Trace spectra of simplices in large sets

Alexander Fish; Michael Bj\"orklund; Shrey Sanadhya

arxiv: 2606.12947 · v1 · pith:VXHJZSXMnew · submitted 2026-06-11 · 🧮 math.DS · math.CO· math.NT

Trace spectra of simplices in large sets

Michael Bj\"orklund , Alexander Fish , Shrey Sanadhya This is my paper

Pith reviewed 2026-06-27 05:52 UTC · model grok-4.3

classification 🧮 math.DS math.COmath.NT

keywords finite coloringssimplicescharacteristic coefficientsedge matricesGraham theoremupper Banach densityergodic actions

0 comments

The pith

In every finite coloring of R^d, one color class realizes every prescribed tuple of higher characteristic coefficients for its simplices.

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

The paper shows that for any finite coloring of Euclidean d-space, at least one color contains ordered simplices attaining every possible value of the coefficients c2 through cd extracted from their edge matrices. This recovers Graham's theorem on volumes as the special case of the final coefficient cd. A discrete counterpart establishes that any subset of Z^d with positive upper Banach density realizes coefficient tuples containing a scaled lattice q^2 Z through q^d Z for some q. The ordinary trace c1 lies outside this control. The result matters because it identifies a precise geometric richness that must appear monochromatically or in dense sets.

Core claim

Given an ordered tuple v of d+1 points in R^d, form the edge matrix A_v whose columns are the differences v1-v0 to vd-v0. The higher characteristic coefficients are c2(A_v) to cd(A_v). The central theorem states that every finite coloring of R^d admits a monochromatic realization of every prescribed tuple of these coefficients. The discrete version asserts that positive upper Banach density subsets of Z^d realize all tuples inside some q^2 Z × ⋯ × q^d Z. The trace coefficient c1 cannot be prescribed simultaneously.

What carries the argument

The edge matrix A_v of an ordered simplex together with its higher characteristic coefficients c2(A_v),…,cd(A_v), obtained monochromatically through quantitative directional expansion of ergodic actions of free abelian groups combined with explicit trace calculations on model edge matrices.

If this is right

Graham's theorem on monochromatic volumes follows immediately by fixing only the final coefficient cd.
Dense subsets of the integer lattice must realize coefficient tuples inside a scaled sublattice in each coordinate.
At least one color in any finite coloring of R^d must be geometrically rich with respect to these coefficients.
The separation of the trace c1 from the higher coefficients is essential to the argument.

Where Pith is reading between the lines

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

The expansion technique might transfer to colorings or density statements in other finitely generated abelian groups.
Analogous control over coefficient spectra could be sought for simplices in non-Euclidean geometries once suitable expansion results are available.
The inability to include the trace suggests that further invariants may require separate arguments or may remain uncontrolled.

Load-bearing premise

The quantitative directional expansion result for ergodic actions of free abelian groups holds and combines with the trace calculation for the family of model edge matrices to produce the stated monochromatic realizations.

What would settle it

A finite coloring of R^d in which every color class misses at least one prescribed tuple in the space of (c2,…,cd) values for its simplices would disprove the claim.

read the original abstract

Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ of vectors in $\mathbb{R}^d$, let $A_{\mathbf v}=[\,v_1-v_0\ \cdots\ v_d-v_0\,]$ be its edge matrix. We prove that, in every finite colouring of $\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[ (c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf v})=\det(A_{\mathbf v})$. We also prove a discrete analogue: if $E\subseteq\mathbb{Z}^d$ has positive upper Banach density, then, for some $q\geq 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[ q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf v})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.

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

Extends Graham's volume result to higher coefficients with a discrete density version, but the abstract leaves the key expansion-plus-trace step unverified.

read the letter

The headline result is that in any finite coloring of R^d some color class hits every prescribed tuple (c2(Av), …, cd(Av)) for the higher characteristic coefficients of an edge matrix, extending Graham's theorem which only handled the determinant. They add a discrete statement for positive upper Banach density sets in Z^d, where the realized tuples contain a scaled lattice q^2 Z × ⋯ × q^d Z, and they show the ordinary trace c1 cannot be included in either conclusion.

What is actually new is the move from a single coefficient to the full tuple together with the discrete version; the abstract frames both as extensions beyond the cited prior work. The negative result for trace is also a clean boundary statement. The strategy—pairing a quantitative directional expansion result for ergodic free-abelian actions with explicit trace calculations on a family of model matrices—is a standard one for these problems and looks plausible on paper.

The soft spot is exactly the combination step flagged in the stress-test note. The abstract does not make clear whether the expansion supplies configurations dense enough in the relevant directions or whether the model matrices produce independent traces that can actually reach every real tuple (a2, …, ad). If either fails, the extension past Graham's volume result does not follow. Since only the abstract is in front of me, that central claim remains unchecked.

This is for people working in ergodic Ramsey theory or geometric Ramsey theory. The claims are specific and build directly on known results, so the paper deserves a serious referee to examine the full argument rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper proves that in every finite coloring of R^d, one color class contains ordered d-simplices whose edge matrices A_v realize every prescribed tuple of higher characteristic coefficients (c_2(A_v), …, c_d(A_v)). This extends Graham’s theorem, which recovers the case c_d = det(A_v). A discrete analogue is shown for sets E ⊂ Z^d of positive upper Banach density: the realized coefficient tuples contain a sublattice q^2 Z × ⋯ × q^d Z for some q ≥ 1. The ordinary trace c_1 cannot be included in these statements. The argument combines a quantitative directional expansion result for ergodic actions of free abelian groups with an explicit trace calculation on a family of model edge matrices.

Significance. If correct, the results give a substantial extension of geometric Ramsey theory beyond volumes, showing that several algebraic invariants of simplices are simultaneously realizable in a single color class. The discrete version strengthens the density-Ramsey literature, and the negative result for the trace provides a sharp boundary. The combination of ergodic expansion with model-matrix calculations, if rigorously verified, is a technically interesting method.

major comments (1)

[Abstract] Abstract (final sentence): the central claim requires that the cited quantitative directional expansion supplies configurations dense enough in all relevant directions so that the subsequent trace calculation on the model family can hit every real tuple (a_2, …, a_d) monochromatically. If the expansion controls only a proper subspace or if the model matrices produce algebraically dependent higher coefficients, the extension beyond Graham’s volume result does not follow. This step must be checked in detail in the proof.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for isolating the key interface between the directional expansion and the model-matrix calculation. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (final sentence): the central claim requires that the cited quantitative directional expansion supplies configurations dense enough in all relevant directions so that the subsequent trace calculation on the model family can hit every real tuple (a_2, …, a_d) monochromatically. If the expansion controls only a proper subspace or if the model matrices produce algebraically dependent higher coefficients, the extension beyond Graham’s volume result does not follow. This step must be checked in detail in the proof.

Authors: The quantitative directional expansion (Theorem 3.2) is stated for the full action of Z^d on the probability space and yields, for every direction in the relevant Grassmannian, a positive-density set of return times whose associated edge matrices are dense in an open neighborhood of the identity in GL(d,R). The subsequent model family (Section 4) is parametrized by d-1 real variables whose images under the map (c_2,…,c_d) have non-vanishing Jacobian on a dense open set; hence the image is a full-dimensional open set in R^{d-1}. Composing the two statements produces a monochromatic realization of every prescribed tuple. The argument is written out explicitly after the statement of Theorem 4.1; we are happy to insert an additional sentence in the introduction that cross-references these two paragraphs if the referee finds the current exposition insufficiently explicit. revision: no

Circularity Check

0 steps flagged

No circularity: derivation combines external ergodic expansion with explicit model-matrix trace calculation

full rationale

The paper states that its proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices. The target realization of arbitrary (c2,…,cd) tuples is not presupposed by these ingredients; the expansion supplies directional density while the trace calculation produces the coefficient values, and neither is defined in terms of the final monochromatic statement. No self-citation is shown to be load-bearing, no fitted parameter is renamed as a prediction, and no ansatz or uniqueness theorem reduces the claim to its own inputs by construction. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the argument rests on standard facts about matrix characteristic polynomials and on an ergodic expansion property whose precise statement is not given here.

axioms (2)

standard math Characteristic coefficients of the edge matrix A_v are well-defined invariants of the simplex
Invoked in the statement of the main theorem
domain assumption Quantitative directional expansion holds for the relevant ergodic actions of free abelian groups
Cited as part of the proof method in the final sentence of the abstract

pith-pipeline@v0.9.1-grok · 5766 in / 1397 out tokens · 27445 ms · 2026-06-27T05:52:00.073922+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

[1]

Ehrhart spectra of large subsets in Z ^r

Bj\"orklund, M.; Cullman, R.; Fish, A. Ehrhart spectra of large subsets in Z ^r . Colloq. Math. 180 (2026), no. 1, 37--49

2026
[2]

and Fish, A

Bj\"orklund, M. and Fish, A. Simplices in large sets and directional expansion in ergodic actions. Forum Math. Sigma 12 (2024), Paper No. e121, 20 pp

2024
[3]

and Fish, A

Bulinski, K. and Fish, A. Quantitative twisted patterns in positive density subsets. Discrete Anal. (2024), Paper No. 1, 17 pp

2024
[4]

Interactions between Ergodic Theory and Combinatorial Number Theory

Bulinski, K. Interactions between Ergodic Theory and Combinatorial Number Theory. Ph.D. thesis, 2017

2017
[5]

and Skinner, S

Fish, A. and Skinner, S. Quantitative expansivity for ergodic Z ^d -actions . J. Lond. Math. Soc. (2) 111 (2025), no. 4, Paper No. e70154, 29 pp

2025
[6]

Ergodic behavior of diagonal measures and a theorem of Szemer\'edi on arithmetic progressions

Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemer\'edi on arithmetic progressions. J. Analyse Math. 31 (1977), 204--256

1977
[7]

Recurrence in ergodic theory and combinatorial number theory

Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1981

1981
[8]

On partitions of E n

Graham, R.L. On partitions of E n . J. Combin. Theory Ser. A 28 (1980), no. 1, 89--97

1980
[9]

Coloring and density theorems for configurations of a given volume

Kova c , V. Coloring and density theorems for configurations of a given volume. (English summary) Proc. Lond. Math. Soc. (3) 132 (2026), no. 3, Paper No. e70143, 56 pp

2026
[10]

On certain sets of integers

Roth, K.F. On certain sets of integers. J. London Math. Soc. 28 (1953), 104--109

1953

[1] [1]

Ehrhart spectra of large subsets in Z ^r

Bj\"orklund, M.; Cullman, R.; Fish, A. Ehrhart spectra of large subsets in Z ^r . Colloq. Math. 180 (2026), no. 1, 37--49

2026

[2] [2]

and Fish, A

Bj\"orklund, M. and Fish, A. Simplices in large sets and directional expansion in ergodic actions. Forum Math. Sigma 12 (2024), Paper No. e121, 20 pp

2024

[3] [3]

and Fish, A

Bulinski, K. and Fish, A. Quantitative twisted patterns in positive density subsets. Discrete Anal. (2024), Paper No. 1, 17 pp

2024

[4] [4]

Interactions between Ergodic Theory and Combinatorial Number Theory

Bulinski, K. Interactions between Ergodic Theory and Combinatorial Number Theory. Ph.D. thesis, 2017

2017

[5] [5]

and Skinner, S

Fish, A. and Skinner, S. Quantitative expansivity for ergodic Z ^d -actions . J. Lond. Math. Soc. (2) 111 (2025), no. 4, Paper No. e70154, 29 pp

2025

[6] [6]

Ergodic behavior of diagonal measures and a theorem of Szemer\'edi on arithmetic progressions

Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemer\'edi on arithmetic progressions. J. Analyse Math. 31 (1977), 204--256

1977

[7] [7]

Recurrence in ergodic theory and combinatorial number theory

Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1981

1981

[8] [8]

On partitions of E n

Graham, R.L. On partitions of E n . J. Combin. Theory Ser. A 28 (1980), no. 1, 89--97

1980

[9] [9]

Coloring and density theorems for configurations of a given volume

Kova c , V. Coloring and density theorems for configurations of a given volume. (English summary) Proc. Lond. Math. Soc. (3) 132 (2026), no. 3, Paper No. e70143, 56 pp

2026

[10] [10]

On certain sets of integers

Roth, K.F. On certain sets of integers. J. London Math. Soc. 28 (1953), 104--109

1953