A strengthening of a theorem of Bourgain-Kontorovich-IV

We give certain generalization of Niederreiter's result concerning famous Zaremba's conjecture on existence of rational numbers with bounded partial quotients.

We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a/q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q = O(p^{C})$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a/q$ are bounded by two.

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with $N$. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of p...

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ...

Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set AB when A and B are sets of positive rational numbers with numerator and denominator less or equal than Q. We extend and slightly improve that lower bound using a different approach.

A theorem of Kurzweil ('55) on inhomogeneous Diophantine approximation states that if $\theta$ is an irrational number, then the following are equivalent: (A) for every decreasing positive function $\psi$ such that $\sum_{q = 1}^\infty \psi(q) = \infty$, and for almost every $s\in\mathbb R$, there exist infinitely many $q\in\mathbb N$ such that $\|q\theta - s\| < \psi(q)$, and (B) $\theta$ is badly approximable. This theorem is not true if one adds to condition (A) the hypoth...

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary.

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_n\theta, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real ...

We prove that a subset $A\subseteq [1, N]$ with \[\sum_{n\in A}\frac{1}{n} \ge (\log N)^{4/5 + o(1)}\] contains a subset $B$ such that \[\sum_{n\in B} \frac{1}{n} = 1.\] Our techniques refine those of Croot and of Bloom. Using our refinements, we additionally consider a number of questions regarding unit fractions due to Erd\H{o}s and Graham.

In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of values of numerators of differences of consecutive elements in this set. After proving an asymptotic result for these frequencies, we use estimates coming from incomplete Kloosterman sums to generalize our result to subintervals of [0,1].