A set of measure zero containing a copy of any finite set (Q1766457)

Let \(S=\{\sum_{n=2}^\infty\frac{b_n}{n!}: 0\leq b_n\leq n-2\), \(b_n\in\mathbb N\}\). The set~\(S\) is a~perfect nowhere dense set of Lebesgue measure zero. The author proves that the set \(\bigcup_{k=1}^\infty k\cdot S= \{k\cdot x:k\in\mathbb N\) and \(x\in S\}\) contains a~translated copy of any finite set of reals. This is a~modification of a~result of P.~Erdős and S.~Kakutani. The author uses this result to answer a~question of R.~E. Svetic. In the second part of the paper the author proves that for a~measurably large set~\(X\) in a Euclidean space, \(X\cap(X-t)\) is measurably large for almost all \(t\in X\). This improves a~lemma of V.~Bergelson, N.~Hindman, and B.~Weiss.