Radical bound for Zaremba's conjecture

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