A strengthening of a theorem of Bourgain-Kontorovich-III

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...

We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.

Motivated by a WhattsApp message, we find out the integers $x> y\ge 1$ such that $(x+1)/(y+1)=(x\circ(y+1))/(y\circ (x+1))$, where $\circ$ means the concatenation of the strings of two natural numbers (for instance $783\circ 56=78356$). The discussion involves the equation $x(x+1)=10y(y+1)$, a slight variation of Pell's equation related to the arithmetic of the Dedekind ring $\mathbb{Z}[\sqrt{10}]$. We obtain the infinite sequence $\mathcal{S}=\{(x_n,y_n)\}_{n\ge 1}$ of all t...

Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $\tau\geq 2$ and any non-empty proper subset $J(b)$ of $\mathcal A$, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series in $b$ with coefficients in $J(b)$ and with the irrationality exponent (in terms of Gaussian int...

We consider a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For this expansion, we apply the method of Rockett and Sz\"usz from [6] and obtained the solution of its Gauss-Kuzmin type problem.

For each integer $b \geq 3$ and every $x \geq 1$, let $\mathcal{N}_{b,0}(x)$ be the set of positive integers $n \leq x$ which are divisible by the product of their nonzero base $b$ digits. We prove bounds of the form $x^{\rho_{b,0} + o(1)} < \#\mathcal{N}_{b,0}(x) < x^{\eta_{b,0} + o(1)}$, as $x \to +\infty$, where $\rho_{b,0}$ and $\eta_{b,0}$ are constants in ${]0,1[}$ depending only on $b$. In particular, we show that $x^{0.526} < \#\mathcal{N}_{10,0}(x) < x^{0.787}$, for ...

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...

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which this partition avoided was uniquely avoidable. We prove that the set of numerators of con...

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of th...

For a real number $q\in(1,2)$ and $x\in[0,1/(q-1)]$, the infinite sequence $(d_i)$ is called a \emph{$q$-expansion} of $x$ if $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i},\quad d_i\in\{0,1\}\quad\textrm{for all}~ i\ge 1. $$ For $m=1, 2, \cdots$ or $\aleph_0$ we denote by $\mathcal{B}_m$ the set of $q\in(1,2)$ such that there exists $x\in[0,1/(q-1)]$ having exactly $m$ different $q$-expansions. It was shown by Sidorov (2009) that $q_2:=\min \mathcal{B}_2\approx1.71064$, and later ask...