Proof of the Finiteness of the Chromatic Number of Two-Dimensional Lacunary Distance Graphs

Nabh Singh

arxiv: 2606.22539 · v1 · pith:44LH6L7Qnew · submitted 2026-06-21 · 🧮 math.CO · cs.DM

Proof of the Finiteness of the Chromatic Number of Two-Dimensional Lacunary Distance Graphs

Nabh Singh This is my paper

Pith reviewed 2026-06-26 10:06 UTC · model grok-4.3

classification 🧮 math.CO cs.DM

keywords chromatic numberdistance graphslacunary sequencesinteger latticegraph coloringmultiplier vectortwo dimensions

0 comments

The pith

Any integer distance graph from a lacunary sequence of vectors in Z squared has finite chromatic number.

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

The paper extends the lonely set method from one dimension to two by applying a lacunary matrix theorem to a sequence of displacement vectors in the integer lattice. The theorem produces a satisfactory multiplier vector that supports an explicit geometric coloring of the resulting graph. This coloring uses only finitely many colors and is proper for all edges defined by the sequence. The result therefore establishes that the chromatic number remains finite for every such graph. Readers interested in plane-coloring problems can see that sparsity in the distance set suffices to avoid the infinite-color requirement that may hold in denser cases.

Core claim

Given any lacunary sequence of displacement vectors in Z², the Broderick-Fishman-Kleinbock lacunary matrix theorem guarantees a satisfactory multiplier vector; this vector then permits an explicit geometric argument that colors the corresponding integer distance graph with finitely many colors, proving the chromatic number is finite.

What carries the argument

Satisfactory multiplier vector produced by the lacunary matrix theorem, which enables the geometric coloring.

If this is right

The chromatic number of the graph is finite.
The same multiplier-vector technique works for every lacunary sequence in two dimensions.
An explicit finite coloring is constructible once the multiplier is known.
The one-dimensional lonely-set method extends directly to this two-dimensional setting.

Where Pith is reading between the lines

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

The same matrix theorem might be checked in three or more dimensions to test whether finiteness persists.
Sequences that are not lacunary could still admit finite chromatic number if an analogous multiplier exists by other means.
The geometric coloring supplies an upper bound on the chromatic number that could be computed for concrete sequences such as exponential vectors.

Load-bearing premise

The lacunary matrix theorem applies to the given sequence of displacement vectors and yields a satisfactory multiplier vector.

What would settle it

A concrete lacunary sequence of vectors in Z² whose integer distance graph requires infinitely many colors.

read the original abstract

We extend the one-dimensional lonely set method to two dimensions for the purpose of studying the chromatic number of integer distance graphs in two dimensions. Given a lacunary sequence of displacement vectors in $Z^{2}$, we use a lacunary matrix theorem given by Broderick, Fishman and Kleinbock, to prove the existence of a satisfactory multiplier vector. We then give an explicit geometric colouring argument. This proves that any integer distance graph generated by a lacunary sequence of vectors in two dimensions has finite chromatic number.

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 sketches a 2D extension of the lonely-set method for lacunary distance graphs by invoking the Broderick-Fishman-Kleinbock theorem, but the abstract supplies no verification that the theorem applies to the sequence or that the multiplier works geometrically.

read the letter

The main point is that this paper claims to prove finite chromatic number for any integer distance graph generated by a lacunary sequence of vectors in Z^2. It does so by lifting the one-dimensional lonely set method, using the cited lacunary matrix theorem to obtain a multiplier vector, and then running a geometric coloring argument.

The extension itself is new relative to the one-dimensional results referenced. The strategy of relying on an external theorem for the multiplier and handling the coloring separately is reasonable on its face, and it avoids reinventing the Diophantine machinery.

The soft spot is exactly the one flagged in the stress-test: the abstract states that the theorem is applied but gives no check that the sequence meets the required lacunarity parameters, rank conditions, or Diophantine properties, nor does it confirm that the resulting multiplier produces a finite coloring in two dimensions. Without those steps shown, the central claim rests on an unverified applicability step.

This is for people working on chromatic numbers of geometric graphs or lacunary sequences in combinatorial number theory. A reader already familiar with the one-dimensional case would see the natural next question being addressed, provided the details hold.

I would send it to peer review. The result would be useful if the theorem application is made explicit and correct, but a referee needs to see the verification before the claim can be accepted.

Referee Report

1 major / 1 minor

Summary. The paper extends the one-dimensional lonely set method to two dimensions. Given a lacunary sequence of displacement vectors in Z^2, it invokes the Broderick-Fishman-Kleinbock lacunary matrix theorem to establish existence of a satisfactory multiplier vector and then supplies an explicit geometric colouring argument. The central claim is that any integer distance graph generated by such a lacunary sequence in two dimensions has finite chromatic number.

Significance. If the argument is correct, the result supplies a two-dimensional extension of known finiteness theorems for chromatic numbers of lacunary distance graphs, linking Diophantine approximation techniques with geometric graph colouring.

major comments (1)

[the section applying the Broderick-Fishman-Kleinbock theorem] The manuscript invokes the Broderick-Fishman-Kleinbock lacunary matrix theorem to produce the multiplier vector but supplies no explicit verification that the given sequence of vectors in Z^2 satisfies the theorem's hypotheses (lacunarity parameters, matrix rank, or Diophantine conditions). This verification is load-bearing for the existence claim and for the subsequent geometric colouring step.

minor comments (1)

[Abstract] The abstract sketches the proof outline but does not indicate where in the manuscript the verification of the theorem's hypotheses appears; a brief pointer would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for explicit verification in the application of the Broderick-Fishman-Kleinbock theorem. We address the major comment below.

read point-by-point responses

Referee: [the section applying the Broderick-Fishman-Kleinbock theorem] The manuscript invokes the Broderick-Fishman-Kleinbock lacunary matrix theorem to produce the multiplier vector but supplies no explicit verification that the given sequence of vectors in Z^2 satisfies the theorem's hypotheses (lacunarity parameters, matrix rank, or Diophantine conditions). This verification is load-bearing for the existence claim and for the subsequent geometric colouring step.

Authors: We agree that the manuscript would benefit from an explicit verification subsection to confirm that the lacunary sequence of displacement vectors in Z^2 meets all hypotheses of the Broderick-Fishman-Kleinbock theorem, including the lacunarity parameters, matrix rank conditions, and relevant Diophantine properties. In the revised version we will insert a dedicated paragraph (or short subsection) immediately preceding the invocation of the theorem, providing this verification for the sequences under consideration. This addition will make the existence of the multiplier vector fully rigorous and will directly support the subsequent geometric colouring argument. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external theorem then independent geometric argument

full rationale

The paper's chain is: apply Broderick-Fishman-Kleinbock lacunary matrix theorem (external authors) to obtain multiplier vector from the given lacunary sequence, then supply an explicit geometric colouring argument to bound the chromatic number. No self-citations appear, no parameter is fitted and renamed as prediction, no ansatz is smuggled, and no step reduces by definition or construction to its own inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of the external lacunary matrix theorem to Z^2 sequences and on the validity of the one-dimensional lonely-set method when lifted to two dimensions.

axioms (1)

domain assumption The lacunary matrix theorem of Broderick, Fishman and Kleinbock applies to the displacement vectors in Z^2 and produces a satisfactory multiplier vector.
Invoked in the abstract to guarantee the multiplier needed for the geometric coloring step.

pith-pipeline@v0.9.1-grok · 5607 in / 1204 out tokens · 57514 ms · 2026-06-26T10:06:31.495010+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 3 canonical work pages

[1]

Katznelson,Chromatic numbers of Cayley graphs onZand recurrence, Combinatorica, 21(2) (2001), 211–219

Y. Katznelson,Chromatic numbers of Cayley graphs onZand recurrence, Combinatorica, 21(2) (2001), 211–219

2001
[2]

Peres and W

Y. Peres and W. Schlag,Two Erd˝ os problems on lacunary sequences: chromatic number and Diophantine approximation, Bulletin of the London Mathematical Society, 42(2) (2010), 295–

2010
[3]

doi:10.1112/blms/bdp126

work page doi:10.1112/blms/bdp126
[4]

Broderick, L

R. Broderick, L. Fishman and D. Y. Kleinbock,Schmidt’s game, fractals, and orbits of toral endomorphisms, Ergodic Theory and Dynamical Systems, 31(4) (2011), 1095–1107. doi:10.1017/S0143385710000374

work page doi:10.1017/s0143385710000374 2011
[5]

Dubickas,On the fractional parts of lacunary sequences, Mathematica Scandinavica, 99(1) (2006), 136–146

A. Dubickas,On the fractional parts of lacunary sequences, Mathematica Scandinavica, 99(1) (2006), 136–146. doi:10.7146/math.scand.a-15004. 6

work page doi:10.7146/math.scand.a-15004 2006

[1] [1]

Katznelson,Chromatic numbers of Cayley graphs onZand recurrence, Combinatorica, 21(2) (2001), 211–219

Y. Katznelson,Chromatic numbers of Cayley graphs onZand recurrence, Combinatorica, 21(2) (2001), 211–219

2001

[2] [2]

Peres and W

Y. Peres and W. Schlag,Two Erd˝ os problems on lacunary sequences: chromatic number and Diophantine approximation, Bulletin of the London Mathematical Society, 42(2) (2010), 295–

2010

[3] [3]

doi:10.1112/blms/bdp126

work page doi:10.1112/blms/bdp126

[4] [4]

Broderick, L

R. Broderick, L. Fishman and D. Y. Kleinbock,Schmidt’s game, fractals, and orbits of toral endomorphisms, Ergodic Theory and Dynamical Systems, 31(4) (2011), 1095–1107. doi:10.1017/S0143385710000374

work page doi:10.1017/s0143385710000374 2011

[5] [5]

Dubickas,On the fractional parts of lacunary sequences, Mathematica Scandinavica, 99(1) (2006), 136–146

A. Dubickas,On the fractional parts of lacunary sequences, Mathematica Scandinavica, 99(1) (2006), 136–146. doi:10.7146/math.scand.a-15004. 6

work page doi:10.7146/math.scand.a-15004 2006