Real numbers with redundant representations (Q1189089)

Let \((P_ j)\) be a sequence of sets \(P_ j\) of real numbers such that each set is countable and has more than one point. Put \(A=\{x\): \(x=\sum_{j=1}^{+\infty} a_ j\) (\(a_ j\in P_ j\)) uniquely\(\}\) and \(B=\{y\): \(y=\sum_{j=1}^{+\infty} b_ j=\sum_{j=1}^{+\infty} c_ j\) (\(b_ j,c_ j\in P_ j\), \((b_ j)\neq (c_ j)\}\). Unifying and generalizing results by M. Petkovsek and M. Starbird and T. Starbird, the author proves that if the set \(A\cup B\) is of the second category, then \(A\cup B\subset \text{cl}(B)\) (the closure of \(B\)).