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The spectral geometry of the apollonian packing

✍ Scribed by Robert Brooks

Publisher
John Wiley and Sons
Year
1985
Tongue
English
Weight
456 KB
Volume
38
Category
Article
ISSN
0010-3640

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✦ Synopsis

Now let E be a fixed constant, and let x be an arbitrary point in supp(T) lying in this copy of Pick. As in [3], it follows from bounded mean curvature that there is a uniform lower bound for the area of an &-ball in T about x. This &-ball may meet at most a fixed number of copies of pick. Hence we have the bound #(&Pick)) 5 cg area(T)

for some c3, so that combining this with our previous estimate, we establish Lemma 4.

Combining Lemmas 3 and 4, we see that vol(int(T)) = vol,(int(T)) + vol,(int(T))

5 c4( area( T ) ) ,

where c4 = 1 + c2, establishing the theorem with h = 1/c4.

Acknowledgments.

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