A sequence \((f_ n)\) of real functions defined on a set X is called quasi-normally convergent to f if there exists a sequence \((\epsilon_ n)\), \(\epsilon_ n\to +0\), such that for every \(x\in X\) there is an index n(x) with \(| f_ n(x)-f(x)| <\epsilon_ n\) for \(n\geq n(x)\). This means that X is the union of a sequence of sets \(X_ k\) such that \((f_ n)\) converges uniformly on each \(X_ k\). The author proves a few simple propositions about Dirichlet sets in which the quasi-normal convergence plays a rôle. A set \(E\subset [0,1]\) is called Dirichlet set if there exists an increasing sequence \((n_ k)\) such that \(e^{2\pi in_ kx}\) tends to 1 uniformly on E. The author calls \(E\subset [0,1]\) a D-set iff there exists an increasing sequence \((n_ k)\) such that \(e^{2\pi in_ kx}\) converges to 1 quasi-normally on E. Thus E is a D-set iff it is the union of an increasing sequence of Dirichlet sets.
Further the interval [0,1] is replaced by \({\mathbb{T}}\) (the torus) and it is proved a.o. (using quasi-normal convergence) that if E is a D-set then the group generated by E is also a D-set. As a corollary one obtains that there exist D-sets E such that not every closed subset \(F\subset E\), \(F\neq E\) is Dirichlet.
The most interesting part of the paper concerns the following problem: let \(\{E_ s\}\) (s\(\in S)\) be a family of Dirichlet sets such that any finite union of them is Dirichlet. How big can one take card S to be sure that \(\cup_{s\in S}E_ s\) is a D-set? Of course that is the case for \(| S| =\omega\). To answer this question one is lead to consider the following property of a family of subsets of \({\mathbb{N}}:\)
Let \({\mathbb{F}}\) be the family. Assume that \(A_ 1\cap...\cap A_ n\) is infinite whenever \(A_ j\in {\mathbb{F}}\), \(n\in {\mathbb{N}}\) and that, for any \(B\subset {\mathbb{N}}\) infinite there exists \(A\in F\) with \(B\setminus A\) infinite. Now let \({\mathfrak p}\) be the least cardinal for which such a family \({\mathbb{F}}\) exists. Then one has \(\omega\leq {\mathfrak p}<2^{\omega}\) [compare D. H. Fremlin, Consequences of Martin’s axiom (Cambridge Tracts Math. 84, 1984; Zbl 0551.03033), p. 2-3]. The author proves (without assuming the continuum hypothesis) that \(| S| <{\mathfrak p}\) is sufficient. There follow some remarks about possible implications between various kinds of “thinness” of a set expressed in terms of Fourier series theory.