In 1955, Roth established that if $\xi$ is an irrational number such that there are a positive real number $\epsilon$ and infinitely many rational numbers $p/q$ with $q\ge 1$ and $|\xi -p/q|\<q^{-2-\epsilon }$, then $\xi$ is transcendental. A few years later, Cugiani obtained the same conclusion with $\epsilon$ replaced by a function $q\mapsto \epsilon \left(q\right)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi -p/q|\<q^{-2-\epsilon \left(q\right)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it...

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $\alpha ̲:=(\alpha ,\alpha ²,...,{\alpha }^d)$. It is well-known that $c\left(\alpha ̲\right):=limin{f}_{q\to \infty }{q}^{1/d}·\left|\right|q\alpha ̲\left|\right|>0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c\left(\alpha ̲\right)n^{-1/d}\le c\left(n\alpha ̲\right)\le nc\left(\alpha ̲\right)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c\left(n\alpha ̲\right)\le C{n}^{-1/d}$ for any integer n ≥ 1.

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...

Let $\alpha \&gt;1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

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 prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f:{ℝ}^s\to ℝ$ can be approximated with arbitrary accuracy by an infinite sum ${\sum }_{r=1}^{\infty }{H}_r(x₁,...,x_s)\in C^{\infty }\left({ℝ}^s\right)$ of analytic functions $H_r$, each solving the same system of universal partial differential equations, namely $P(x_{\sigma };H_r,\partial H_r/\partial x_{\sigma },...,\partial ⁵H_r/\partial x{⁵}_{\sigma }⁵)=0$ (σ = 1,..., s).

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m,p}\left(\Omega \right)$ into $L^S\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m,p}\left(\Omega \right)$ into $L^{\phi }\left(\Omega \right)$...............................................................