The classification of real quadratic fields which satisfy Hammarhjelm's condition

Zeev Rudnick

arxiv: 2606.31544 · v1 · pith:M4LZFNYZnew · submitted 2026-06-30 · 🧮 math.NT · math.DS

The classification of real quadratic fields which satisfy Hammarhjelm's condition

Zeev Rudnick This is my paper

Pith reviewed 2026-07-01 04:11 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords real quadratic fieldsHammarhjelm's conditionunique factorizationMinkowski latticefundamental unitclass number onegenus theorydiscriminant

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The pith

Exactly seven real quadratic fields satisfy Hammarhjelm's condition.

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

Real quadratic fields satisfy Hammarhjelm's condition when their ring of integers has unique factorization and the Minkowski lattice contains no point in a rectangle fixed by the fundamental unit. The paper proves only seven such fields exist, with discriminants 8, 5, 13, 29, 53, 173 and 293. A reader cares because these fields appear in constructions of visible points in algebraic cut-and-project sets. The argument first forces the fundamental unit to be small relative to the discriminant, then exhausts the remaining cases with genus theory and prior classifications of class-number-one fields.

Core claim

A real quadratic field satisfies Hammarhjelm's condition if its ring of integers has unique factorization and its Minkowski lattice contains no point in a certain rectangle determined by the fundamental unit. The paper proves there are exactly seven such fields, namely those with discriminant 8, 5, 13, 29, 53, 173, 293. The proof shows that for such fields the fundamental unit is small relative to the discriminant, then applies genus theory and Biro's classification of class number one fields in Yokoi's family to finish the list.

What carries the argument

Hammarhjelm's condition, which requires both unique factorization in the ring of integers and emptiness of a specific rectangle in the Minkowski lattice of the ring.

If this is right

The seven listed fields are the complete set that can appear in visible-point constructions for cut-and-project sets.
Any further example must violate the small-fundamental-unit bound.
Genus theory combined with existing class-number classifications is sufficient to rule out all other discriminants.
The geometric rectangle condition is only possible when the unit is unusually small.

Where Pith is reading between the lines

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

The same small-unit restriction might be used to classify fields satisfying related lattice conditions in other families of quadratic fields.
One could check whether the seven fields produce visibly denser or sparser point sets than generic quadratic fields in cut-and-project constructions.
The method suggests that adding geometric constraints to unique-factorization problems often reduces the search to a finite list that existing tables can finish.

Load-bearing premise

Every field satisfying Hammarhjelm's condition must have its fundamental unit small relative to the discriminant.

What would settle it

Finding an eighth real quadratic field with unique factorization whose Minkowski lattice has no points in the Hammarhjelm rectangle.

read the original abstract

A real quadratic field satisfies Hammarhjelm's condition if its ring of integers has unique factorization, and the Minkowski lattice of its ring of integers contains no point in a certain rectangle determined by the fundamental unit. Such fields have recently appeared in the study of visible points in algebraic cut-and-project sets. We prove that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition, namely those with discriminant 8, 5, 13, 29, 53, 173, 293. The proof is based on showing that for such fields, the fundamental unit is small relative to the discriminant, together with genus theory and Biro's classification of class number one fields in Yokoi's family.

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

Rudnick classifies exactly seven real quadratic fields meeting Hammarhjelm's condition by reducing to small fundamental units then applying genus theory and Biro's list.

read the letter

The key takeaway is that this paper produces an explicit list of the seven real quadratic fields satisfying Hammarhjelm's condition, namely those with discriminants 8, 5, 13, 29, 53, 173, and 293. The argument shows the condition forces the fundamental unit to be small relative to the discriminant, after which genus theory and Biro's classification of class-number-one fields in Yokoi's family finish the job.

What the paper does well is deliver a previously unavailable complete enumeration by combining a standard internal reduction with existing external results. The listed discriminants align with the known small-unit class-number-one fields, and the proof strategy follows the usual effective approach for these problems without visible circularity or invented quantities.

The soft spots are limited. The reduction to small units is the load-bearing step, and the result inherits whatever completeness Biro's list actually has; any gap there would propagate. The abstract gives the outline but no derivation details, so one would check the unit bound carefully in the full text. Nothing in the stated strategy looks unjustified or mismatched to the claim.

This is for number theorists working on real quadratics, class numbers, or their appearance in cut-and-project sets and visible points. A reader already tracking Yokoi's family or similar classifications would get direct use from the list. It deserves a serious referee to verify the reduction and the application of the cited results.

Referee Report

2 major / 2 minor

Summary. The paper claims to classify all real quadratic fields satisfying Hammarhjelm's condition (unique factorization in the ring of integers together with no Minkowski lattice points in a rectangle determined by the fundamental unit). It proves there are exactly seven such fields, with discriminants 8, 5, 13, 29, 53, 173 and 293, by first establishing that the condition forces the fundamental unit to be small relative to the discriminant and then exhausting the remaining possibilities via genus theory and Biro's classification of class-number-one fields in Yokoi's family.

Significance. If the central reduction holds, the result supplies a complete explicit list of fields relevant to visible points in algebraic cut-and-project sets. The approach follows the standard effective method for class-number-one problems in real quadratics and correctly invokes an external classification (Biro) rather than re-deriving it, which is a strength. The finite list is falsifiable by direct verification for the listed discriminants and supplies a concrete input for applications in lattice theory.

major comments (2)

[Proof strategy paragraph] The reduction that Hammarhjelm's condition implies the fundamental unit is small relative to the discriminant is load-bearing for the entire classification; the manuscript must state the explicit bound obtained and confirm that the seven listed fields satisfy it while all larger candidates are excluded.
[Genus theory paragraph] The application of genus theory to rule out class-number-one fields outside Yokoi's family is invoked without visible detail on which genera are eliminated; a concrete reference to the genus characters or a short table of the relevant quadratic characters would be needed to verify exhaustiveness.

minor comments (2)

The term 'Yokoi's family' is used without a definition or citation in the abstract; a parenthetical description or reference to the original definition should be added for readers outside the immediate subfield.
The seven discriminants are listed in an order that is neither strictly increasing nor grouped by family; reordering them (e.g., 5,8,13,29,53,173,293) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments on clarity. We respond to each major comment below.

read point-by-point responses

Referee: The reduction that Hammarhjelm's condition implies the fundamental unit is small relative to the discriminant is load-bearing for the entire classification; the manuscript must state the explicit bound obtained and confirm that the seven listed fields satisfy it while all larger candidates are excluded.

Authors: We agree that the explicit bound is essential for transparency. The proof obtains a concrete inequality relating the fundamental unit to the discriminant; the revised manuscript will state this bound explicitly, verify it holds for the seven fields with discriminants 8, 5, 13, 29, 53, 173, 293, and confirm that it excludes all larger candidates when combined with the subsequent genus-theory and Biro arguments. revision: yes
Referee: The application of genus theory to rule out class-number-one fields outside Yokoi's family is invoked without visible detail on which genera are eliminated; a concrete reference to the genus characters or a short table of the relevant quadratic characters would be needed to verify exhaustiveness.

Authors: We accept that the current presentation is too terse on this point. The revised version will add a short table or explicit list of the genus characters (the quadratic characters modulo the discriminant) used to eliminate class-number-one fields outside Yokoi's family, thereby making the exhaustiveness of the argument directly verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by proving that Hammarhjelm's condition forces the fundamental unit to be small relative to the discriminant, after which genus theory and Biro's independent external classification of class-number-one fields in Yokoi's family are invoked to list the seven cases. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim rests on external benchmarks rather than internal renaming or ansatz smuggling. This is the standard effective classification approach and yields score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard tools of algebraic number theory plus one specific external classification result; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Genus theory for quadratic fields
Invoked in the proof strategy described in the abstract.
domain assumption Biro's classification of class number one fields in Yokoi's family
Explicitly used together with genus theory to finish the classification.

pith-pipeline@v0.9.1-grok · 5644 in / 1342 out tokens · 30289 ms · 2026-07-01T04:11:35.794482+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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

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Degert, G¨ unter. ¨Uber die Bestimmung der Grundeinheit gewisser reell- quadratischer Zahlk¨ orper. Abh. Math. Sem. Univ. Hamburg 22 (1958), 92–97. 10

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[3]

On the error bounds for visible points in some cut-and-project sets

Gringlaz, Ilya; Kumar, Rishi and Weiss, Barak. On the error bounds for visible points in some cut-and-project sets. Israel J. Math., to appear. arXiv:2505.02631 [math.DS]. 2

work page arXiv
[4]

The density and minimal gap of visible points in some planar quasicrystals

Hammarhjelm, Gustav. The density and minimal gap of visible points in some planar quasicrystals. Discrete Math. 345 (2022), no. 12, Paper No. 113074, 22 pp. 2

2022
[5]

On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets

Kumar, Rishi and Ospina, Carlos. On the diffraction spectrum of the set of visible points in lattices and certain cut and project sets. arXiv:2606.30132 [math.NT] 2

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Quadratics

Mollin, Richard A. Quadratics. CRC Press, 2018. 10

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and Reichardt, H

R´ edei, L. and Reichardt, H. Die Anzahl der durch vier teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlk¨ orpers. J. Reine Angew. Math. 170 (1934), 69–74. 5

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[8]

Arithmetischer Beweis des Satzes ¨ uber die Anzahl der durch vier teil- baren Invarianten der absoluten Klassengruppe im quadratischen Zahlk¨ orper

R´ edei, L. Arithmetischer Beweis des Satzes ¨ uber die Anzahl der durch vier teil- baren Invarianten der absoluten Klassengruppe im quadratischen Zahlk¨ orper. J. Reine Angew. Math. 171 (1934), 55–60. 5 School of Mathematical Sciences, Tel A viv University, Tel A viv 69978, Israel Email address:rudnick@tauex.tau.ac.il

1934

[1] [1]

Yokoi’s conjecture

Bir´ o, Andr´ as. Yokoi’s conjecture. Acta Arith. 106 (2003), no. 1, 85–104. 2, 9

2003

[2] [2]

¨Uber die Bestimmung der Grundeinheit gewisser reell- quadratischer Zahlk¨ orper

Degert, G¨ unter. ¨Uber die Bestimmung der Grundeinheit gewisser reell- quadratischer Zahlk¨ orper. Abh. Math. Sem. Univ. Hamburg 22 (1958), 92–97. 10

1958

[3] [3]

On the error bounds for visible points in some cut-and-project sets

Gringlaz, Ilya; Kumar, Rishi and Weiss, Barak. On the error bounds for visible points in some cut-and-project sets. Israel J. Math., to appear. arXiv:2505.02631 [math.DS]. 2

work page arXiv

[4] [4]

The density and minimal gap of visible points in some planar quasicrystals

Hammarhjelm, Gustav. The density and minimal gap of visible points in some planar quasicrystals. Discrete Math. 345 (2022), no. 12, Paper No. 113074, 22 pp. 2

2022

[5] [5]

On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets

Kumar, Rishi and Ospina, Carlos. On the diffraction spectrum of the set of visible points in lattices and certain cut and project sets. arXiv:2606.30132 [math.NT] 2

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Quadratics

Mollin, Richard A. Quadratics. CRC Press, 2018. 10

2018

[7] [7]

and Reichardt, H

R´ edei, L. and Reichardt, H. Die Anzahl der durch vier teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlk¨ orpers. J. Reine Angew. Math. 170 (1934), 69–74. 5

1934

[8] [8]

Arithmetischer Beweis des Satzes ¨ uber die Anzahl der durch vier teil- baren Invarianten der absoluten Klassengruppe im quadratischen Zahlk¨ orper

R´ edei, L. Arithmetischer Beweis des Satzes ¨ uber die Anzahl der durch vier teil- baren Invarianten der absoluten Klassengruppe im quadratischen Zahlk¨ orper. J. Reine Angew. Math. 171 (1934), 55–60. 5 School of Mathematical Sciences, Tel A viv University, Tel A viv 69978, Israel Email address:rudnick@tauex.tau.ac.il

1934