Continued Fractions with Partial Quotients Bounded in Average

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator.

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...

We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infini...

We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main diffculty is to control the length of these subinter...

It is shown that for some explicit constants $c>0, A>0$, the asymptotic for the number of positive non-square discriminants $D<x$ with fundamental solution $\varepsilon_D< x^{\frac 12+\alpha}, 0<\alpha <c$, remains preserved if we require moreover $\mathbb Q(\sqrt D)$ to contain an irrational with partial quotients bounded by $A$.

This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab == c (mod n) with a,b,c in the set. In a previous paper we constructed an infinite sequence of integers (n_i)_{i > 0} and product-free sets S_i in Z/n_iZ such that the density |S_i|/n_i tends to 1 as i tends to infinity, where |S_i|$ denotes the cardinality of S_i. Here we obtain matching, up to constants, upper and lower bounds on the maximal atta...

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant A. Several new theorems concerning this conjecture were proved by Bourgain and Kontorovich in 2011. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A = 50 has positive proportion in natural numbers. In...

Let $\psi: \mathbb{N} \to [0,1/2]$ be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi(q)/q$, provided that the series $\sum_{q=1}^\infty \varphi(q) \psi(q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $\alpha$ the number of ...

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.

For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsets of $$ \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}. $$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in 2013 by Cilleruelo, Kumchev, Luca, Ru\'{e} and Shparlinski..