Thin sets, differentiable functions, and the category method (Q1378968)

The author considers some properties of \(M_0\)- sets. Let \((m_k)_1^\infty\) be a sequence of integers exceeding 1, let \(y_k=(m_1m_2\dots m_k)^{-1}\), and let \(E\) be the set of all sums \(\sum_1^\infty\delta_k y_k\) with \(\delta_k=0, 1\). Suppose that \(\sup m_k<\infty\). The main results of this paper are the next three interesting theorems: Theorem 1. Let \(F\) be a compact perfect linear set; then \(E\cdot F\) is an \(M_0\)-set. Theorem 2. Let \(f\) be a function in \(C^1[0,1]\) such that \(f'\) has continuous distribution relative to the product measure \(\mu\) in \(E\). Then \(f(E)\) is an \(M_0\)-set. Theorem 3. Suppose that \(m_k\geq 3\) infinitely often, \(F\) is compact and perfect. Then there is a continuous map of \(E\) into \(F\) such that the set of products \(\{xh(x): x\in E\}\) is an \(M_0\)-set.