We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$ , where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$ . The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$,

$$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$

$L^{p}$

$p>1$

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$ , a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages

$$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

$f$

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$

We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$ . A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$ , the group of $p$ -adic integers. However, if the dual group, $\widehat{G}$ , contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$ , $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$ ; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$ .

Consider an irrational rotation of a unit circle and a real continuous function. A point is declared ‘maximizing’ if the growth of the ergodic sums at this point is maximal up to an additive constant. In the case of two-sided ergodic sums, the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build, for the ‘usual’ one-sided ergodic sums, examples in Hölder or smooth classes, indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but also for the transformation 2x mod 1. For this latter transformation, it is known that any Hölder continuous function has a maximizing point.

In this paper we construct a compact set K of zero Hausdorff dimension that satisfies certain ‘arithmetic-type’ thickness properties. The concept of ‘arithmetic thickness’ has its origins in applications to harmonic analysis, introduced in a paper by Lebedev and Olevskiĭ. For example, there are no spectral sets whose ‘essential boundary’ can contain the above set K.

Questions of Haight and of Weizsäcker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non-empty intervals IFI∞⊂[½,1) such that Σ∞n = 1f(nx) = +∞ for every x εI∞, and Σ∞n = 1f(nx) >+∞ for almost every x εIf. The function f may be taken to be the characteristic function of an open set E.

We give examples of non-integrable measurable functions for which there are ‘many’ rotations such that the arithmetic (ergodic) averages exist for almost every x. We also show that if the above ergodic averages exist for almost every x for a set of rotations of positive measure, then the function should be integrable on [0, 1].

We show that for a Henstock-Kurzweil integrable function f for every ∈ > 0 one can choose an upper semicontinuous gage function δ, used in the definition of the HK-integral if and only if |f| is bounded by a Baire 1 function. This answers a question raised by C. E. Weil.

We show that there exists an open set H⊆[0, 1] × [0, 1] with λ2(H) = 1 such that for any ε > 0 there exists a set E satisfying and H contains the product set E × E but there is no set S with and S × S ⊆ H. Especially this property is verified for sets of the form H = where the sets Ei are independent and . The results of this paper answer questions of M. Laczkovich and are related to a paper of D. H. Fremlin.

We prove that for every dense Gδ set H, there exists a continuous function f, such that f intersects every analytic function in finitely many points and f is infinitely differentiable exactly at the points of H. This answers a problem of S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss. They proved a result which implies that every continuous function with finite intersections with analytic functions is infinitely differentiable at the points of a dense Gδ set.