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arxiv: 1907.05825 · v1 · pith:PHV2WJJBnew · submitted 2019-07-12 · 🧮 math.CO

Patterns in sets of positive density in trees and affine buildings

M. Bj\"orklund , A. Fish , J. Parkinson This is my paper

Pith reviewed 2026-05-24 22:20 UTC · model grok-4.3

classification 🧮 math.CO
keywords pinned distancespositive densityhomogeneous treesaffine buildingsdistance realizationBourgain analogue
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The pith

Sets of positive density in homogeneous trees and affine buildings realize all large pinned distances.

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

The paper proves an analogue of Bourgain's pinned distance theorem that applies to homogeneous trees and certain affine buildings. In these spaces a subset with positive upper density contains every sufficiently large distance from some point inside the subset. The proof uses the symmetry and regularity built into homogeneous trees and the chosen buildings. A separate construction shows that the conclusion fails in a non-homogeneous tree even when the subset has positive Hausdorff dimension.

Core claim

In homogeneous trees and the affine buildings considered, every set of positive upper density realizes all sufficiently large distances from at least one pinned point inside the set; the same fails in non-homogeneous trees.

What carries the argument

The metric and combinatorial regularity of homogeneous trees and affine buildings that transfers the pinned-distance argument from Euclidean space.

If this is right

  • Positive-density subsets of homogeneous trees pin all large distances.
  • The same distance realization holds inside the affine buildings studied.
  • Homogeneity is required; the property collapses once the tree loses regularity.

Where Pith is reading between the lines

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

  • Distance patterns in positive-density sets are sensitive to global symmetry rather than local geometry alone.
  • The counterexample technique may extend to other non-regular graphs or buildings.
  • The result suggests that pinned-distance theorems are stable under mild deformations that preserve homogeneity.

Load-bearing premise

The space must be homogeneous or satisfy the exact regularity conditions of the affine buildings used in the argument.

What would settle it

Exhibit a homogeneous tree and a positive-density subset that misses infinitely many large distances.

Figures

Figures reproduced from arXiv: 1907.05825 by A. Fish, J. Parkinson, M. Bj\"orklund.

Figure 1
Figure 1. Figure 1: Illustration for Claim 1 Claim 2: Suppose there are integers 0 < t1 < t2 < · · · < tk such that for each 1 ≤ j ≤ k there exist no vertices x, y ∈ X with d(o, x) = d(o, y) and d(x, y) = 2tj . Then for all n > tk we have |X ∩ Sn| |Sn| < q−k . Proof of Claim 2: Let vk ∈ Sn−tk . By Claim 1 (with v = v2), the proportion of atoms of Fn,tk−1 contained in C(vk, tk) with the property that they intersect nontriviall… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration for Claim 2 (with k = 2) 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A balanced (k, t, r)-star. Theorem 2 follows immediately from the following theorem. Theorem 1.3. Let X ⊆ V with d ∗ (X) > 0. For each k > 0 and each r ∈ N k there exists a constant K = K(X, k, r) > 0 such that X contains a balanced (k, t, r)-star for all sequences t = (ti) k i=1 ∈ Mk with t1 > K. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates a type A vertex x (with r ≤ 4, and where elements of X are denoted by •). n − s + 1 n n − s x y1 y2 • • • • • • • • [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Coxeter complex of type C˜ 2 The root system of type C2 is Φ = ±{α1, α2, α1 + α2, 2α1 + α2}, where α1 = e1 − e2 and α2 = 2e2. We have α ∨ 1 = α1 and α ∨ 2 = e2, and the dual root sys￾tem is Φ∨ = ±{α ∨ 1 , α∨ 2 , α∨ 1 + α ∨ 2 , α∨ 1 + 2α ∨ 2 }. The fundamental coweights are ω1 = e1 and ω2 = 1 2 e1 + 1 2 e2. The coroot lattice Q is the set of • vertices, and the coweight lattice P is the union of the • a… view at source ↗
Figure 6
Figure 6. Figure 6: The chamber c(z, z′ ) Let cx = c(x, o), and let O(x) denote the set of all chambers c of ∆ with x ∈ c such that dist(cx, c) = ℓ(w0). These are the chambers “opposite” cx in the “residue” of x, and we have |O(x)| = q ℓ(w0) . Then, for each c ∈ O(x) define C(x, c, λ) = {y ∈ C(x, λ) | c(x, y) = c}. (3.4) This situation is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Decomposing into atoms Lemma 3.4. Let λ, µ ∈ P ++, and write ν = λ + µ. The members of Aν,λ = {C(x, c, λ) | x ∈ Sµ and c ∈ O(x)} form a partition of the sphere Sν. Moreover we have (1) |Aν,λ| = |Sµ|q ℓ(w0) , and (2) |C(x, c, λ)| = q hλ,2ρi−ℓ(w0) , independent of µ ∈ P ++, x ∈ Sµ, and c ∈ O(x). Proof. If y ∈ Sν and x ∈ Sµ then y ∈ C(x, λ) if and only if x = πν,µ(y). Thus we have a disjoint union Sν = S x∈Sµ… view at source ↗
Figure 8
Figure 8. Figure 8: Illustration for Lemma 3.5 Since dist(c1, c2) = ℓ(w0) the affine geometry of the Coxeter complex (c.f. [1, §11.5]) implies that if γ1 is a minimal length gallery joining c ′ 1 to c1, and γ2 is a minimal length gallery joining c1 to c2, and γ3 is a minimal length gallery joining c2 to c ′ 2 , then the concatenation γ = γ1 ·γ2 ·γ3 is a minimal length gallery joining c ′ 1 to c ′ 2 (see [PITH_FULL_IMAGE:figu… view at source ↗
read the original abstract

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.

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

0 major / 2 minor

Summary. The paper proves an analogue of Bourgain's pinned-distance theorem for sets of positive density, showing that in homogeneous trees and certain affine buildings all sufficiently large distances are realized as pinned distances from a positive-density subset. It additionally constructs an explicit counterexample on a non-homogeneous tree of positive Hausdorff dimension in which a positive-density subset misses infinitely many even distances, demonstrating the necessity of the homogeneity hypothesis.

Significance. If the proofs hold, the result meaningfully extends pinned-distance phenomena from Euclidean spaces to regular discrete geometries, clarifying the role of transitivity and apartment structure. The counterexample is a concrete, falsifiable illustration of sharpness and strengthens the main theorem. The approach via automorphism groups and Weyl-group actions appears independent of fitted parameters.

minor comments (2)
  1. The abstract states the result for 'certain affine buildings' but does not name the precise class (e.g., which root systems or thickness conditions); a one-sentence clarification in the introduction would help readers locate the exact setting.
  2. Notation for the distance function and the pinned-distance set is introduced without an early displayed equation; adding a short displayed definition early in §2 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, their assessment of its significance, and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an analogue of Bourgain's pinned-distance theorem for homogeneous trees and specific affine buildings by exploiting their regularity properties (automorphism transitivity, apartment structure, and Weyl-group action) together with an explicit counterexample construction on non-homogeneous trees. The central derivation relies on combinatorial and geometric arguments internal to the structures considered; no step reduces by definition or construction to a fitted parameter, self-citation chain, or ansatz imported from the authors' prior work. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no details on free parameters, axioms, or invented entities used in the proof or counterexample.

pith-pipeline@v0.9.0 · 5572 in / 931 out tokens · 18686 ms · 2026-05-24T22:20:11.990455+00:00 · methodology

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Reference graph

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