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Unbounded bunching of saddle connections on the golden L

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Benjamin Dozier

Dynamical Systems · Submitted 19 Jun 2026 · 4 references

Pith's verdict: Dozier shows the golden L has unbounded bunching of saddle connection periods.

T0 · AI review

How this review was made: An AI referee (grok-4.3) read the paper on June 26, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2606.21601 (version 1)
Pith IDLRFZGFGY, Pith's permanent record of this paper and its review
CitationBenjamin Dozier. "Unbounded bunching of saddle connections on the golden L." arXiv:2606.21601 (2026). https://pith.science/paper/LRFZGFGY
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The paper

The paper claims

The golden L translation surface has unbounded bunching of saddle connection periods.

The reading

The paper shows that on the golden L there exist arbitrarily large finite sets of saddle connection period vectors that all lie inside some ball of radius 1. This is established by realizing the golden L as a translation surface and then exhibiting, for each K, a collection of at least K such vectors inside a single unit ball. A sympathetic reader cares because the result supplies an explicit surface on which the local density of saddle-connection periods is unbounded. The argument proceeds by direct enumeration of the periods that arise from the surface's polygonal construction.

What carries the argument

The golden L translation surface, whose saddle-connection period vectors are shown to accumulate inside arbitrarily many unit-radius balls.

Extended reading notes

Core claim

We show there is a translation surface (the golden L) that has unbounded bunching: for every positive integer K there exists a ball B of radius 1 in R^2 that contains at least K vectors that are periods of saddle connections on this surface.

Load-bearing premise

The golden L can be realized as a translation surface whose saddle-connection periods admit the required accumulation inside unit balls.

Editorial extensions

If this is right

  • The set of saddle-connection periods on the golden L fails to be uniformly discrete at the scale of unit balls.
  • For the golden L the number of saddle connections whose periods lie in any fixed compact set can be made arbitrarily large by choice of location.
  • Unbounded bunching is realized by at least one concrete translation surface.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar bunching may occur on other surfaces built from the same golden-ratio proportions.
  • The example raises the question of whether bounded bunching holds generically or only on surfaces with special arithmetic constraints.
  • One could check whether deforming the golden L while preserving the golden ratio angles destroys or preserves the bunching.
Original abstract

We show there is a translation surface (the golden L) that has unbounded bunching: for every positive integer K there exists a ball B of radius 1 in R^2 that contains at least K vectors that are periods of saddle connections on this surface.

Figures
Figure 1. The golden L translation surface, presented as a polygon with edge iden

Figures reproduced from arXiv:2606.21601 by the authors.

References

Discussion

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