Construction techniques for some thin sets in duals of compact abelian groups (Q1071991)

Various techniques are presented for constructing \(\Lambda\) (p) sets which are not \(\Lambda (p+\epsilon)\) for all \(\epsilon >0\). The main result is that there is a \(\Lambda\) (4) set in the dual of any compact abelian group which is not \(\Lambda (4+\epsilon)\) for all \(\epsilon >0\). Along the way to proving this, new constructions are given in dual groups in which constructions were already known of \(\Lambda\) (p) not \(\Lambda (p+\epsilon)\) sets, for certain values of p. The main new constructions in specific dual groups are: - there is a \(\Lambda\) (2k) set which is not \(\Lambda (2k+\epsilon)\) in \({\mathbb{Z}}(2)\otimes {\mathbb{Z}}(2)\otimes..\). for all \(2\leq k\), \(k\in {\mathbb{N}}\) and \(\epsilon >0\), and in \({\mathbb{Z}}(p)\otimes {\mathbb{Z}}(p)\otimes..\). (p a prime, \(p>2)\) for \(2\leq k<p\), \(k\in {\mathbb{N}}\) and \(\epsilon >0\) - there is a \(\Lambda\) (2k) set which is not \(\Lambda (4k-4+\epsilon)\) in \({\mathbb{Z}}(p^{\infty})\) for \(2\leq k\), \(k\in {\mathbb{N}}\) and all \(\epsilon >0.\) It is also shown that random infinite integer sequences are \(\Lambda\) (2k) but not \(\Lambda (2k+\epsilon)\) for \(2\leq k\), \(k\in {\mathbb{N}}\) and \(\epsilon >0\).