Non-colliding billiards in the plane

Alexander Shamov; Itai Benjamini

arxiv: 2605.23575 · v1 · pith:OKQPGYRAnew · submitted 2026-05-22 · 🧮 math.DS

Non-colliding billiards in the plane

Itai Benjamini , Alexander Shamov This is my paper

Pith reviewed 2026-05-25 02:44 UTC · model grok-4.3

classification 🧮 math.DS

keywords non-colliding billiardshard disksvelocity assignmentinteger latticecontinuous vector fieldseparation inequalitydynamical systems

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The pith

Bounded non-colliding velocity assignments exist for the integer lattice, but no bounded continuous vector field works for the entire plane.

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

The paper poses an open problem about whether bounded velocities can be assigned to points in the plane so that freely moving hard disks never collide. It gives a positive answer for the integer lattice by exhibiting a bounded injective assignment that meets the separation condition for all pairs. It gives a negative answer by showing that any bounded continuous vector field on the plane must violate the separation inequality for some pair of points farther than distance one apart. The results leave the existence question open for general configurations of points.

Core claim

A bounded injective velocity assignment on the integer lattice satisfies the separation inequality for every pair of points at distance greater than one, ensuring non-collision of their trajectories, while no bounded continuous vector field on the whole plane can satisfy the same inequality for all such pairs.

What carries the argument

The separation inequality, a condition on relative velocities of any two points more than unit distance apart that prevents their straight-line trajectories from bringing them closer than distance one in the future.

If this is right

The integer lattice admits non-colliding motions with uniformly bounded speeds.
Any velocity assignment that works for all points in the plane must be discontinuous.
The open problem of existence for arbitrary point sets remains unresolved by the continuous prohibition.

Where Pith is reading between the lines

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

A positive solution for the full plane would likely require the axiom of choice and produce a non-measurable assignment.
The problem connects to the dynamics of infinitely many non-interacting particles in unbounded domains.
Numerical checks on large finite lattices could indicate whether the discrete construction scales.

Load-bearing premise

The separation inequality is assumed to be the only constraint needed to guarantee non-collision.

What would settle it

An explicit construction of a bounded continuous vector field on the plane that satisfies the separation inequality for every pair of points at distance greater than one would disprove the negative result.

read the original abstract

We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The positive deterministic result gives a bounded, injective, non-colliding velocity assignment for the integer lattice. The negative result shows that no bounded continuous vector field on the whole plane can serve as a universal assignment satisfying the same separation inequality for all pairs of points at distance greater than one.

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

Paper states an open problem on non-colliding point velocities with a lattice construction and a continuous no-go result.

read the letter

The main takeaway is that Benjamini and Shamov pose a new open problem on assigning bounded velocities to points in the plane so unit disks never collide under straight-line motion, and they supply two scoped partial results: an explicit construction on the integer lattice and a proof that no bounded continuous vector field works universally on R^2. Both are stated with a clear separation inequality that keeps points at least distance one apart after any time t. The lattice assignment is bounded and injective, which gives a concrete positive example. The negative result rules out continuous fields as a universal solution, which is a useful obstruction. These pieces look new relative to the cited literature and are presented without overclaiming generality. The lattice result is limited to Z^2 and does not extend to arbitrary dense sets. The continuous prohibition leaves room for discontinuous or non-universal assignments in the full plane, and the paper does not explore whether other global constraints (topological or otherwise) might block solutions even if continuity is dropped. The statements are internally consistent and scoped precisely, with no circularity or unfalsifiable claims. This is for readers in dynamical systems and billiards who follow open problems on infinite configurations. It deserves peer review because the problem is well-posed and the partial results are precise enough to be checked and built on.

Referee Report

0 major / 2 minor

Summary. The manuscript presents an open problem on non-colliding freely moving hard disks in the Euclidean plane. It supplies two partial results: an explicit bounded injective non-colliding velocity assignment on the integer lattice satisfying the separation inequality, and a proof that no bounded continuous vector field on R^2 satisfies the same inequality for every pair of points at distance greater than one.

Significance. If the results hold, they contribute to dynamical systems by delineating constraints on non-colliding motions, with an explicit lattice construction and a continuous prohibition that together separate discrete and continuum settings. The paper supplies an explicit construction and a proof for the scoped claims.

minor comments (2)

The separation inequality should be stated explicitly with its precise form early in the introduction (e.g., near the statement of the open problem) to make the positive and negative results immediately comparable.
A brief remark on whether the lattice construction can be extended or approximated in the continuum, even if only negatively, would clarify the scope without altering the central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the open problem and the two partial results, as well as the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; direct existence/non-existence statements

full rationale

The paper states an open problem on non-colliding billiards and supplies two scoped partial results: an explicit bounded injective velocity assignment on the integer lattice satisfying the separation inequality, and a proof that no bounded continuous vector field on R^2 satisfies the same inequality for all pairs at distance >1. Both are existence and non-existence claims with no fitted parameters, no self-definitional reductions, and no load-bearing self-citations that collapse the central claims to their own inputs. The derivation chain consists of direct constructions and impossibility arguments that remain independent of any prior fitted quantities or circular renamings.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The separation inequality is treated as given background.

pith-pipeline@v0.9.0 · 5580 in / 1006 out tokens · 25156 ms · 2026-05-25T02:44:36.991354+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Solomon and B

Y. Solomon and B. Weiss, Dense forests and Danzer sets,Annales Scientifiques de l’ENS49 (2016), 1049–1070

work page 2016
[2]

Some remarks on the lonely runner conjecture

T. Tao, Some remarks on the lonely runner conjecture, arXiv:1701.02048, https://arxiv. org/abs/1701.02048. 7

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

Solomon and B

Y. Solomon and B. Weiss, Dense forests and Danzer sets,Annales Scientifiques de l’ENS49 (2016), 1049–1070

work page 2016

[2] [2]

Some remarks on the lonely runner conjecture

T. Tao, Some remarks on the lonely runner conjecture, arXiv:1701.02048, https://arxiv. org/abs/1701.02048. 7

work page internal anchor Pith review Pith/arXiv arXiv