On Zaremba’s conjecture

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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Authors

Jean Bourgain

School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540

Alex Kontorovich

Stony Brook University, Stony Brook, NY

Current address:

Department of Mathematics, Yale University, New Haven, CT 06511