Let L be a lattice and let F be a fundamental domain for L. This exercise sketch

Question

Let L be a lattice and let F be a fundamental domain for L. This exercise sketches a proof that lim R rightarrow infinity #(B_R(0) intersection L)/Vol(B_R(0)) = 1/Vol(F). (a) Consider the translations of F that are entirely contained within B_R (0), and also those that have nontrivial intersection with B_R (0). Prove the inclusion of sets union_ v element L F + v B_R(0) (F + v) B_R(0) union_nu element L (F + nu) intersection B_R(0) notequalto (F +v) (b) Take volumes in (a) and prove that # {v element L: F + v B_R (0)} middot Vol (F) lessthanorequalto Vol(B_R(0)) lessthanorequalto # {element L: (F + nu) intersection B_R (0) notequalto 0} middot Vol(F). (c) Prove that the number of translates F + nu that intersect B_R(0) without being entirely contained within B_R(0) is comparatively small compared to the number of translates F_v that are entirely contained within B_R(0). (This is the hardest part of the proof.) (d) Use (b) and (c) to prove that Vol (B_R (0)) = #(B_R(0) intersection L). Vol (F) + (smaller term). Divide by vol(B_R(0)) and let R rightarrow infinity to complete the proof of (7.63).

Explanation / Answer

I have written the proof for all the equations for calculating the functional domain by having the Lattice point for their respective points. I have added the comments for each part of the section and the final output of it.

Let me explain you in more simpler and step-by-step procedure:-

A)

Step-1:

The given equation is UVL(F+v)   BR (0)   UVL (F+v).

B)

Step-2:

C)

Step-3:

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.