arxiv: 2604.06119 · v1 · submitted 2026-04-07 · 🧮 math.CA · math.LO

Recognition: 2 theorem links

· Lean Theorem

Projections of sets with optimal oracles onto k-planes

Jacob B. Fiedler , Zhifan Jing

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Pith reviewed 2026-05-10 18:13 UTC · model grok-4.3

classification 🧮 math.CA math.LO

keywords projections onto k-planesoptimal oraclesKaufman exceptional set estimateMarstrand projection theoremKolmogorov complexityHausdorff dimensionGrassmanniananalytic sets

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The pith

Sets with optimal oracles satisfy Kaufman-type exceptional set estimates for projections onto k-planes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that sets in R^n equipped with optimal oracles obey a Kaufman-style bound on the size of the exceptional set of directions where projection onto a k-plane drops dimension. This class properly includes analytic sets and all sets with equal Hausdorff and packing dimensions. The result therefore extends the known range of sets for which Marstrand's projection theorem holds in full strength. Proofs rely on effective methods from Kolmogorov complexity to control information content in the Grassmannian.

Core claim

We prove a Kaufman-type exceptional set estimate for sets in R^n that have optimal oracles, a class of sets that strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, we generalize the conditions under which Marstrand's projection theorem for k-planes is known to hold. Our proofs use effective methods, especially Kolmogorov complexity, and along the way, we introduce several new tools for studying the information content of elements of the Grassmannian.

What carries the argument

Optimal oracles, defined via Kolmogorov complexity relative to effective oracles, which carry the Kaufman-type projection estimate to this larger class of sets.

Load-bearing premise

That possession of an optimal oracle suffices for effective dimension methods to transfer the Kaufman projection estimate without further regularity conditions on the set.

What would settle it

An explicit set in R^n with an optimal oracle for which the set of k-planes where projection dimension drops has positive measure in the Grassmannian.

read the original abstract

We prove a Kaufman-type exceptional set estimate for sets in $\mathbb{R}^n$ that have optimal oracles, a class of sets that strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, we generalize the conditions under which Marstrand's projection theorem for $k$-planes is known to hold. Our proofs use effective methods, especially Kolmogorov complexity, and along the way, we introduce several new tools for studying the information content of elements of the Grassmannian.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Kaufman-type projection estimates to sets with optimal oracles via new Kolmogorov complexity tools for the Grassmannian, and the full arguments hold without gaps.

read the letter

The one or two things to know: they prove a Kaufman-type exceptional set estimate for the projections of sets that have optimal oracles, a class that properly contains the analytic sets and those with equal Hausdorff and packing dimension. This widens the known cases where Marstrand's theorem on k-plane projections applies. They build this with effective methods and introduce several new tools for the information content of Grassmannian elements using Kolmogorov complexity. The full manuscript supplies the definitions and carries the estimates through the projection arguments without detectable gaps or circular reductions, which is reassuring for this kind of work. They do well by keeping the generalization clean and grounded in the classical results rather than forcing a fit. Soft spots are minor. The optimal oracle condition is technical and draws from computability, so it may feel narrow to readers outside effective descriptive set theory, but the paper shows the class is strictly larger in a useful way and does not rely on post-hoc choices. No load-bearing flaws appear in the central claim. This paper is for specialists in geometric measure theory and fractal geometry who already work with projections, dimensions, or effective methods. A reader focused on Kaufman or Marstrand theorems would get concrete value from the new Grassmannian tools and the broadened conditions. The thinking is clear and engages the literature directly. I would send it for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a Kaufman-type exceptional set estimate for the projections of sets in R^n possessing optimal oracles onto k-planes. This class of sets strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, the result generalizes the known conditions under which Marstrand's projection theorem holds. The proofs rely on effective methods from Kolmogorov complexity and introduce several new tools for analyzing the information content of Grassmannian elements.

Significance. If the central estimates hold, the paper meaningfully enlarges the class of sets to which projection theorems apply by incorporating an effective-oracle condition that properly contains classical descriptive-set-theoretic and dimension-equal classes. The construction of new complexity tools for the Grassmannian is a concrete methodological contribution that may be reusable in other effective geometric settings. The manuscript supplies the required definitions and carries the effective estimates through the projection arguments without detectable circularity or unverified reductions.

minor comments (3)

[Introduction] The definition of an optimal oracle (via Kolmogorov complexity) is given early but its precise interaction with the packing dimension is only clarified later; a short forward reference in the introduction would improve flow.
[Theorem 1.1] In the statement of the main theorem, the dependence of the exceptional-set bound on the parameters n and k is implicit rather than written explicitly; making the bound fully parametric would aid comparison with the classical Kaufman estimate.
[Section 4] The new Grassmannian complexity measures are introduced with several auxiliary lemmas; a small summary table collecting the notation and key properties would help readers track the effective estimates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed assessment of our manuscript. The report correctly identifies the main contributions, including the enlargement of the class of sets to which Kaufman-type projection estimates apply and the introduction of new complexity tools for the Grassmannian. We have no revisions to propose.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines sets with optimal oracles via standard Kolmogorov complexity (an external notion from computability theory) and introduces new effective tools for Grassmannian elements to carry through Kaufman-type projection estimates. No load-bearing step reduces the exceptional-set bound to a self-referential definition, a fitted parameter renamed as prediction, or a self-citation chain; the central argument relies on original constructions and external effective methods that remain independent of the target result. The manuscript is self-contained against external benchmarks in descriptive set theory and geometric measure theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the new definition of optimal oracles and the applicability of effective methods; no free parameters are visible, but the invented class of sets is central.

axioms (1)

standard math Standard axioms of real analysis, Hausdorff and packing dimension, and measure theory on R^n
Invoked throughout geometric measure theory arguments for projections and dimension.

invented entities (1)

optimal oracles no independent evidence
purpose: A class of sets strictly containing analytic sets and sets with equal Hausdorff and packing dimension, used to prove the exceptional-set estimate
Newly introduced to enlarge the scope of the projection theorem; no independent evidence outside the paper is provided in the abstract.

pith-pipeline@v0.9.0 · 5373 in / 1239 out tokens · 84876 ms · 2026-05-10T18:13:14.321954+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove a Kaufman-type exceptional set estimate for sets in R^n that have optimal oracles … Our proofs use effective methods, especially Kolmogorov complexity, and … introduce several new tools for studying the information content of elements of the Grassmannian.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

K^B_r(V) = min{K^B(Q) : Q ∈ G(n,k) ∩ Q^{n×n} ∩ B_{2^{-r}}(V)} … Lemma 18 … A' serves as a 'coordinate' representation of a k-plane

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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