mathematics

Zimmert set

Zimmert set is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Zimmert set rather than just read about it. In short: In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group. Definition Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d).

Key takeaways

Zimmert set belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Zimmert set to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Zimmert set from memory before moving on to harder problems.

Reference excerpt

In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.

Definition Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).

Property For all but a finite number of d we have z(d) > 1: indeed this is true for all d > 10476.

Application Let Γd denote the Bianchi group PSL(2,Od), where Od is the ring of integers of. As a subgroup of PSL(2,C), there is an action of Γd on hyperbolic 3-space H3, with a fundamental domain. It is a theorem that there are only finitely many values of d for which Γd can contain an arithmetic subgroup G for which the quotient H3/G is a link complement. Zimmert sets are used to obtain results in this direction: z(d) is a lower bound for the rank of the largest free quotient of Γd and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.

References

Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.

Worked examples

Example 1 — a first encounter with Zimmert set

Start with the simplest possible case. Write down what Zimmert set claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Zimmert set before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Zimmert set ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Zimmert set

In research: Zimmert set appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Zimmert set in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Zimmert set is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hyperbolic geometry, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Zimmert set outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Zimmert set in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Zimmert set means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Zimmert set out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Zimmert set in simple terms?

In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group. Definition Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d).

Why does Zimmert set matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Zimmert set?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Zimmert set.