ID: 1503.06132

A strengthening of a theorem of Bourgain-Kontorovich-IV

March 20, 2015

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Quantitative generalizations of Niederreiter's result concerning continuants

September 8, 2011

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Igor D. Kan, Natalia A. Krotkova
Number Theory

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

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On a modular form of Zaremba's conjecture

November 18, 2019

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Nikolay G. Moshchevitin, Ilya D. Shkredov
Number Theory
Combinatorics

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.

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On the distribution of partial quotients of reduced fractions with fixed denominator

October 25, 2022

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Christoph Aistleitner, Bence Borda, Manuel Hauke
Number Theory

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...

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On the Lebesgue measure of sum-level sets for continued fractions

January 13, 2009

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Marc Kesseböhmer, Bernd O. Stratmann
Dynamical Systems
Number Theory

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 ...

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A note on product sets of rationals

May 6, 2014

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Javier Cilleruelo
Number Theory
Combinatorics

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.

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An analogue of a theorem of Kurzweil

December 18, 2014

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David Simmons
Number Theory

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...

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A note on the number of Egyptian fractions

October 11, 2022

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Noah Lebowitz-Lockard, Victor Souza
Number Theory

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.

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Levy-Khintchin Theorem for best simultaneous Diophantine approximations

June 26, 2019

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Yitwah Cheung, Nicolas Chevallier
Number Theory
Dynamical Systems

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 ...

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On further questions regarding unit fractions

April 10, 2024

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Yang P. Liu, Mehtaab Sawhney
Number Theory

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.

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A Note on Farey Fractions With Odd Denominators

July 13, 2009

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Alan K. Haynes
Number Theory

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].

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