On concentrated sets and N-sets.(English)Zbl 0795.03066

A set of reals $$E\subseteq \mathbb{R}$$ is an N-set if there exists a trigonometric series $$\sum^ \infty_{n=1} b_ n\sin 2\pi nx$$ absolutely convergent on $$E$$, but $$\sum| b_ n|=\infty$$. An R- set is defined similarly, except it is required that $$b^ 2_ n$$ does not converge to 0. A set of reals, $$X$$, is a Luzin set iff it is uncountable but it meets every meager set in a countable set. In this paper, it is shown that if there is a Luzin set of cardinality $$\kappa$$, then there exists a set of reals of cardinality $$\kappa$$ which is neither an N-set nor an R-set.
A set of reals $$X$$ is concentrated on a set of reals $$A$$ iff every open set $$G\supseteq A$$ contains all but countably many elements of $$X$$. It is shown that if $${\mathfrak b}> \omega_ 1$$, $$A$$ is a coanalytic set of reals, and $$X$$ is concentrated on $$A$$, then $$X\backslash A$$ is countable.

MSC:

 3e+15 Descriptive set theory
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