A strengthening of a theorem of Bourgain-Kontorovich-IV

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$ being bounded by an absolute constant $A.$ Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 ...

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$ being bounded by an absolute constant $A.$ Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 ...

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,\ldots,d_{k}],$ with all partial quotients $d_1,d_2,\ldots,d_{k}$ being bounded by an absolute constant $A.$ Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with...

Zaremba's conjecture (1971) states that every positive integer number can be represented as a denominator (continuant) of a finit continued fraction with all partial quotients being bounded by an absolute constant A. Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A = 50 has positive proportion in N. In 2014 the author with D. A. ...

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ whose partial quotients $d_1,d_2,...,d_{k}$ belong to a finite alphabet $\A\subseteq\N.$ In this paper it is proved for an alphabet $\A,$ such that the Hausdorff dimension $\delta_{\A}$ of the set of infinite continued fractions whose partial quotients belong to $\A,$ that the set of number...

It is shown that there is a constant A and a density one subset S of the positive integers, such that for all q in S there is some 1<=p<q, (p, q)=1, so that p/q has all its partial quotients bounded by A.

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.

We prove there exists a density one subset $\dd \subset \N$ such that each $n \in \dd$ is the denominator of a finite continued fraction with partial quotients bounded by 5.

We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.

Famous Zaremba's conjecture (1971) states that for each positive integer $q\geq2$, there exists positive integer $1\leq a <q$, coprime to $q$, such that if you expand a fraction $a/q$ into a continued fraction $a/q=[a_1,\ldots,a_n]$, all of the coefficients $a_i$'s are bounded by some absolute constant $\mathfrak k$, independent of $q$. Zaremba conjectured that this should hold for $\mathfrak k=5$. In 1986, Niederreiter proved Zaremba's conjecture for numbers of the form $q=2...