Construction of normal numbers by classified prime divisors of integers
For positive integers \(d\) and \(n\) with \(d\geq2\) we denote by \[ n=\varepsilon_0(n)+\varepsilon_1(n)d+\cdots+\varepsilon_t(n)d^t \] the \(d\)-ary representation with \(\varepsilon_i(n)\in A_d:=\{0,1,\ldots,d-1\}\) for \(0\leq i\leq t\) and \(\varepsilon_t(n)\neq0\). Similarly, for a real \(x\in[0,1)\) we denote by \[ x=0.a_1a_2a_3\ldots \] with \(a_i\in A_d\) its \(d\)-ary expansion. We call a real \(x\in[0,1)\) normal if for any positive integer \(k\) and any block \(B\in A_d^k\) of digits of length \(k\) the number of occurrences of this block within the \(d\)-ary expansion is equal to the expected limiting frequency, namely \(d^{-k}\). In the present paper the authors construct normal numbers using classified prime divisors. Let \(\mathbb{P}\) be the set of all primes. Given an integer \(d\geq2\) they consider partitions of \(\mathbb{P}\) in disjoint sets such that \[ \mathbb{P}=\mathcal{R}\cup\mathbb{P}_0\cup\cdots\cup\mathbb{P}_{d-1}. \] In the next step they associate with every subset \(\mathbb{P}_j\) the digit \(j\) and the empty word \(\Lambda\) with the reste (the elements in \(\mathcal{R}\)), \textit{i.e.} they define \(H:\mathbb{P}\to A_d\) by \(H(p)=j\) if \(p\in\mathbb{P}_j\), (\(j=0,1,\ldots,d-1\)) and \(H(p)=\Lambda\) if \(p\in\mathcal{R}\). Let \(A_d^*\) be the set of finite words over \(A_d\). Now the prime factor decomposition of every positive integer \(n\) induces a function \(R:\mathbb{N}\to A_d^*\) as follows. We set \(R(1)=\Lambda\) and if \(n=p_1^{a_1}\cdots p_r^{a_r}\), where \(p_1<\cdots<p_r\) are primes and \(a_i\in\mathbb{N}\), then we set \[ R(n)=H(p_1)\ldots H(p_r). \] Their first kind of results states, that if the primes are sufficiently uniformly distributed in the sets \(\mathbb{P}_0,\ldots,\mathbb{P}_{d-1}\), then the number \[ \xi=0.R(1)R(2)R(3)\ldots \] is normal to base \(d\). Another construction, they provide, is the following. Let \(k\geq1\) be a fixed integer and set \(E(n):=n(n+1)\cdots(n+k-1)\). Moreover for \(n\in\mathbb{N}\) we define \[ e(n)=\prod_{p^a\| E(n),\,p\leq k-1}p^a. \] As above they associate with every positive integer a word over \(A_k\) as follows: \(h_n(p)=\Lambda\) if \(p\mid e(n)\) and \(h_n(p)=\ell\) if \(p\mid n+\ell\) and \(\mathrm{gcd}(p,e(n))=1\). If \(E(n)=p_1^{a_1}\cdots p_r^{a_r}\) with \(p_1<\cdots<p_r\) primes and \(a_i\in\mathbb{N}\), then we set \[ S(E(n))=h_n(p_1)\ldots h_n(p_r). \] Their second kind of results states that the number \[ \xi=0.S(E(1))S(E(2))S(E(3))\ldots \] is normal to base \(k\).
scientific article; zbMATH DE number 5989421