We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on \([0,1]^d\) for \(d \in \mathbb{N}\) and use this to obtain results on the ‘horizon problem’ for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.

Some types of compactness in the ideal context are defined and relations between ideal exhaustiveness and equicontinuity of measures are investigated. As applications, some versions of limit theorems involving ideal pointwise convergence of measure sequences and some weak compactness results related to integral functionals are presented.

We show that there is no addition preserving Erd\H{o}s-Sierpi\'nski mapping on any uncountable locally compact abelian Polish group. This generalizes results of Bartoszy\'nski and Kysiak.

We say that $f: \mathR\to\mathR$ is a Hamel function if $f$, considered as a subset of $\mathR^2$, is a Hamel basis of $\mathR^2$. For a Cantor set $C\subset\mathR$ we construct a quasi-continuous Hamel function such that $f\restr(\mathR\setminus C)$ is of Baire class one.

New sufficient conditions for representation of a function as an absolutely convergent Fourier integral are obtained in the paper.

Consider a fixed differentiable curve $K$ (which is the boundary of a convex domain) and a family indexed by $\lambda\in [0,1]$ of differentiable curves $L = L_\lambda$ such that they are the boundary of convex (connected) domains $A_\lambda$. Suppose that for $\lambda_1<\lambda_2$ we have $A_{\lambda_1}\subset A_{\lambda_2}$. Then, the number of $n$-Poncelet pairs is given by $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd$(m,n)=1$. The curve $K$ does not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion.

An almost isometry-invariant set $A\subset \mathbb{R}$ satisfies $|gA \Delta A| < \mathbf{c}$ for any isometry $g$ acting on $\mathbb{R}$, where ${\bf c}$ is the cardinality of the continuum. A shading is any set $S\subseteq \mathbb{R}$ in which $\frac{\mu(S\cap I)}{\mu(I)}$ has the same constant value for every finite interval $I$, for any Banach measure $\mu$. (A Banach measure is a finitely additive, isometry-invariant extension of the Lebesgue measure to $2^{\mathbb{R}}$.) In this paper we prove several theorems that show how these two types of sets are related. We also prove several sum and difference set results for almost isometry-invariant sets. Finally, we completely solve a problem involving subsets of Archimedean sets first posed by R. Mabry and partially solved by K. Neu.

We investigate continuous selfmappings of a real compact interval, which are, in a sense, close to iterable ones, that is embeddable into continuous iteration semigroup. This is a continuation of a research initiated by W.~Jarczyk. We present some necessary and sufficient conditions of weak almost iterability and $M-$weak almost iterability. Classes of functions investigated here are generalizations of a class of almost iterable functions introduced by W. Jarczyk. This refers to the problem posed by E.~Jen.