REVIEW 5 major objections 4 minor 9 references
Exact Hausdorff dimension of some sofic self-affine fractals
T0 review · 5 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper claims that for certain sofic self-affine fractals, the Hausdorff dimension solves an explicit infinite-degree equation.
desk verdict Novel tower-decomposition approach for sofic self-affine fractals, but the worked examples contain algebraic errors that undermine the paper's main claim; needs major revision. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The tower-decomposition: insert a distinguished string s whose matrix product has 1-dimensional image, record the scalar factor J_u multiplying a fixed left vector v^top for every possible placement of that string, and group the matrix-product sum by the position of the last such factor. This converts the original sum into iteration of an infinite matrix M (a shift plus a return map), whose spectral radius gives the dimension. In $R^{3}$ the analogous operator acts on the direct sum indexed by the rooted tree Gamma of finite words over non-removable indices, with return weights C_s(u), and the dimension becomes log_{m1} of the spectral radius of the product operator.
What would settle it
For one of the paper's own $R^{3}$ examples, such as Example 4.3 or 4.4, compute both sides of Claim 3.7 for increasing finite N -- the tower sum sum_s ||Phi_N(s)||$_1^{{a_1}}$ and the original double sum in equation (1.1) -- and compare their exponential growth rates; if they differ, the reduction fails and the claimed dimension does not follow.
Extended reading notes
Core claim
The central discovery is that the matrix-product limit for the Hausdorff dimension can be collapsed, under suitable degeneracy conditions on the adjacency matrices, into iteration of a single linear operator. In the planar case, if some string of matrices has 1-dimensional image, the dimension is log_{m1} r where r is the unique positive solution of r^L = C_0 + C_1/r + C_2/$r^{2}$ + ... with explicitly given non-negative constants C_k. In $R^{3}$, when the sofic set has a recursive structure (for every first-level symbol s there is a second-level symbol t with A_{(s,t)} having image spanned by a fixed vector), the same reduction works with an operator on a tree-indexed space, and the dimension is again log_{m1} of a spectral radius. The author's three-dimensional examples are the first non-trivial exact Hausdorff dimension calculations for sofic sets in $R^{3}$.
Load-bearing premise
The proof of the three-dimensional reduction (Claim 3.7) assumes that the $\ell^1$ norms of the tower vectors grow at exactly the same exponential rate as the original matrix-product sums, and the justification for one inequality step in that claim is not fully spelled out.
Editorial extensions
If this is right
- For planar sofic sets satisfying the 1-dimensional-image condition, the Hausdorff dimension is exactly log_{m1} r with r the unique positive root of the explicit infinite-degree equation.
- For R^3 sofic sets with recursive structure and primitive sum matrix, the dimension is the exponential growth rate of ||M_{s_N} ... M_{s_1} Phi_0||_1^{a_1} summed over first-level strings, computable as a spectral radius.
- When a removable index exists, the R^3 dimension satisfies a one-variable infinite-degree equation r = b_0 + b_1/r + b_2/r^2 + ..., as in Proposition 3.8.
- The worked examples give explicit numerical dimensions: log_2 3.1201... = 1.6416... and log_3 6.4693... = 1.6994... in the plane; log_2 2.2894... = 1.1950... and log_2 4.673... = 2.224... in R^3.
Reading between the lines
- If the tower-decomposition extends to higher dimensions, the same tree-indexed operator formalism could yield explicit dimension equations for sofic self-affine sponges in R^d whenever a recursive-structure analogue holds.
- The infinite-degree equation formulation suggests a numerical scheme: truncate the operator to finite submatrices and apply Perron-Frobenius to approximate r, yielding certified bounds on the dimension.
- The spectral-radius viewpoint may allow perturbation estimates: small changes to the adjacency matrices should change r continuously, giving stability statements for the dimension without recomputing the full limit.
- The success under the 1-dimensional-image condition hints that other degenerate conditions, such as rank-one factors appearing in longer products, might also collapse the matrix-product limit to a single operator.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a 'tower-decomposition' technique for computing the Hausdorff dimension of sofic affine-invariant sets. For planar sofic sets whose summed adjacency matrix is primitive and which contain a block with one-dimensional image, it claims the dimension is log_{m1} r where r solves an explicit infinite-degree equation r^L = C_0 + C_1/r + C_2/r^2 + ... (§3.1, Theorem 3.2). For sofic sets in R^3 with a recursive structure, it claims a reduction to products of tree-indexed operators and, under an l1-increasing hypothesis, a similar explicit equation (§3.2, Theorem 3.6 and Proposition 3.8). Four worked examples in §4 are presented, including a claimed first non-trivial exact dimension calculation for a sofic set in R^3. The conceptual framework is attractive and the derivation is parameter-free, but the worked examples contain algebraic errors and one key proof step in Claim 3.7 is incomplete.
Significance. If the main theorems are correct, the tower-decomposition method would be a valuable extension of Kenyon and Peres' work, turning a generally intractable limit of matrix products into the spectral radius of an explicit operator. The coefficients are derived from the adjacency matrices rather than fitted, and I found no circularity. However, the advertised contribution is substantiated only through the four examples in §4, and at least three of them contain algebraic errors that change the displayed infinite-degree equations and numerical values. The R^3 claim therefore is not supported as written. The paper's core idea is worth pursuing, but the manuscript needs substantial correction and re-verification before the results can be accepted.
major comments (5)
- [Section 4.1, Example 4.1] The displayed identity for the planar example is false. For N=1 and k=0, the string is (0), and direct multiplication using the displayed matrices gives (1,1,1) A0^2 A1 = (4,4,4) = 4(1,1,1). The displayed formula gives C_{1,0} = 2^2((2+2√2)-(2-2√2)) = 16√2, which is not the scalar 4. Since the coefficients in the infinite-degree equation for r are exactly these scalars raised to log_3 2, the stated value r = 3.1201... is not derived from the displayed computation.
- [Section 4.2, Example 4.2] The formula (1,0) A1^k A2^{N-k} A0 = ((3^k 5^{N-k}-1)/2)(1,0) is incorrect. For k=0, N=1, direct computation gives (1,0) A2 A0 = (7,0), not (2,0). Diagonalizing A1 and A2 yields (1,0) A1^k A2^{N-k} A0 = ((3^{k+1}5^{N-k}-1)/2)(1,0). This missing factor of 3 changes the coefficients C_k and hence the numerical value r = 6.4693... claimed in the example.
- [Section 4.3, Example 4.3] The coefficient b_2 is miscomputed. From the displayed matrix M1, M1 Φ0 = e1 and M1^2 Φ0 = e_∅ + e2, so ||M1^2 Φ0||_1 = 2 and therefore b_2 = 2^{log_3 2}, not √2. The expansion displayed in the example, 1 + 1/r + √2/r^2 + ..., is accordingly inconsistent with the definition of b_k and with the stated value r = 2.2894... for the R^3 example.
- [Section 3.2, proof of Claim 3.7] The final inequality in the proof of Claim 3.7 is not justified as written. The sum over W_2^{N+d}(s) and Q(0) is bounded by a constant times a sum over strings of length N+3d+1, but the displayed expression uses Φ_N(s) rather than Φ_{N+3d+1}(s), and the reindexing of the inserted factors A^d and A_p into the tower decomposition is not supplied. Since this estimate is the bridge between Theorem 2.2 and the products M_{u_N}...M_{u_1}, it is load-bearing and needs a complete proof.
- [Section 4.3, Section 4.4, and Proposition 3.8] Examples 4.3 and 4.4 invoke Proposition 3.8, but hypothesis (3) of that proposition is not verified. The required string (t_1,...,t_L) is never identified, and the footnote 'This is easily satisfied in most cases' is not a proof of the l1-increasing property for the specific operators M_u. Because the b_k formulas in these examples are derived from Proposition 3.8, this missing verification affects the validity of the R^3 dimension calculations.
minor comments (4)
- [Section 2, Definition 2.1] The definition of a sofic system says 'an additional condition on G is required, which we omit here.' This omitted condition should be stated explicitly or cited precisely, since the paper's main objects are sofic systems.
- [Section 3.2, proof of Claim 3.7] In the last displayed inequality of the proof, the tower vector is written as Φ_N(s) although the summation is over I_1^{N+3d+1}; this should be Φ_{N+3d+1}(s), and the surrounding text should be corrected accordingly.
- [Section 4.3, Example 4.3] The example defines b_k only for 'natural number k' but then uses b_0 = 1 in the expansion; the definition should explicitly include k=0.
- [Introduction, first R^3 example] The introductory statement of the R^3 example uses an operator notation with b_k defined by an expression containing nested norms; the later §4.3 definition is clearer, but the two displays should be reconciled to avoid confusion.
Circularity Check
No significant circularity: the derived dimension equations are explicit reductions of the matrix-product formula, and the numerical values are solved from those equations rather than fitted.
full rationale
The paper's derivation chain is self-contained with respect to its central claim. The starting dimension formula in Theorem 2.2 is attributed to Barral-Feng [BF12] and Feng [Fe24], and the own-paper citation [Ali24] is offered only as an alternative proof route after the fact; that self-citation is not load-bearing. In Theorems 3.2, 3.6, and Proposition 3.8, the coefficients C_k, C_s(u), c(s), and b_k are defined by explicit matrix-vector products using the given adjacency matrices and the fixed vector v spanning the one-dimensional image. Nothing is fitted to the target Hausdorff dimension: the unknowns r are defined as the unique positive solutions of the displayed infinite-degree equations and then computed numerically. The operator M and its spectral radius are derived from the same coefficients, so the spectral-radius formulation is a reorganization of the same equation, not a separate empirical input. Consequently, no step reduces a predicted quantity to a fitted parameter or to a self-citation. The paper itself notes that the three-dimensional tower technique is a 'fortunate discovery' without a deeper underlying picture, and the proof of Claim 3.7 compresses a substantial inequality chain; these are limitations in explanatory depth and proof completeness, but they do not make the theorem's conclusion an input of its derivation. The algebraic errors alleged in Section 4, if present, would undermine the correctness of the computations, but they do not constitute circularity under the definitions used here.
Assumptions & free parameters
assumptions (4)
- domain assumption Dimension formula of Theorem 2.2 (from [BF12], [Fe24]) is assumed as a black box.
- domain assumption The sum matrix A = Σ A_i (planar) or A = Σ A_w (R^3) is primitive.
- domain assumption Recursive structure in R^3: for every s ∈ I1 there exists t ∈ I2 with the row space of A(s,t) spanned by a fixed vector v.
- ad hoc to paper The l1-increasing condition in Proposition 3.8: M_u M_{t_L}...M_{t_1} is increasing with respect to the l1-norm.
Cite this review
Pith. "Pith review of Exact Hausdorff dimension of some sofic self-affine fractals." pith.science (2026). https://pith.science/paper/6JETH3WQ
@misc{pith2026241205805,
author = {Pith},
title = {Pith review of: Exact Hausdorff dimension of some sofic self-affine fractals},
year = {2026},
howpublished = {\url{https://pith.science/paper/6JETH3WQ}},
note = {Machine review of arXiv:2412.05805}
}
abstract
Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial cases, we demonstrate that the dimension can in fact be calculated explicitly. Specifically, the dimension is expressed as the solution to an infinite-degree equation with explicit coefficients, which also corresponds to the spectral radius of a certain linear operator. Our result provides the first non-trivial calculation of the exact Hausdorff dimension of sofic sets in $\mathbb{R}^3$. This is achieved by developing a new technique inspired by the work of Kenyon and Peres (1998).
Reference graph
Works this paper leans on
-
[1]
Alibabaei, Weighted topological pressure revisited, Ergod
N. Alibabaei, Weighted topological pressure revisited, Ergod. Th. & Dynam. Sys. , 45 (2025), 34-70
work page 2025
-
[2]
J. Barral and D. J. Feng, Weighted thermodynamic formalism on subshifts and applications, Asian J. Math. 16 (2012), 319–352
work page 2012
- [3]
-
[4]
On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets
Z. Feng, On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets, arXiv:2405.03213
-
[5]
D.-J. Feng, W. Huang, Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016), 411-452
work page 2016
- [6]
- [7]
-
[8]
E. Olivier, Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus. Ergod. Th. & Dynam. Sys. 30 (2010), 1503-1528
work page 2010
Show all 9 references
-
[9]
B Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973), 462-474
1973
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.