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Construction of Normal Numbers Via Pseudo Polynomial Prime Sequences

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Construction of Normal Numbers Via Pseudo Polynomial Prime Sequences CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES MANFRED G. MADRITSCH Abstract. In the present paper we construct normal numbers in base q by concatenating q-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author. 1. Introduction Let q ≥ 2 be a positive integer. Then every real θ 2 [0; 1) admits a unique expansion of the form X k θ = akq (ak 2 f0; : : : ; q − 1g) k≥1 called the q-ary expansion. We denote by N (θ; d1 ··· d`;N) the number of occurrences of the block d1 ··· d` amongst the first N digits, i.e. N (θ; d1 ··· d`;N) := #f0 ≤ i < n: ai+1 = d1; : : : ; ai+` = d`g: Then we call a number normal of order ` in base q if for each block of length ` the frequency of occurrences tends to q−`. As a qualitative measure of the distance of a number from being normal we introduce for integers N and ` the discrepancy of θ by N (θ; d1 ··· d`;N) −k RN;`(θ) = sup − q ; d1:::d` N where the supremum is over all blocks of length `. Then a number θ is normal to base q if for each ` ≥ 1 we have that RN;`(θ) = o(1) for N ! 1. Furthermore we call a number absolutely normal if it is normal in all bases q ≥ 2. Borel [2] used a slightly different, but equivalent (cf. Chapter 4 of [3]), definition of normality to show that almost all real numbers are normal with respect to the Lebesgue measure.p Despite their omnipresence it is not known whether numbers such as log 2, π, e or 2 are normal to any base. The first construction of a normal number is due to Champernowne [4] who showed that the number 0:1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ::: is normal in base 10. Date: July 10, 2014. 2010 Mathematics Subject Classification. Primary 11N37; Secondary 11A63. Key words and phrases. normal number, pseudo-polynomial. 1 2 M. G. MADRITSCH The construction of Champernowne laid the base for a class of normal numbers which are of the form σq = σq(f) = 0: bf(1)cq bf(2)cq bf(3)cq bf(4)cq bf(5)cq bf(6)cq :::; where b·cq denotes the expansion in base q of the integer part. Davenport and Erd}os[6] showed that σ(f) is normal for f being a polynomial such that f(N) ⊂ N. This construction was extended by Schiffer [19] to polynomials with rational coefficients. Furthermore he −1 showed that for these polynomials the discrepancy RN;`(σ(f)) (log N) and that this is best possible. These results where extended by Nakai and Shiokawa [17] to polynomials having real coefficients. Madritsch, Thuswaldner and Tichy [12] considered transcendental entire functions of bounded logarithmic order. Nakai and Shiokawa [16] used pseudo- polynomial functions, i.e. these are function of the form β0 β1 βd (1.1) f(x) = α0x + α1x + ··· + αdx with α0; β0; α1; β1; : : : ; αd; βd 2 R, α0 > 0, β0 > β1 > ··· > βd > 0 and at least one βi 62 Z. Since we often only need the leading term we write α = α0 and β = β0 for short. They were also able to show that the discrepancy is O((log N)−1). We refer the interested reader to the books of Kuipers and Niederreiter [11], Drmota and Tichy [7] or Bugeaud [3] for a more complete account on the construction of normal numbers. The present method of construction by concatenating function values is in strong con- nection with properties of q-additive functions. We call a function f strictly q-additive, if f(0) = 0 and the function operates only on the digits of the q-ary representation, i.e., ` ` X X h f(n) = f(dh) for n = dhq : h=0 h=0 A very simple example of a strictly q-additive function is the sum of digits function sq, defined by ` ` X X h sq(n) = dh for n = dhq : h=0 h=0 Refining the methods of Nakai and Shiokawa [16] the author obtained the following result. Theorem 1.1 ( [14, Theorem 1.1]). Let q ≥ 2 be an integer and f be a strictly q-additive function. If p is a pseudo-polynomial as defined in (1.1), then there exists η > 0 such that X 1−η f (bp(n)c) = µf N logq(p(N)) + NF logq(p(N)) + O N ; n≤N where q−1 1 X µ = f(d) f q d=0 and F is a 1-periodic function depending only on f and p. CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES 3 In the present paper, however, we are interested in a variant of σq(f) involving primes. As a first example, Champernowne [4] conjectured and later Copeland and Erd}os[5] proved that the number 0:2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 ::: is normal in base 10. Similar to the construction above we want to consider the number τq = τq(f) = 0: bf(2)cq bf(3)cq bf(5)cq bf(7)cq bf(11)cq bf(13)cq :::; where the arguments of f run through the sequence of primes. Then the paper of Copeland and Erd}os corresponds to the function f(x) = x. Nakai and Shiokawa [18] showed that the discrepancy for polynomials having rational coefficients is O((log N)−1). Furthermore Madritsch, Thuswaldner and Tichy [12] showed, that tran- scendental entire functions of bounded logarithmic order yield normal numbers. Finally in a recent paper Madritsch and Tichy [13] considered pseudo-polynomials of the special form αxβ with α > 0, β > 1 and β 62 Z. The aim of the present paper is to extend this last construction to arbitrary pseudo- polynomials. Our first main result is the following Theorem 1.2. Let f be a pseudo-polynomial as in (1.1). Then −1 RN (τq(f)) (log N) : In our second main result we use the connection of this construction of normal numbers with the arithmetic mean of q-additive functions as described above. Known results are due to Shiokawa [20] and Madritsch and Tichy [13]. Similar results concerning the moments of the sum of digits function over primes have been established by K´atai[10]. Let π(x) stand for the number of primes less than or equal to x. Then adapting these ideas to our method we obtain the following Theorem 1.3. Let f be a pseudo-polynomial as in (1.1). Then X q − 1 s (bf(p)c) = π(P ) log P β + O(π(P )); q 2 q p≤P where the sum runs over the primes and the implicit O-constant may depend on q and β. Remark 1.4. With simple modifications Theorem 1.3 can be extended to completely q-additive functions replacing sq. The proof of the two theorems is divided into four parts. In the following section we rewrite both statements in order to obtain as a common base the central theorem { Theorem 2.1. In Section 3 we start with the proof of this central theorem by using an indicator function and its Fourier series. These series contain exponential sums which we treat by different methods (with respect to the position in the expansion) in Section 4. Finally, in Section 5 we put the estimates together in order to proof the central theorem and therefore our two statements. 4 M. G. MADRITSCH 2. Preliminaries Throughout the rest p will always denote a prime. The implicit constant of and O may depend on the pseudo-polynomial f and on the parameter " > 0. Furthermore we fix a block d1 ··· d` of length ` and N, the number of digits we consider. In the first step we want to know in the expansion of which prime the N-th digit occurs. This can be seen as the translation from the digital world to the world of blocks. To this end let `(m) denote the length of the q-ary expansion of an integer m. Then we define an integer P by X X ` (bf(p)c) < N ≤ ` (bf(p)c) ; p≤P −1 p≤P where the sum runs over all primes. Thus we get the following relation between N and P X N = `(bf(p)c) + O(π(P )) + O(β logq(P )) p≤P (2.1) β P = P + O : log q log P Here we have used the prime number theorem in the form (cf. [21, Th´eor`eme4.1]) x (2.2) π(x) = Li x + O ; (log x)G where G is an arbitrary positive constant and Z x dt Li x = : 2 log t Now we show that we may neglect the occurrences of the block d1 ··· d` between two expansions. We write N (f(p)) for the number of occurrences of this block in the q-ary expansion of bf(p)c. Then (2.1) implies that X N (2.3) N (τ (f); d ··· d ; N) − N (f(p)) : q 1 ` log N p≤P In the next step we use the polynomial-like behavior of f. In particular, we collect all the values having the same length of expansion. Let j0 be a sufficiently large integer. Then for each integer j ≥ j0 there exists a Pj such that j−2 j−1 j q ≤ f(Pj) < q ≤ f(Pj + 1) < q with j Pj q β : Furthermore we set J to be the greatest length of the q-ary expansions of f(p) over the primes p ≤ P , i.e., J := max `(bf(p)c) = logq(f(P )) + O(1) log P: p≤P CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES 5 Now we show that we may suppose that each expansion has the same length (which we reach by adding leading zeroes).
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