Construction of normal numbers via pseudo-polynomial prime sequences (Q2925471)

Every real number \(\theta \in [0,1)\) has a unique \(q\)-ary expansion \(\theta = \sum_{k\geq 1} a_k q^k\) with \(a_k\in \{0,1,\ldots,q-1 \}\), where \(q\geq 2\) is a fixed integer. Let \({\mathcal N}(\theta,d_1\cdots d_{\ell},N)\) denote the number of occurrences of the block \(d_1\cdots d_{\ell}\) amongst the first \(N\) digits of this expansion. Then the number \(\theta\) is called normal of order \(\ell\) in base \(q\) if for each block of length \(\ell\) the frequency of occurrences \({\mathcal N}(\theta,d_1\cdots d_{\ell},N)\) tends to \(q^{-\ell}\).NEWLINENEWLINEIn the present paper, the author constructs normal numbers in base \(q\) by concatenating \(q\)-ary expansions of pseudo-polynomials evaluated at primes, extending a recent result of the same author and \textit{R. F. Tichy} [J. Integer Seq. 16, No. 2, Article 13.2.12, 17 p. (2013; Zbl 1294.11123)].