Some remarks on difference sets of Bernstein sets (Q1332586)

There are alternate ways to classify a set \(A\) as ``thick'' or ``thin''; one way would be to examine its difference set \(D(A)= \{x- y\mid x\in A,y\in A\}\). In his paper, the author examines the difference set of a Bernstein set; recall that a subset \(C\) of the real line \(\mathbb{R}\) is called a Bernstein set if both \(C\) and \(\widetilde C= \mathbb{R}\backslash C\) have nonempty intersection with every perfect set. For more information on Bernstein sets, see \textit{J. C. Oxtoby} [Measure and category (1971; Zbl 0435.28011)]. Among other results, the author gets: i) (Theorem 1). There exists a Bernstein set such that both \(D(C)\) and \(D(\widetilde C)\) do not contain an interval. ii) (Theorem 2). There exists a Bernstein set such that \(D(C)\) and \(D(\widetilde C)\) both contain intervals. iii) (Corollary 10). Let \(B\) be any rationally independent set, then \(D(\widetilde B)= \mathbb{R}\).