Just give me time to veil my Apollonian form in a pair of trousers, and I appear.
"Within an Inch of His Life" by Emile Gaboriau
The Apollonians came out to oppose him, but he drove them, terrified and dismayed, within their walls.
"The History of Rome; Books Nine to Twenty-Six" by Titus Livius
Nietzsche divided art into Apollonian and Dionysian.
"The Literature of Ecstasy" by Albert Mordell
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These include the local-global problem for integral Apollonian gaskets and Zaremba’s Conjecture on ﬁnite continued fractions with absolutely bounded partial quotients.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
In fact, Apollonian gaskets are rigid, in the sense that one can be mapped to any other by M¨obius transformations.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
Apollonian group Γ consists of all quadruples corresponding to curvatures of four mutually tangent circles in the gasket G .
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
But there are standard methods (smoothing and later unsmoothing) which overcome these technical irritants. A version of the above works with the Apollonian group Γ in place of Z, once one overcomes a number of further technical obstructions.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
This conjecture is stated by Graham-Lagarias-Mallows-Wilks-Yan [GLM+03, p. 37], in the ﬁrst of a lovely series of papers Apollonian gaskets and generalizations.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
Recall from (3.15) that the orbit O = Γ · v0 of the root quadruple v0 under the Apollonian group Γ contains all quadruples of curvatures, and in particular its entries consist of all curvatures in G .
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
O . A heuristic analogy between Zaremba and the Apollonian problem is actually already given in [GLM+03, p. 37], but it is crucial for us that both problems are exactly of the form (3.28); compare to (2.28).
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
For Theorem A, one needs the theory of automorphic representations for the full Apollonian group, as hinted to at the end of §3.1.6.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
We now leave the discussion of the Apollonian problem, returning to it again in §5.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
As in the Apollonian case, the Pythagorean form Q has a special (determinant one) orthogonal group preserving it: SOQ := {g ∈ SL3 : Q(g · x) = Q(x)}.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
We are ﬁnally in position to relate this Pythagorean problem to the Apollonian and Zaremba’s.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
We use different strategies to prove (5.17) for the Pythagorean and Zaremba settings X = P or Z, versus the Apollonian setting X = A, so we present them individually.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
The above strategy fails for the Apollonian problem, because the Hausdorff dimension (3.4) is a ﬁxed invariant which refuses to be adjusted to suit our needs.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
Instead, we recall that the Apollonian group Γ contains the special (arithmetic) subgroup Ξ from (3.35).
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
Jean Bourgain and Elena Fuchs. A proof of the positive density conjecture for integer Apollonian circle packings. J.
From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits
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