The Eisenpint Schmidt arrangement for Q(sqrt(-3)) consists exactly of all primitive Eisenstein circle packings, satisfying strong approximation, a density-one local-global principle, and quadratic reciprocity obstructions but no cubic ones.
On Zaremba's Conjecture
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abstract
Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.
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Eisenstein circle packings and the Eisenpint Schmidt arrangement
The Eisenpint Schmidt arrangement for Q(sqrt(-3)) consists exactly of all primitive Eisenstein circle packings, satisfying strong approximation, a density-one local-global principle, and quadratic reciprocity obstructions but no cubic ones.