On Zaremba’s conjecture


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.


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