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arxiv: 2606.01040 · v1 · pith:QMHWU655new · submitted 2026-05-31 · 🧮 math.NT · math.MG

Topological and Diophantine properties of lattice subset projections

Wayne M Lawton This is my paper

Pith reviewed 2026-06-28 16:38 UTC · model grok-4.3

classification 🧮 math.NT math.MG
keywords lattice subsetk-denseGrassmannianG_delta setDiophantine propertiesKhintchine-Groshev theoremorthogonal projectioncrystalline measures
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The pith

A lattice subset L of Z^m is k-dense exactly when the set of n-planes where zero is a limit point of its projection forms a G_delta set.

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

The paper proves an equivalence: L subset of the integer lattice is k-dense if and only if the set L_{n,lim} of n-dimensional subspaces in which the orthogonal projection of L accumulates at zero is a G_delta set inside the Grassmannian Gr(n,m). It then invokes the Khintchine-Groshev theorem to tie the Diophantine character of points in L_{n,lim} to lacunary features of L and produces explicit k-dense examples in which L_{n,lim} has rotation-invariant measure zero and others in which the measure equals one. The topological characterization therefore supplies a route to control the measure of the limit set. Readers see a direct link between a category-theoretic density condition and measurable Diophantine behavior under projection.

Core claim

L is k-dense iff L_{n,lim} is a G_delta set. Khintchine-Groshev's theorem characterizes the Diophantine properties of L_{n,lim} by the lacunary properties of L. There exist k-dense L with sigma_n(L_{n,lim}) = 0 and with sigma_n(L_{n,lim}) = 1.

What carries the argument

The set L_{n,lim} of n-planes in Gr(n,m) where zero is a limit point of P_W(L), together with the Baire-category equivalence to k-density.

If this is right

  • k-dense lattice subsets exist whose associated limit set L_{n,lim} has measure zero.
  • k-dense lattice subsets exist whose associated limit set L_{n,lim} has measure one.
  • Diophantine properties of points in L_{n,lim} are controlled by lacunary properties of L.
  • The equivalence raises explicit questions about the existence of multidimensional crystalline measures and Fourier quasicrystals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The measure-zero and measure-one constructions may supply concrete candidates for Fourier quasicrystals whose diffraction spectra can be checked numerically.
  • Lacunary sequences such as exponential gaps could be tested directly to produce explicit low-measure examples without invoking the full strength of Khintchine-Groshev.
  • The topological equivalence might extend to uniform distribution questions for projections onto subspaces of varying dimension.

Load-bearing premise

Baire's category theorem applies directly to the Grassmannian to equate k-density of any lattice subset L with the G_delta property of L_{n,lim}.

What would settle it

An explicit lattice subset L that is k-dense yet L_{n,lim} fails to be G_delta, or one for which L_{n,lim} is G_delta yet L is not k-dense.

read the original abstract

Fix $1 \leq n < m, k = m-n.$ The Grassmannian $Gr(n,m)$ is a compact $kn$-dimensional manifold with a unique rotation invariant probability measure $\sigma_n.$ For $W \in Gr(n,m)$, $P_W : \mathbb R^m \mapsto W$ is orthogonal projection. A lattice subset $L \subset \mathbb Z^m \subset \mathbb R^m$ is called $k$-dense if it intersects $C(O) := \bigcup_{V \in O} V\backslash \{0\}$ for every nonempty open $O \subset Gr(k,m)$. We use Baire's category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim} := \{W \in Gr(n,m) : 0 \mbox{ is a limit point of } P_W(L) \}$ is a $G_\delta$ set. We use Khintchine-Groshev's theorem [5,13,20] to characterize Diophantine properties of $L_{n,lim}$ by lacunary properties of $L$ and construct $k$-dense $L$ with $\sigma_n(L_{n,lim}) = 0$ and with $\sigma_n(L_{n,lim}) = 1.$ We pose related questions about the construction of multidimensional crystalline measures and Fourier quasicrystals.

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.

Referee Report

1 major / 0 minor

Summary. The paper defines a lattice subset L ⊂ ℤ^m to be k-dense if it intersects every nonempty open set in the union of lines from Gr(k,m). It claims to prove, via Baire's category theorem, that L is k-dense if and only if the set L_{n,lim} = {W ∈ Gr(n,m) : 0 is a limit point of P_W(L)} is G_δ in the Grassmannian. It then invokes Khintchine-Groshev to relate Diophantine properties of L_{n,lim} to lacunary properties of L, constructs k-dense examples with σ_n(L_{n,lim}) = 0 and =1, and poses questions on crystalline measures and Fourier quasicrystals.

Significance. The claimed topological equivalence would connect density in Grassmannians to measure-theoretic Diophantine properties, but the equivalence is false. The subsequent constructions and questions rest on this foundation and therefore do not establish the stated results.

major comments (1)
  1. [Abstract] Abstract (and the unnumbered theorem invoking Baire's category theorem): the asserted equivalence 'L is k-dense iff L_{n,lim} is G_δ' does not hold. For any countable L ⊂ ℤ^m the sets {W : ||P_W(l)|| < 1/n} are open (by continuity of the projection map), so U_n = ∪_{l∈L\{0}} {W : ||P_W(l)|| < 1/n} is open and L_{n,lim} = ∩_n U_n is always G_δ. No appeal to Baire or to any special property of L is required. Consequently the equivalence would imply that every countable L is k-dense or that none are; both statements are false (e.g., any finite nonzero L fails to be k-dense yet still yields a G_δ set L_{n,lim}). The citation [4] therefore cannot establish the claimed equivalence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying an error in our claimed equivalence. We agree that the statement does not hold and will revise the manuscript accordingly. The constructions based on Khintchine-Groshev may be presented independently.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the unnumbered theorem invoking Baire's category theorem): the asserted equivalence 'L is k-dense iff L_{n,lim} is G_δ' does not hold. For any countable L ⊂ ℤ^m the sets {W : ||P_W(l)|| < 1/n} are open (by continuity of the projection map), so U_n = ∪_{l∈L\{0}} {W : ||P_W(l)|| < 1/n} is open and L_{n,lim} = ∩_n U_n is always G_δ. No appeal to Baire or to any special property of L is required. Consequently the equivalence would imply that every countable L is k-dense or that none are; both statements are false (e.g., any finite nonzero L fails to be k-dense yet still yields a G_δ set L_{n,lim}). The citation [4] therefore cannot establish the claimed equivalence.

    Authors: We agree with the referee's analysis. For any countable L ⊂ ℤ^m, each set {W : ||P_W(l)|| < 1/n} is open by continuity of the projection, so each U_n is open and L_{n,lim} = ∩_n U_n is always G_δ. No appeal to Baire's theorem is needed, and the equivalence cannot hold because not every countable L is k-dense. We will revise the abstract and the relevant theorem to remove the incorrect equivalence claim and the misplaced citation to [4]. The constructions of k-dense examples with σ_n(L_{n,lim}) equal to 0 or 1 rely on the Khintchine-Groshev theorem and can stand independently of the topological equivalence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external theorem application

full rationale

The paper's central equivalence is derived by direct invocation of Baire's category theorem via external citation [4], with no reduction of the G_δ property or k-density definition to self-referential inputs, fitted parameters, or self-citation chains. The subsequent Diophantine characterizations rely on Khintchine-Groshev results [5,13,20] as independent external benchmarks. No ansatz smuggling, renaming of known results, or load-bearing self-citations appear in the derivation chain. The constructions of k-dense lattices with prescribed measures are presented as consequences of these cited theorems rather than tautological restatements of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on two standard theorems from topology and Diophantine approximation; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Baire's category theorem
    Invoked to prove the equivalence of k-density with L_n,lim being G_delta.
  • standard math Khintchine-Groshev's theorem
    Invoked to characterize Diophantine properties of L_n,lim via lacunary properties of L and to enable the measure 0/1 constructions.

pith-pipeline@v0.9.1-grok · 5782 in / 1456 out tokens · 30844 ms · 2026-06-28T16:38:38.844409+00:00 · methodology

discussion (0)

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