Type 1 and 2 sets for series of translates of functions (Q2322491)

Let \(\Lambda\subseteq [0,+\infty)\) be an infinite set without finite accumulation point. \(\Lambda\) is called type 1 if \(\sum_{\lambda\in\Lambda}f(x+\lambda)<\infty\) a.e. for every measurable function \(f:\mathbb{R}\rightarrow [0,+\infty)\); otherwise \(\Lambda\) is type 2. The authors prove besides others: If \(\limsup \frac{a_n}{a_{n-1}}=\infty\) where \(a_n\) denotes the cardinality of \(\Lambda\cap [n,n+1)\), then \(\Lambda\) is type 2. There exists a collection \((\Lambda_n)_{n\in\mathbb{Z}}\) such that \(\Lambda_{n+1}\subseteq\Lambda_n\) for all \(n\in\mathbb{Z}\) and \(\Lambda_n\) is type 1 if \(n\) is odd and type 2 if \(n\) is even. If \(\Lambda=\{\alpha_1, \alpha_2,\dots\}\) where the \(\alpha_i\)s are linear independent over the rationals, then \(\Lambda\) is type 2. There exists a type 1 set \(\Lambda\) containing infinitely many numbers linear independent over the rationals. If \(\Lambda\) and \(\Lambda'\) are type 1, then \(\Lambda+\Lambda'\) is also type 1.