Radical bound for Zaremba's conjecture

Abstract

AbstractFamous Zaremba's conjecture (1971) states that for each positive integer , there exists a positive integer , coprime to , such that if you expand a fraction into a continued fraction , all of the coefficients ’s are bounded by some absolute constant , independent of . Zaremba conjectured that this should hold for . In 1986, Niederreiter proved Zaremba's conjecture for numbers of the form with and for with . In this paper, we prove that for each number , there exists , coprime to , such that all of the partial quotients in the continued fraction of are bounded by , where is the radical of an integer number, that is, the product of all distinct prime numbers dividing . In particular, this means that Zaremba's conjecture holds for numbers of the form with , generalizing Neiderreiter's result. Our result also improves upon the recent result by Moshchevitin, Murphy and Shkredov on numbers of the form , where is an arbitrary prime and is sufficiently large.