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Normality of numbers generated by the values of polynomials at primes. (English) Zbl 0881.11062

Let \(r\geq 2\) be a fixed integer and let \(\theta=0.a_1a_2a_3\dots\) be the \(r\)-adic expansion of a real number \(\theta\) with \(0<\theta<1\). Then \(\theta\) is said to be normal to base \(r\) if, for any block \(b_1\dots b_l\in\{0,1,\dots,r-1\}^l\), \(\lim_{n\to\infty} n^{-1}N(\theta;b_1\dotsb_l;n)= r^{-l}\), where \(N(\theta,b_1\dots b_l;n)\) is the number of indices \(i\leq n-l+1\) such that \(a_i=b_1, a_{i+1}=b_2,\dots, a_{i+l+1}=b_l\). Let \((m)_r\) denote the \(r\)-adic expansion of an integer \(m\geq 1\). For any infinite sequence \(\{m_1,m_2,\dots\}\) of positive integer, we consider the number \(\theta=0.(m_1)_r (m_2)_r\dots\) whose \(r\)-adic expansion is obtained by the concatenation of the strings \((m_1)_r, (m_2)_r,\dots\) of \(r\)-adic digits; which will be written simply as \(\theta=0.m_1 m_2m_3\dots(r)\).
Copeland and Erdös proved that the number \(\theta\) defined above is normal to base \(r\) for any increasing sequence \(\{m_1,m_2,\dots\}\) of positive integers such that, for every positive \(\rho<1\), the number of \(m_i\)’s up to \(x\) exceeds \(x^\rho\) provided \(x\) is sufficiently large. In particular, the normality of the number \(\alpha=0. 23571113\dots(r)\) defined by the prime was established for the first time. Davenport and Erdös proved that the number \(0.f(1) f(2)\dots f(n)\dots(r)\) is normal to base \(r\), where \(f(x)\) is any nonconstant polynomial taking positive integral values at all positive integers. We prove the following: Theorem. Let \(f(x)\) be as above. Then, the number \[ \alpha(f)= 0.f(2)f(3) f(5)f(7) f(11)f(13)\dots(r) \] defined by the values of \(f(x)\) at primes is normal to base \(r\). More precisely, for any block \(b_1\dots b_l\in\{0,1,\dots, r-1\}^l\), we have \[ n^{-1} N(\alpha(f); b_1\dots b_l;n)= r^{-l}+o((\log n)^{-1}) \] as \(n\to\infty\), where the implied constant depends possibly on \(r\), \(f\), and \(l\).
Reviewer: Y.Nakai (Kofu)

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A63 Radix representation; digital problems
Full Text: DOI EuDML