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Universal Sets for Projections

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Pith's one-line read This paper proves that for AD-regular planar sets there is a universal set of directions of lower box dimension zero, and for weakly regular sets there is one of arbitrarily small lower box dimension.

desk verdict Genuinely new constructions and a sharp gap in the weakly-regular theorem; the paper earns a referee but Theorem 3 needs a repaired proof. read the letter →

arxiv 2411.16001 v2 pith:CKI6C6R5 submitted 2024-11-24 math.CA math.LO

Jacob B. Fiedler , D. M. Stull This is my paper · ORCID
classification math.CAmath.LO MSC 28A7828A8068Q30
keywords projectiontheoremsuniversalsetsofdirectionsHausdorffdimensionboxAD-regularweaklyregulareffectiveKolmogorovcomplexity
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

Classical projection theorems guarantee that for an analytic plane set, projections in almost every direction have maximal Hausdorff dimension. This paper asks the converse: how small a set of directions can be while still containing, for every set in a class, at least one direction whose projection is maximal. The answer depends on regularity: for AD-regular sets (roughly, sets that look the same size at every scale) a single direction set of lower box dimension $0$ is universal, and for weakly regular sets (those whose Hausdorff and packing dimensions agree) a direction set of lower box dimension $\varepsilon$ is universal for every $\varepsilon > 0$. The proofs translate the geometry into statements about the Kolmogorov complexity of projected points, using the point-to-set principle and direction sets built by zeroing selected bits of algorithmically random angles.

What carries the argument

The load-bearing construction is a family of direction sets $D_s \subset S^1$ ($0 \le s < 1$) defined from algorithmically random angles $\theta$ by setting blocks of binary digits to zero. The counting scale $r_n$ grows extremely fast, with $r_{n+1} = 2^{2^{r_n}}$, and bits in the blocks $(r_n, n r_n]$ (for $D_0$) or $(r_n, r_n/s]$ (for $D_s$) are replaced by zeros; this gives lower box dimension $0$ or $s$ while retaining high effective complexity on all sufficiently long precision intervals. The proofs then rest on 'teal/yellow' interval lemmas (Lemmas 11–13), which bound the conditional complexity $K(x \mid p_e x, e)$ on intervals where the direction $e$ has uniformly high complexity and the point $x$ has either low ('teal') or high ('yellow') complexity growth; these bounds let the argument sum pointwise complexity over a partition of $[1, r]$. The point-to-set principle converts the resulting pointwise statements into the classical Hausdorff-dimension theorems.

What would settle it

Complete the omitted proofs of Lemmas 12 and 13 and explicitly sum their error terms over the $M \approx r/r_n$ intervals of Proposition 20 in the case $r_n \le r < (1/\varepsilon^2)\lfloor r_n/s \rfloor$; if the total error is not $o(r)$, then Theorem 3's pointwise estimate for weakly regular sets fails.

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Extended reading notes

Core claim

The central claim is that strong regularity shrinks the dimension of a universal direction set to zero, while weak regularity only forces an arbitrarily small positive dimension. Specifically, the paper constructs a set $D_0 \subseteq S^1$ of lower box dimension zero that is universal for the class of AD-regular sets, and, for every $\varepsilon > 0$, a set $D_\varepsilon \subseteq S^1$ of lower box dimension $\varepsilon$ that is universal for the class of weakly regular sets, those whose Hausdorff and packing dimensions coincide. The same construction, with the same $D_0$, yields a Bourgain-universal set for analytic sets, meaning every analytic $E$ has a direction $e \in D_0$ with $\dim_H(p_e E) \ge \dim_H(E)/2$. The paper also proves an exceptional set estimate for sets with optimal oracles (a broad class that includes analytic, weakly regular, and metric-outer-measure sets): for $0 < s \le \min\{\dim_H(E), 1\}$, the set of directions whose projection of $E$ has dimension $<s$ has Hausdorff dimension at most $s$.

Load-bearing premise

The load-bearing premise is that the omitted proofs of Lemmas 12 and 13 are correct and that the error accumulated across the partition in Proposition 20 is $o(r)$; if either fails, the weakly regular universal set theorem (Theorem 3) is unsupported.

Editorial extensions

If this is right

  • For AD-regular planar sets, a single fixed zero-dimensional direction set is universal: every AD-regular $E$ has some $e \in D_0$ with $\dim_H(p_e E) = \min\{\dim_H(E), 1\}$.
  • For weakly regular sets, every $\varepsilon > 0$ admits a universal direction set of lower box dimension $\varepsilon$, so weakening AD-regularity to weak regularity costs at most an arbitrarily small positive dimension.
  • The exceptional set estimate for optimal-oracle sets ($\dim_H\{e : \dim_H(p_e E) < s\} \le s$) recovers and extends the classical exceptional set estimate, giving universal direction sets for any essentially $s$-dimensional set of directions.
  • There exists a zero-dimensional set $D_0$ that is Bourgain universal for analytic sets: every analytic $E$ has a direction with projection dimension at least half its own dimension.

Reading between the lines

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

  • If Lemmas 12 and 13 hold, the bit-zeroing template likely generalizes to other classes of sets defined by equality of two dimensions, producing universal direction sets of arbitrarily small but positive dimension.
  • The method indicates that the boundary between zero-dimensional and positive-dimensional universal sets is controlled by whether the pointwise complexity bound holds with logarithmic rather than linear error; identifying which regularity classes admit logarithmic errors would map that boundary.
  • A concrete test of the omitted proofs is to replace Lemma 13's teal-case error $4\varepsilon b$ with the explicit enumeration error from Lemma 14 and check whether the sum over the partition in Proposition 20 stays $o(r)$ in the critical range $r < r_n/(\varepsilon^2 s)$.
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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. This paper studies universal sets of directions for projection theorems in R^2. A direction set D is universal for a class C of planar sets if for every E in C some e in D satisfies dim_H(p_e E) = min{1, dim_H(E)}. The authors construct a family of small direction sets D_s, s in (0,1), of lower box dimension s, and prove (Theorem 21) that D_s is universal for the class of weakly regular sets; for s=0, D_0 has lower box dimension zero and is claimed to be universal for Ahlfors-David regular sets (Theorem 24). They also prove a Kaufman-type exceptional set estimate for sets with optimal oracles (Theorem 27), deduce existence of essentially s-dimensional universal sets for optimal-oracle classes (Corollary 28), and construct zero-dimensional Bourgain universal sets for analytic sets (Theorem 26). The proofs are based on effective dimension, the point-to-set principle, and a sequence of 'teal/yellow' interval decomposition lemmas from earlier work of the second author.

Significance. If the main theorems are correct, they give a sharp qualitative picture: stronger regularity assumptions on the class of sets shrink the dimension of universal direction sets to zero, while weak regularity still allows universal sets of arbitrarily small positive dimension. The construction of D_0 and D_s is elegant and the pointwise-to-classical transfer is well organized. The paper also generalizes Kaufman's exceptional set estimate to a broad 'optimal oracles' class, which is a substantial contribution in itself. However, the proof of Theorem 21 relies on Proposition 20, whose first case contains a concrete arithmetic error in the accumulation of errors from Lemma 13; as printed, Theorem 3 is not established. The paper also omits proofs of Lemmas 12 and 13, one of which drives the flawed step. The framework is promising and the gap appears local, so the result is likely repairable.

major comments (3)
  1. [Section 5.1, Proposition 20, Eq. (15)] The bound in (15) does not follow from the preceding estimate. In the first case, after applying Lemma 13 to each [r_i, r_{i+1}], the total error is at least 4ε Σ_{i=1}^M r_{i+1} (and the author's intermediate quantity 4Mεr is of the same order), which is of order ε r^2/r_n. Under the case assumption r < ε^{-2}⌊r_n/s⌋, this error can be as large as 4r/(sε), which is not bounded by 5εr; for example, with s=1/2, ε=10^{-2}, and r=2·10^4 r_n, the error is approximately 800r. Hence the inequality K^{A,B,e}_r(p_e x) ≥ min{α,1} - 5εr - O(log r) is arithmetically false under the stated hypotheses. Since Theorem 21 is proved directly from Proposition 20, Theorem 3 is not established by the present proof.
  2. [Section 3, Lemma 13] Lemma 13 is stated without proof, as a 'simple modification' of Lemmas 11 and 12. This is not merely a presentation issue: the error term 4εb on each interval is exactly what causes the summation failure in Proposition 20. If the correct statement had error 4ε(b-a), the total error over a partition of [r_n, r] would be O(εr), and the argument could be repaired. The authors should provide the proof of Lemma 13, or correct its statement and verify the corrected version, since Theorem 21 depends on it.
  3. [Section 5.1, Proposition 20] The assertion that each interval [r_i, r_{i+1}] is both (α, ε)-almost teal and (α, ε)-almost yellow is not justified. The hypotheses give only global bounds on the liminf and limsup of r^{-1}K^A_r(x); they do not directly imply uniform bounds on K^A_{b,s}(x|x) for all s in [a,b]. Moreover, Lemma 13 is applied with σ = min{α,1}, but if α > 1 the intervals would need to be (1,ε)-almost teal (upper bound with slope 1), which is not implied by an (α,ε)-almost teal bound (larger slope). This step should be proved carefully, as it is used to convert complexity differences into projection bounds.
minor comments (4)
  1. [Section 5.1, Proposition 20, condition (3)] In the statement of Proposition 20, condition (3) reads 'K^{A,B,e}_s(x) ≥ K^A_r(x) - εr for every sufficiently large r ∈ N'; the subscripts on the two sides do not match. It should presumably be K^{A,B,e}_r(x) ≥ K^A_r(x) - εr.
  2. [Section 5.1, Proposition 20, proof, second case] In the second case, the line 'K^{A,B,e}_r(p_e x) ≥ min{r, K^A_r(x)} - 25εr - O(log r)' appears to contain a typo: the right-hand side should be of the form min{dim^A(x),1} r - 25εr, as in Theorem 9. As written, the expression min{r, K^A_r(x)} compares a precision with a complexity and is not the intended lower bound.
  3. [Section 4, Lemma 19] In the proof of Lemma 19, the notation for the condition 'r_n/s ≤ r ≤ r_{n+1}' uses real division but the definition of D_s uses ⌊r_n/s⌋; the floors should be handled consistently to make the inequalities exact.
  4. [Global] There are several typos, e.g., 'Hasudorff' in Section 2.2, 'Oc(log b)2' in Lemma 11, and 'K A,D b,b,a (pe | e, x )' where 'pe' should read 'p_e x' in the proof of Lemma 16. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the universal-set theorems are proved from explicit constructions and independent effective-dimension tools; the flagged gaps are correctness/support issues, not reductions of conclusions to hypotheses.

full rationale

The paper's central claims, Theorems 2 and 3, assert the existence of small universal direction sets for AD-regular and weakly regular sets. The derivation does not build these conclusions into the hypotheses: the sets D0 and Ds are constructed explicitly in Section 4 from Martin-Löf random angles by bit deletion, and their dimensions are established independently by covering estimates and effective dimension arguments (Lemmas 18 and 19). Universality is then proved pointwise in Propositions 23 and 20 and lifted to the classical theorems via the point-to-set principle. The imported material from prior work by the second author (effective projection theorems, optimal oracles, Lemmas 7 and 15, and Lemma 36 of [29]) has hypotheses that do not mention universal sets or the specific Ds construction; these are independent tools, not restatements of the main claims. No fitted parameter is renamed as a prediction, and no exceptional-set estimate is defined in terms of the universal set whose existence it is used to prove. The text itself flags two concerns, and both are non-circular. Section 3.1 says "the proofs for Lemma 12 and Lemma 13 are simple modifications of the proof, and are omitted," which is an omitted proof rather than a circular step. In Proposition 20, the first case obtains K^{A,B,e}_r(p_e x) ≥ σr − (4M+1)εr − O(log r) and then asserts (15) "≥ min{α,1} − 5εr"; under the stated case assumption r < (1/ε^2)⌊r_n/s⌋, M = ⌊r/r_n⌋ can be on the order of 1/(sε^2), so (4M+1)εr need not be dominated by 5εr. That is an arithmetic gap in the written proof, not a definitional identification of the conclusion with its inputs. Accordingly, no circular step is established by the quoted text.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical or metaphysical entities are introduced; the universal direction sets D_s and D_0 are explicit mathematical constructions. The ledger instead highlights that the central claims rest on a set of effective-dimension tools, several of which come from the second author's prior work, and on two new lemmas whose proofs are omitted.

free parameters (3)
  • block sequence r_n = r_{n+1} = 2^{2^{r_n}}
    The rapidly increasing sequence defines the bit-deletion blocks and must grow fast enough for error terms to be sublinear; the choice is explicit, not fitted.
  • dimension parameter s = chosen in (0,1)
    D_s is defined for any s; the universal set for weakly regular sets uses this s as its lower box dimension.
  • accuracy parameter epsilon = arbitrary positive constant
    Appears in all effective statements and controls the pointwise error; it is not fitted to data.
assumptions (5)
  • standard math Point-to-set principle (Lutz-Lutz 2018)
    Used throughout to pass from classical Hausdorff/packing dimension to effective dimension; cited as Theorem 8.
  • domain assumption Optimal oracles exist for weakly regular sets and can be assumed to be packing oracles
    Invoked in Theorem 21 without proof; attributed to [28] and the weak regularity equality.
  • ad hoc to paper Lemma 13 (almost teal/yellow projection bound)
    New lemma central to Proposition 20; proof is omitted ('simple modification'), but its correctness is essential to the main weakly-regular theorem.
  • domain assumption Lemma 22 (AD-regular oracles exist, and joins preserve the property)
    Used in Theorem 24 to choose an oracle relative to which an AD-regular set is effectively compact; cited from [7].
  • domain assumption Every analytic set has compact approximants with computable measures relative to some oracle
    Used in the proof of Theorem 26 (Bourgain universal) to reduce to effective compact sets; standard in computable analysis but not proved in the paper.

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Cite this review

Pith. "Pith review of Universal Sets for Projections." pith.science (2026). https://pith.science/paper/CKI6C6R5

@misc{pith2026241116001,
  author       = {Pith},
  title        = {Pith review of: Universal Sets for Projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKI6C6R5}},
  note         = {Machine review of arXiv:2411.16001}
}
abstract

We investigate variants of Marstrand's projection theorem that hold for sets of directions and classes of sets in $\mathbb{R}^2$. We say that a set of directions $D \subseteq\mathcal{S}^1$ is $\textit{universal}$ for a class of sets if, for every set $E$ in the class, there is a direction $e\in D$ such that the projection of $E$ in the direction $e$ has maximal Hausdorff dimension. We construct small universal sets for certain classes. Particular attention is paid to the role of regularity. We prove the existence of universal sets with arbitrarily small positive Hausdorff dimension for the class of weakly regular sets. We prove that there is a universal set of zero Hausdorff dimension for the class of AD-regular sets.

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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. Algorithmic Information Bounds for Distances and Orthogonal Projections

    cs.CC 2025-09 conditional novelty 7.0 of 10

    A new proof technique shows distances and orthogonal projections retain at least half of a planar point's Kolmogorov complexity, improving pinned distance dimension bounds to 3/4 s and generalizing Bourgain's theorem.

Reference graph

Works this paper leans on

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