Some results on mid-point sets of sets of natural numbers
For two sets \(P\), \(Q\subset\mathbb{R}\) the mid-point set is defined as \[ M(P,Q):= \Biggl\{{x+y\over 2}\mid x\in P,\,y\in Q\Biggr\}. \] In particular, for \(P= Q\) is written \(M(P,P)= M(P)\). With the counting function \(A(n)=\sum_{a\in A,a\leq n}1\) of a set \(A\subset\mathbb{N}\) the upper asymptotic density of \(A\) is given by \(\limsup_{n\to\infty}{A(n)\over n}:=\overline d(A)\) and, if it exists, \(\lim_{n\to\infty}{A(n)\over n}:= d(A)\) the asvmptotic density of \(A\). In this note are studied some properties of mid-point sets. We mention the following two results: (1) Let \(A\subset\mathbb{N}\) with \(\overline d(A)= 1\). Then each rational number \(>0\) can be represented as the mid-point for infinite number of pairs \((g, h)\) with \(g,h\in A\). (2) Let \(A\subset\mathbb{N}\) with \(d(A)> 0\). Then the mid-point set \(M(A)\) is not nowhere dense in \([1,\infty]\).
scientific article; zbMATH DE number 5009698