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On the packing dimension of unions and extensions of $k$-planes

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Pith's one-line read The paper shows that extending every hyperplane that meets a set in full Hausdorff dimension to the whole hyperplane does not increase packing dimension, and proves an improved lower bound for unions of such hyperplane slices.

desk verdict Genuinely new packing-dimension results for unions and extensions of k-planes, anchored by a useful effective-dimension framework on Grassmannians; the hyperplane theorems rest on an under-verified Proposition 27 that needs a full proof. read the letter →

arxiv 2508.18257 v1 pith:FHHMUZ3N submitted 2025-08-25 math.CA math.LO

Jacob B. Fiedler This is my paper · ORCID
classification math.CAmath.LO MSC 28A7828A8068Q30
keywords packingdimensionHausdorffk-planesaffineGrassmannianeffectiveKolmogorovcomplexityFurstenberg-typesetshyperplaneextensions
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

Fractal size can be measured in several ways; packing dimension records the largest scales at which a set looks big, in contrast to Hausdorff dimension, which records the smallest scales. The paper asks how large a union of subsets of $k$-planes must be when the family of planes has packing dimension $t$ and each subset has dimension at least $s$, and also how much packing dimension can grow when such subsets are fattened to entire planes. Using a new effective dimension on the spaces of $k$-planes, it proves bounds for general $k$ and two stronger hyperplane results: equality under full-dimensional hyperplane extension, and an improved union bound for full-dimensional hyperplane slices.

What carries the argument

The machinery is effective dimension defined on the Grassmannian $G(n,k)$ and the affine Grassmannian $A(n,k)$, meaning Kolmogorov complexity at precision $r$ of rational orthogonal-projection matrices that approximate a plane. A point-to-set principle on these computable metric spaces converts per-point complexity bounds into classical packing-dimension bounds. The load-bearing pieces are Lemma 22, which shows a $k$-plane is computable from approximations of $k+1$ points on it; Corollary 23, which gives the orthogonal complement; Lemma 24, which bounds the complexity of points on a known plane by $kr+O(\log r)$; and Lemma 30, which reduces the hyperplane intersection estimate to a two-dimensional statement.

What would settle it

Exhibit, for some $n$ and $0<t<n$, a family $\mathcal{P}\subseteq A(n,n-1)$ with packing dimension $t$ and full Hausdorff-dimensional subsets whose union $F$ satisfies $\dim_P(F)<n-1+\frac{nt}{(n-1)t+n}$; such a construction would refute Theorem 6. Alternatively, a set $E$ whose full-hyperplane-slice extension has $\dim_P(F)>\dim_P(E)$ would refute Theorem 5.

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Extended reading notes

Core claim

The paper's central claim is that a single pointwise algorithmic inequality drives both hyperplane theorems. For a hyperplane written as $(x_1,\dots,x_{n-1})\mapsto a\cdot x+b$, if $x$ has effective dimension at least $n-1-\varepsilon/4$ relative to $(a,b)$, then for every precision $r$, $$K^A_r(x,a\cdot x+b)\ge (n-1)r+K^A_{c_r}(a,b)+(r-c_r)-C\sqrt{\varepsilon r},$$ where $c_r$ is the precision that maximizes $K^A_t(a,b)-t$ over $t\le r$. Feeding this inequality into the point-to-set principle yields the extension equality $\dim_P(F)=\dim_P(E)$ and the improved hyperplane union bound $\dim_P(F)\ge n-1+\frac{nt}{(n-1)t+n}$.

Load-bearing premise

The strongest hyperplane theorems rest on Lemma 29, a stated-but-unproved adaptation of a two-dimensional lemma; the author says the proof is essentially the same and omits it, so the main results' support depends on that adaptation being valid.

Editorial extensions

If this is right

  • If Theorem 5 is correct, the packing dimension of a set is unchanged when all hyperplanes meeting it in Hausdorff dimension $n-1$ are extended to full hyperplanes, so full-dimensional slices cannot inflate packing dimension.
  • If Theorem 6 is correct, the general union bound $n-1+t/n$ for hyperplanes is improved to $n-1+\frac{nt}{(n-1)t+n}$ for every $n$ and every $0<t<n$.
  • For general $k$, the Hausdorff-slice and packing-slice versions of the union problem have different bounds, reflecting that the scales on which the family of planes is large need not align with the scales on which the slices are large.
  • For $k$-planes, extending all positive-measure slices cannot push packing dimension above $2\dim_P(E)-k$; for lines, the same holds even when only full Hausdorff-dimensional slices are extended.

Reading between the lines

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

  • Beyond the paper's claims, the same effective-dimension machinery should apply to other families of geometric objects with computable rational approximations, such as spheres or algebraic submanifolds, yielding analogous union bounds.
  • The paper leaves open whether the gap between its packing-dimension bound and the known Hausdorff-dimension bound for hyperplane unions is an artifact of the proof; constructing sharp examples for the packing case would settle this.
  • The proof's use of oracle-dependent precision plateaus suggests that the improvement in Theorem 6 comes from exploiting periods where the complexity of the parameter $(a,b)$ grows slowly; varying the oracle construction may yield further refinements.
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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

3 major / 4 minor

Summary. The paper develops tools from algorithmic information theory to bound the packing dimension of unions of subsets of k-planes and of extensions of such subsets to full k-planes. It introduces effective dimension on the Grassmannian and affine Grassmannian, proves point-to-set principles and a symmetry-of-information result in that setting, and uses these to obtain Furstenberg-type bounds (Theorems 1 and 2), extension bounds (Theorem 3 and Proposition 4), and sharper hyperplane results (Theorems 5 and 6). The central new claims are an analog of Héra's Hausdorff-dimension bound for packing dimension, a generalization of Fraser's line result, a hyperplane extension equality, and a packing-dimension bound for unions of full-dimension subsets of hyperplanes that improves the general bound in the case k=n-1.

Significance. If correct, the paper would make a real contribution: Theorem 1 gives a packing-dimension analog of Héra's bound, Theorem 2 generalizes Fraser's line result to k-planes, and the Grassmannian effective-dimension machinery in Section 3 is a genuinely new and reusable tool. Theorems 5 and 6 are the headline results for hyperplanes, and Theorem 6's bound n-1+nt/((n-1)t+n) improves the general bound for all n and 0<t<n. The development is mostly careful and self-contained for Sections 3-5. However, the hyperplane results rest on Proposition 27, whose proof is supported by an unproved lemma (Lemma 29), a sign error in a displayed inequality, and an incorrect statement about projections in the proof of Lemma 30. These issues are load-bearing but appear repairable, so the paper merits revision rather than rejection.

major comments (3)
  1. [§6.2, Lemma 29] Lemma 29 is stated without proof; the text explicitly says 'This is just Lemma 6 in [25] modified for hyperplanes. Since the proof of this lemma is essentially the same as the proof in [25]; we omit it.' This lemma is the enumeration step on which Proposition 27, and therefore Theorems 5 and 6, rest. The passage from lines to hyperplanes is not purely notational: the constraint u·x+v=a·x+b couples the parameters (u,v) to x, and the proof must control both the precision t=−log|(a,b)−(u,v)| and the family of hyperplanes through the point (x,a·x+b). Please include a complete proof, or a precise step-by-step reduction to [25, Lemma 6], rather than an omission.
  2. [§6.2, proof of Proposition 27] After applying Lemma 30 and (32), the manuscript asserts K^{A,D}_r(u,v) ≥ K^A_r(a,b)+(1−√ε)(r−t)+K^{A,a,b}_{r−t}(x)−(n−2)(r−t)−O(log r). This sign is inconsistent with (32), which gives K^{A,D}_t(a,b) ≥ K^{A,D}_r(a,b)−(1−√ε)(r−t). The displayed inequality should contain a minus sign in front of (1−√ε)(r−t). As written, the subsequent deduction that condition (2) of Lemma 29 holds is not supported. Please correct the sign and re-verify the lower bound.
  3. [§6.2, proof of Lemma 30] The proof of (29) uses the claim 'Projection onto V is the composition of projection onto H1 and projection onto H2.' This is false in general: for two hyperplanes in R^n, the composition of the two projection operators has rank n−1 (not n−2) and does not agree with the orthogonal projection onto their intersection even approximately, as can be seen already in R^3 for H1=span(e1,e2) and H2=span(e1,e2+e3). Consequently the estimate ρ(V, ¯V) ≤ O(2^{-r}) is not justified. This step is essential for (29). Please replace it with a correct computation of the intersection projection from the hyperplane data (for example, by solving the two linear equations defining the hyperplanes) and re-derive the error bound.
minor comments (4)
  1. [§3.2, Lemma 14] Condition (1) should read 1/2 ≤ |q_i| ≤ 2 rather than |q|.
  2. [§6.2, proof of Lemma 30] The notation K^A_{r,t}(u,v|u,v) in the first displayed equation of the proof appears to be a typo; as written it is unclear what conditioning on (u,v) at precision t means in the self-referential identity. Please clarify or correct.
  3. [§6.1, proof of Theorem 5] The pointwise inequality (20) is asserted for every r sufficiently large; the passage from the two displayed facts to n(r−c_r)+K^A_{c_r}(x,a,b) ≥ K^A_{r,t}(x, a·x+b) omits an O(log r) term. Please make the error terms explicit so that the claimed constant C is independent of x.
  4. [§1.1] In the sentence 'This is a packing dimension analog of a result of Héra [15], who proved the same bound but with both instances of packing dimension replaced by Hausdorff dimension,' the word 'both' is inaccurate because the bound involves packing dimension only in the hypothesis on the set of planes and the conclusion on the union; please rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bounds are derived from point-to-set principles and algorithmic lemmas; self-citations are to prior published work and are not used as the target conclusion itself.

full rationale

The derivation chain is not circular in the sense of reducing a claimed prediction to its own inputs. Theorems 1 and 2 are proved directly from the point-to-set principle, Lemma 22, symmetry of information, and dimension estimates; none of these assume the Furstenberg-type bound being established. Theorems 3 and 4 similarly reduce the extension problem to pointwise Kolmogorov complexity inequalities derived from Fubini and randomness, not from the target dimension identity. The hyperplane results (Theorems 5 and 6) do rest on Proposition 27, but Proposition 27 is proved in Section 6.2 from Lemmas 28-30, with Lemma 28 quoted from [25]/[34] and Lemma 30 reduced to Lemma 7 of [25]; these are external published results, not restatements of Theorems 5 or 6. The self-citations to [5] are to a published prior paper on the planar case; the paper explicitly notes the n>2 content and does not cite [5] as the sole justification for the main theorem. The unproved Lemma 29 statement ('This is just Lemma 6 in [25] modified for hyperplanes. Since the proof of this lemma is essentially the same as the proof in [25]; we omit it.') is a missing proof, and the application of (32) after Lemma 30 appears to have a sign discrepancy (the displayed lower bound has a plus where substituting (32) yields a minus); these are correctness risks, not circular reductions. No fitted parameter is relabeled as a prediction, and no effective-dimension definition assumes the classical dimension it is used to prove.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard algorithmic information theory results plus several lemmas imported from prior papers. The largest unpaid-for assumption is Lemma 29, whose proof is omitted. Lemma 28 is also used without proof but is attributed to prior work. No free parameters or invented entities appear: the constants in the bounds are explicit in terms of n, k, s, and t, and all epsilons tend to zero.

assumptions (6)
  • standard math Point-to-set principle (Lutz and Lutz): dim_H(E)=min_A sup_{x in E} dim^A(x), dim_P(E)=min_A sup_{x in E} Dim^A(x).
    Used in Sections 2, 4, and 5; extended to the Grassmannian in Propositions 13 and 17 using [23].
  • standard math Symmetry of information at precision r (Lemma 7 of [25]) and its Grassmannian analog (Proposition 18).
    Used throughout Sections 4 and 6 to split complexity of tuples; the Euclidean version is a known theorem and the Grassmannian version is proved in Section 3.3.
  • standard math Case and Lutz bound on complexity growth over precision intervals (Proposition 8).
    Used to shift precisions with logarithmic error; proved in [6].
  • standard math Oracle flattening lemma (Lemma 28), stated in the form used in [34].
    Used in the proof of Theorem 6 to control the complexity of (a,b) at chosen precisions; cited to [25] and [34] without proof.
  • domain assumption Enumeration lemma (Lemma 29), adapted from Lemma 6 of [25] to hyperplanes.
    The proof is omitted in this paper with the note that it is essentially the same as in [25]; the statement is load-bearing for Proposition 27.
  • standard math Two-dimensional line intersection lemma (Lemma 7 of [25]).
    Used in the proof of Lemma 30 to reduce the high-dimensional intersection bound to R^2.

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Pith. "Pith review of On the packing dimension of unions and extensions of $k$-planes." pith.science (2026). https://pith.science/paper/FHHMUZ3N

@misc{pith2026250818257,
  author       = {Pith},
  title        = {Pith review of: On the packing dimension of unions and extensions of $k$-planes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FHHMUZ3N}},
  note         = {Machine review of arXiv:2508.18257}
}
abstract

We study the packing dimension of unions of subsets of $k$-planes in $\mathbb{R}^n$ using tools from algorithmic information theory, obtaining an analog of a result of H\'era and a mild generalization of a recent result of Fraser. Along the way, we introduce a notion of effective dimension on the Grassmannian and affine Grassmannian, and we establish several useful algorithmic and geometric tools in this setting. Additionally, we consider how the packing dimension of the union of certain subsets of $k$-planes changes when the subsets are extended to the entire $k$-plane. Finally, we improve the above bounds for unions and extensions in the special case that $k=n-1$.

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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

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