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.


03E15 Descriptive set theory
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