4 case is bounded by I_3(N)=O(N^8) times O(N^2) diagonal choices, the rank-3 case by r(0)^3=O(N^6 log^3 N) times O(N^3), and the rank-2 cases by direct counting of parallel direction vectors. Theorem 1.2 is obtained by exact counting in Section 2 using Lemmas 2.1 and 2.2 together with standard Euler-totient estimates; no fitted parameter is later renamed as a prediction. Theorem 1.3 is likewise proved by an independent p-adic lifting argument (Lemma 5.1). The only self-citation, to the second author's [17], appears in introductory context or in the arbitrary-set Theorem 1.6, whose proof invokes Solymosi's external sum-product bound [20] and the paper's own Section 4 scheme; it is not load-bearing for Theorems 1.1 or 1.2. No equation in the paper reduces to its own conclusion by construction, and the paper explicitly marks the d>=4 discussion as illustrative rather than a claimed theorem.

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

No free parameters or invented entities appear. Theorems 1.1 and 1.2 depend only on standard analytic number theory inputs and on lemmas proved in the paper. Theorem 1.6 additionally imports Solymosi's sum-product estimate.

assumptions (3)

standard math Euler totient summatory estimates: sum_{u<=N} phi(u)/u^3 = zeta(2)/zeta(3) - 1 + O(1/N) and sum_{u<=N} phi(u)/u^2 = log N / zeta(2) + O(1).
Used in the proof of Theorem 1.2, around equation (2.9), to evaluate the main constant K.
standard math Divisor function bound tau(n) <<_epsilon n^epsilon.
Used in the proof of Lemma 1.5 to dispose of solutions with a zero entry or with equal off-diagonal entries.
standard math Solymosi's sum-product bound: for a finite set A with |A+A| = K|A|, the multiplicative energy r_A(0) is O(K^2 |A|^2 log(2|A|)).
External theorem from reference [20], used only for Theorem 1.6 and not for Theorems 1.1, 1.2, or 1.3.

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Pith. "Pith review of On commuting integer matrices." pith.science (2026). https://pith.science/paper/JBXCHY7V

@misc{pith2026250415839,
  author       = {Pith},
  title        = {Pith review of: On commuting integer matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBXCHY7V}},
  note         = {Machine review of arXiv:2504.15839}
}

abstract

Given $d, N \in \mathbb{N}$, we define $\mathfrak{C}_d(N)$ to be the number of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N] \cap \mathbb{Z}$ such that $AB = BA$. We prove that $$ N^{10} \ll \mathfrak{C}_3(N) \ll N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further establish that $$ \mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity.

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Reference graph

Works this paper leans on

20 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [4]

Browning, W

T. Browning, W. Sawin, V. Y. Wang, Pairs of commuting integer matrices , arXiv:2409.01920

arXiv

[2] [17]

Mudgal, On commuting pairs in arbitrary sets of 2 × 2 matrices, arXiv:2411.10404

A. Mudgal, On commuting pairs in arbitrary sets of 2 × 2 matrices, arXiv:2411.10404

arXiv

[3] [1]

Aﬁfurrahman, A uniform formula on the number of integer matrices with give n determinant and height, arXiv:2407.08191

M. Aﬁfurrahman, A uniform formula on the number of integer matrices with give n determinant and height, arXiv:2407.08191

arXiv

[4] [2]

Apostol, Introduction to analytic number theory , Undergraduate Texts in Mathematics, Springer- Verlag, New York-Heidelberg, 1976

T. Apostol, Introduction to analytic number theory , Undergraduate Texts in Mathematics, Springer- Verlag, New York-Heidelberg, 1976

work page 1976

[5] [3]

Basili, On the irreducibility of varieties of commuting matrices , J

R. Basili, On the irreducibility of varieties of commuting matrices , J. Pure Appl. Algebra 149 (2000), no. 2, 107–120

work page 2000

[6] [5]

W. Duke, Z. Rudnick, P. Sarnak, Density of integer points on aﬃne homogeneous varieties , Duke Math. J. 71 (1993) 143–179

work page 1993

[7] [6]

Erd˝ os, P

P. Erd˝ os, P. Tur´ an,On some problems of a statistical group-theory. IV , Acta Math. Acad. Sci. Hungar. 19 (1968), 413–435

1968

[8] [7]

W. Feit, N. J. Fine, Pairs of commuting matrices over a ﬁnite ﬁeld , Duke Math. J. 27 (1960), 91-94

1960