Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let $\omega \left(\alpha \right):=sup\theta \ge 1:limin{f}_{n\to \infty }{n}^{\theta }\left|\right|n\alpha |=0$ $.Forany\alpha withagiven\omega \left(\alpha \right),wegivesomesharpestimatesfortheHausdorffdimensionoftheset$ $E_{\phi }\left(\alpha \right)$ := y ∈ ℝ: ||nα -y|| < φ(n) for infinitely many n, where ||·|| denotes the distance to the nearest integer.

We study the Diophantine equations ${(k!)}^n-k^n={(n!)}^k-n^k$ and ${(k!)}^n+k^n={(n!)}^k+n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.

Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A+B\surd (-1)={(m+\surd (-1))}^r$. We prove that the Diophantine equation ${\left|A\right|}^x+{\left|B\right|}^y={(m²+1)}^z$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r>{10}^{74}$ or $m>{10}^{34}$. Our result is an explicit refinement of a theorem due to F. Luca.

Let g ≥ 2 be an integer and ${}_g\subset \mathbb{N}$ be the set of repdigits in base g. Let ${}_g$ be the set of Diophantine triples with values in ${}_g$; that is, ${}_g$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set ${}_g$. We prove effective finiteness results for the set ${}_g$.

The Littlewood conjecture in Diophantine approximation claims that $$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·\parallel q\beta \parallel =0$$ holds for all real numbers $\alpha$ and $\beta$, where $\parallel ·\parallel$ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·{\left|q\right|}_p=0$$ holds for every real number $\alpha$ and every prime number $p$, where ${|·|}_p$ denotes the $p$-adic absolute value normalized by ${\left|p\right|}_p=p^{-1}$. We survey the known results on these conjectures and highlight recent developments. ...

Let $W(m,n;\underline{\psi })$ denote the set of ${\psi }_1,...,{\psi }_n$–approximable points in ${ℝ}^{mn}$. The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underline{\psi }$. Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$. It can not be removed for $m=n=1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m=2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

We consider systems of equations of the form $x_i+x_j=x_k$ and $x_i·x_j=x_k$, which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation ${(-1)}^{u}r{a}^x+{(-1)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a\&gt;1$, $b\&gt;1$, $c\&gt;0$, $r\&gt;0$ and $s\&gt;0$. When $gcd(ra,sb)=1$, we show that $N\le 3$ except for a finite number of cases all of which satisfy $max(a,b,r,s,x,y)\&lt;2·{10}^{15}$ for each solution; when $gcd(a,b)\&gt;1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.

Let $F_n$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$: 1) ${inf}_{n\in \mathbb{N}}\left|\right|F_n\alpha \left|\right|\le \frac{\varphi -1}{\varphi +2}$, 2) ${lim\; inf}_{n\to \infty }\left|\right|F_n\alpha \left|\right|\le \frac{1}{5}$, 3) ${lim\; inf}_{n\to \infty }\left|\right|{\varphi }^n\alpha \left|\right|\le \frac{1}{5}$. These results are the best possible.

Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{\sigma }$. We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0,\sigma }$ is continuous at σ₀ as the function of the parameter $\sigma \in \overline{₀}$ if and only if $HD\left(J_{0,\sigma ₀}\right)\ge 4/3$. Since $HD\left(J_{0,\sigma }\right)>4/3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $HD\left(J_{0,\sigma }\right)$ on an open and dense subset of...

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0,\sigma }$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0,\sigma }$, given by Urbański and Zinsmeister. The closure of the limit set of our IFS ${\varphi }_{\sigma ,\alpha }^{n,k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

In this paper, we study triples $a,b$ and $c$ of distinct positive integers such that $ab+1,ac+1$ and $bc+1$ are all three members of the same binary recurrence sequence.

Let $Q={\left(q_n\right)}_{n=1}^{\infty }$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties. ...