Trigonometric series with gaps
Let a, \(m_ 1,...,m_ n\in {\mathbb{Z}}\) and \(m_ 1>0\), \(m_ k>m_ 1+...+m_{k-1}\), \(k=1,...,n\). Then the set \(\{a+\epsilon_ 1m_ 1+...+\epsilon_ nm_ n,\quad \epsilon_ i=0\quad or\quad 1,\quad i=1,...,n\}\) is called a tracing segment of length \(2^ n\). For instance, any arithmetic progression of \(2^ n\) terms is a tracing segment of length \(2^ n\). The author investigates the sequences containing arbitrarily long tracing segments, and obtains some general results. As a consequence he proves that the sequence of primes contains arbitrarily long tracing segments. These results are then used in order to give an affirmative answer to a conjecture of \textit{W. Rudin} [J. Math. Mech. 9, 203-227 (1960; Zbl 0091.058)] concerning the density of the so- called \(\Lambda\) (p) sequences, in the case \(0<p\leq 2\).
sequences containing arbitrarily long tracing segments
scientific article; zbMATH DE number 3865420