Let $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ the composition (1) is a contraction....

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/(d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right))+...$, where $d_j\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}\left(x\right)\ge d_j\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty }{d}_n^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_H$ to denote...

We show that if $X$ is a subspace of a linearly ordered space, then $C_k\left(X\right)$ is a Baire space if and only if $C_k\left(X\right)$ is Choquet iff $X$ has the Moving Off Property.

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{\alpha }+1):={\sum }_{x{\in }_q}\chi \left(a{x}^{p^{\alpha }+1}\right)$ where χ is a non-trivial additive character of the finite field ${}_q$, $q=p^e$ odd, and $a\in {*}_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{\alpha }+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate...

Let B(κ,λ) be the subalgebra of P(κ) generated by ${\left[\kappa \right]}^{\le \lambda }$. It is shown that if B is any homomorphic image of B(κ,λ) then either $\left|B\right|<2^{\lambda }$ or $\left|B\right|={\left|B\right|}^{\lambda }$; moreover, if X is the Stone space of B then either $\left|X\right|\le 2^{2^{\lambda }}$ or $\left|X\right|=\left|B\right|={\left|B\right|}^{\lambda }$. This implies the existence of 0-dimensional compact $T_2$ spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...