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On a problem on normal numbers raised by Igor Shparlinski. (English) Zbl 1231.11086

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 \(\varepsilon_i(n)\in E:=\{0,1,\ldots,d-1\}\) for \(0\leq i\leq t\) and \(\varepsilon_t(n)\neq0\). To this representation we associate the word \(\overline{n}=\varepsilon_0(n)\varepsilon_1(n)\ldots\varepsilon_t(n)\in E^{t+1}\). Furthermore for a real \(x\in[0,1)\) we denote by \[ x=0.a_1a_2a_3\ldots \] with \(a_i\in E\) the \(d\)-ary expansion of \(x\). We call a real \(x\in[0,1)\) normal if for any positive integer \(k\) and any block \(B\in E^k\) 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 \(d^{-k}\).
The authors of the present answer questions raised by Igor Shparlinski in connection with the construction of normal numbers by describing their \(d\)-ary expansion. Let \(f\in\mathbb Z[X]\) be a polynomial with positive leading coefficient and \(P(n)\) denote the largest prime factor of \(n\). Then they are able to show that \[ 0.\overline{f(P(2))}\,\overline{f(P(3))}\,\overline{f(P(4))}\ldots\overline{f(P(n))}\ldots \] and \[ 0.\overline{f(P(2+1))}\,\overline{f(P(3+1))}\,\overline{f(P(5+1))}\ldots\overline{f(P(p+1))}\ldots, \] where \(p\) runs over the primes, are normal numbers.

MSC:

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A41 Primes
11N37 Asymptotic results on arithmetic functions
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References:

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