arXiv:1412.3285v1 [math.NT] 10 Dec 2014 nqeepnino h form the of expansion unique θ omli l bases all in normal length of blocks for all over is supremum the where aldthe called nei sntkonwehrnmessc slg2, log as such numbers whether omnip known their not Despite ar is numbers measure. real it Lebesgue all the ence almost to that respect show with to normality normal of definition [3]), of flength of focrecso h block the of occurrences of esr ftedsac fanme rmbignra eintroduc we normal being from number a integers of for distance the of measure hnw alanme omlo order of normal number a call we Then snra obase to normal is e od n phrases. and words Key oe 2 sdasihl ieet u qiaet( equivalent but different, slightly a used [2] Borel 2010 Date Let N OSRCINO OMLNMESVIA NUMBERS NORMAL OF CONSTRUCTION N SUOPLNMA RM SEQUENCES PRIME POLYNOMIAL PSEUDO ∞ → uut2,2018. 24, August : q ahmtc ujc Classification. Subject Mathematics base Abstract. vlae ttepie.Ti xed eetrsl yTcyand Tichy by result recent a author. extends This the primes. the at evaluated ( ≥ ,d θ, ℓ q eapstv nee.Te vr real every Then integer. positive a be 2 ayepnin ednt by denote We expansion. -ary h rqec focrecstnsto tends occurrences of frequency the q 1 utemr ecl ubrasltl omli tis it if normal absolutely number a call we Furthermore . N · · · yconcatenating by R and d N,ℓ ℓ θ ntepeetpprw osrc omlnmesin numbers normal construct we paper present the In N , ( ℓ = q θ q sup = ) h iceac of discrepancy the =# := ) ≥ X k AFE .MADRITSCH G. MANFRED ffreach for if ≥ omlnme,pseudo-polynomial. number, normal 2. 1 d a 1 1. ...d k d { q Introduction ℓ 1 q k 0 ayepnin fped polynomials pseudo of expansions -ary
· · · ≤ N ( a ( n < i d ,d θ, ℓ k ℓ 1 { ∈ mns h first the amongst ≥ 1 rmr 13;Scnay11A63. Secondary 11N37; Primary N · · · : ehv that have we 1 0 θ a q , . . . , i N ℓ d +1 by ℓ nbase in ( N , ,d θ, = ) d − 1 1 − · · · a , . . . , 1 q − } q ℓ q θ ) hnanumber a Then . ℓ − d saqualitative a As . k ℓ ∈ ffrec block each for if N
N , R π cf. , i [0 + , digits, N,ℓ h number the ) ℓ , e )amt a admits 1) = ( hpe 4 Chapter or θ d = ) ℓ √ } i.e. . are 2 o res- (1) e e 2 M. G. MADRITSCH normal to any base. The first construction of a normal number is due to Champernowne [4] who showed that the number 0.1234567891011121314151617181920 ... is normal in base 10. The construction of Champernowne laid the base for a class of normal numbers which are of the form σ = σ (f)=0. f(1) f(2) f(3) f(4) f(5) f(6) ..., q q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q where denotes the expansion in base q of the integer part. Daven- ⌊·⌋q port 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 showed that 1 for these polynomials the discrepancy N,ℓ(σ(f)) (log N)− and that this is best possible. These resultsR 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 (1.1) f(x)= α xβ0 + α xβ1 + + α xβd 0 1 ··· d with α0, β0,α1, β1,...,αd, βd R, α0 > 0, β0 > β1 > > βd > 0 and at least one β Z. Since∈ we often only need the··· leading term i 6∈ we write α = α0 and β = β0 for short. They were also able to show 1 that the discrepancy is ((log N)− ). We refer the interested reader to the books of Kuipers andO 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 connection 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., ℓ ℓ h f(n)= f(dh) for n = dhq . Xh=0 Xh=0 A very simple example of a strictly q-additive function is the sum of digits function sq, defined by ℓ ℓ h sq(n)= dh for n = dhq . Xh=0 Xh=0 Refining the methods of Nakai and Shiokawa [16] the author obtained the following result. CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES3
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
1 η f ( p(n) )= µ N log (p(N)) + NF log (p(N)) + N − , ⌊ ⌋ f q q O n N X≤ where q 1 1 − µ = f(d) f q Xd=0 and F is a 1-periodic function depending only on f and p.
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 0.2357111317192329313741434753596167 ... is normal in base 10. Similar to the construction above we want to consider the number τ = τ (f)=0. f(2) f(3) f(5) f(7) f(11) f(13) ..., q q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q ⌊ ⌋q 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 1 polynomials having rational coefficients is ((log N)− ). Furthermore Madritsch, Thuswaldner and Tichy [12] showed,O that transcendental 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 Z. The aim of the present paper is to extend this last construction6∈ 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 (τ (f)) (log N)− . RN q ≪ 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 4 M. G. MADRITSCH
Theorem 1.3. Let f be a pseudo-polynomial as in (1.1). Then q 1 s ( f(p) )= − π(P ) log P β + (π(P )), q ⌊ ⌋ 2 q O p P X≤ where the sum runs over the primes and the implicit -constant may depend on q and β. O 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 expan- sion) in Section 4. Finally, in Section 5 we put the estimates together in order to proof the central theorem and therefore our two statements.
2. Preliminaries Throughout the rest p will always denote a prime. The implicit constant of and may depend on the pseudo-polynomial f and on ≪ O the parameter ε > 0. Furthermore we fix a block d1 dℓ of length ℓ 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 ℓ(m) denote the length of the q-ary expansion of an integer m. Then we define an integer P by ℓ ( f(p) ) < N ℓ ( f(p) ) , ⌊ ⌋ ≤ ⌊ ⌋ p P 1 p P ≤X− X≤ where the sum runs over all primes. Thus we get the following relation between N and P N = ℓ( f(p) )+ (π(P ))+ (β log (P )) ⌊ ⌋ O O q p P (2.1) X≤ β P = P + . log q O log P Here we have used the prime number theorem in the form (cf. [21, Th´eor`eme 4.1]) x (2.2) π(x) = Li x + , O (log x)G CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES5 where G is an arbitrary positive constant and x dt Li x = . log t Z2 Now we show that we may neglect the occurrences of the block d1 dℓ between two expansions. We write (f(p)) for the number of··· occurrences of this block in the q-ary expansionN of f(p) . Then (2.1) implies that ⌊ ⌋ N (2.3) (τq(f); d1 dℓ; N) (f(p)) . N ··· − N ≪ log N p P X≤ In the next step we use the polynomial-like behavior of f. In par- ticular, we collect all the values having the same length of expansion. Let j be a sufficiently large integer. Then for each integer j j there 0 ≥ 0 exists a Pj such that j 2 j 1 j q − f(P ) < q − f(P + 1) < q ≤ j ≤ j with j P q β . j ≍ 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 ℓ( f(p) ) = logq(f(P ))+ (1) log P. p P ≤ ⌊ ⌋ O ≍ Now we show that we may suppose that each expansion has the same length (which we reach by adding leading zeroes). For Pj 1 < p Pj we may write f(p) in q-ary expansion, i.e., − ≤ j 1 j 2 1 (2.4) f(p)= bj 1q − + bj 2q − + + b1q + b0 + b 1q− + .... − − ··· − Then we denote by ∗(f(p)) the number of occurrences of the block N d1 dℓ in the string 0 0bj 1bj 2 b1b0, where we filled up the ex- pansion··· with leading zeroes··· − such− that··· it has length J. The error of doing so can be estimated by
0 ∗(f(p)) (f(p)) ≤ N − N p P p P X≤ X≤ J 1 − (J j)(π(P ) π(P )) + (1) ≤ − j+1 − j O j=j +1 X0 J J qj/β P N π(P )+ (1) . ≤ j O ≪ j ≪ log P ≪ log N j=j +2 j=j +2 X0 X0 6 M. G. MADRITSCH
In the following three sections we will estimate this sum of indicator functions ∗ in order to prove the following theorem. N Theorem 2.1. Let f be a pseudo polynomial as in (1.1). Then
ℓ β P (2.5) ∗ ( f(p) )= q− π(P ) log P + N ⌊ ⌋ q O log P p P X≤ Using this theorem we can simply deduce our two main results.
Proof of Theorem 1.2. We insert (2.5) into (2.3) and get the desired result.
Proof of Theorem 1.3. For this proof we have to rewrite the statement. In particular, we use that the sum of digits function counts the number of 1s, 2s, etc. and assigns weights to them, i.e.,
q 1 − s (n)= d (n; d). q ·N Xd=0 Thus q 1 q 1 − − β β β P s ( p )= d (p )= d ∗(p )+ q ·N ·N O log P p P p P d=0 p P d=0 X≤ X≤ X X≤ X q 1 P = − π(P ) log (P β)+ 2 q O log P and the theorem follows.
In the following sections we will prove Theorem 2.1 in several steps. First we use the “method of little glasses” in order to approximate the indicator function by a Fourier series having smooth coefficients. Then we will apply different methods (depending on the position in the expansion) for the estimation of the exponential sums that appear in the Fourier series. Finally we put everything together and get the desired estimate.
3. Proof of Theorem 2.1, Part I We want to ease notation by splitting the pseudo-polynomial f into a polynomial and the rest. Then there exists a unique decomposition of the following form: (3.1) f(x)= g(x)+ h(x) CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES7 where h R[X] is a polynomial of degree k (where we set k = 0 if h is the zero∈ polynomial) and
r θj g(x)= αjx j=1 X with r 1, αr = 0, αj real, 0 < θ1 < < θr and θj Z for 1 j r. Let γ≥and ρ6 be two parameter which··· we will frequently6∈ use≤ in≤ the sequel. We suppose that 1 θ 0 <γ<ρ< min , r . 4(k + 1) 2 The aim of this section is to calculate the Fourier transform of ∗. N In order to count the occurrences of the block d1 dℓ in the q-ary expansion of f(p) (2 p P ) we define the indicator··· function ⌊ ⌋ ≤ ≤ ℓ i ℓ i ℓ 1, if d q− t t < d q− + q− ; (t)= i=1 i ≤ − ⌊ ⌋ i=1 i I 0, otherwise; ( P P which is a 1-periodic function. Indeed, we have
j (3.2) (q− f(p))=1 d1 dℓ = bj 1 bj ℓ, I ⇐⇒ ··· − ··· − where f(p) has an expansion as in (2.4). Thus we may write our block counting function as follows
J j (3.3) ∗(f(p)) = q− f(p) . N I j=ℓ X In the following we will use Vinogradov’s “method of little glasses” (cf. [23]). We want to approximate from above and from below by two 1-periodic functions having smallI Fourier coefficients. To this end we will use the following Lemma 3.1 ( [23, Lemma 12]). Let α, β, ∆ be real numbers satisfying 1 0 < ∆ < , ∆ β α 1 ∆. 2 ≤ − ≤ − Then there exists a periodic function ψ(x) with period 1, satisfying (1) ψ(x)=1 in the interval α + 1 ∆ x β 1 ∆, 2 ≤ ≤ − 2 (2) ψ(x)=0 in the interval β + 1 ∆ x 1+ α 1 ∆, 2 ≤ ≤ − 2 (3) 0 ψ(x) 1 in the remainder of the interval α 1 ∆ x ≤ ≤ − 2 ≤ ≤ 1+ α 1 ∆, − 2 8 M. G. MADRITSCH
(4) ψ(x) has a Fourier series expansion of the form
∞ ψ(x)= β α + A(ν)e(νx), − ν= Xν=0−∞ 6 where 1 1 (3.4) A(ν) min , β α, . | |≪ ν − ν2∆ We note that we could have used Vaaler polynomials [22], however, we do not gain anything by doing so as the estimates we get are already best possible. Setting (3.5) ℓ ℓ λ 1 λ ℓ 1 α = dλq− + (2δ)− , β = dλq− + q− (2δ)− , − − − γ λ=1 λ=1 δ = P − , X X ℓ ℓ λ 1 λ ℓ 1 α = d q− (2δ)− , β = d q− + q− + (2δ)− . + λ − + λ Xλ=1 Xλ=1 and an application of Lemma 3.1 with (α,β,δ) = (α , β , δ) and − − (α,β,δ)=(α+, β+, δ), respectively, provides us with two functions I− and +. By our choice of (α , β , δ) it is immediate that I ± ±
(3.6) (t) (t) +(t) (t R). I− ≤ I ≤ I ∈ Lemma 3.1 also implies that these two functions have Fourier expan- sions
∞ ℓ γ (3.7) (t)= q− P − + A (ν)e(νt) ± ± I ± ν= Xν=0−∞ 6 satisfying
1 γ 2 A (ν) min( ν − ,P ν − ). | ± |≪ | | | |
In a next step we want to replace by + in (3.3). For this purpose we observe, using (3.6), and (3.7) thatI I
∞ ℓ γ (t) q− P − + A (ν)e(νt). ± I − ≪ ν= Xν=0−∞ 6 CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES9
j Thus setting t = q− f(p) and summing over p P yields ≤ (3.8)
∞ j π(P ) γ ν (q− f(p)) π(P )P − + A (ν) e f(p) . I − qℓ ≪ ± qj p P ν= p P X≤ Xν=0−∞ X≤ 6
Now we consider the coefficients A (ν). Noting (3.4) one observes that ± 1 γ ν− , for ν P ; A (ν) γ 2 | | ≤ γ ± ≪ P ν− , for ν >P . ( | | Estimating all summands with ν >P γ trivially we get | | P γ ∞ ν 1 ν γ A (ν)e f(p) ν− e f(p) + P − . ± qj ≪ qj ν= ν=1 Xν=0−∞ X 6 Using this in (3.8) yields P γ j π(P ) γ 1 (q− f(p)) π(P )P − + ν− S(P, j, ν), I − qℓ ≪ p P ν=1 X≤ X where we have set
ν (3.9) S(P, j, ν) := e f(p) . qj p P X≤ 4. Exponential sum estimates In the present section we will focus on the estimation of the sum S(P, j, ν) for different ranges of j. Since j describes the position within the q-ary expansion of f(p) we will call these ranges the “most signifi- cant digits”, the “least significant digits” and the “digits in the middle”, respectively. Now, if θr > k 0, i.e the leading coefficient of f origins from the pseudo polynomial≥ part g, then we consider the two ranges
j θr ρ θr ρ j θr 1 q P − and P − < q P . ≤ ≤ ≤ For the first one we will apply Proposition 4.3 and for the second one Proposition 4.1. On the other hand, if k > θr > 0, meaning that the leading coefficient of f origins from the polynomial part h, then we have an additional part. In particular, in this case we will consider the three ranges
j θr ρ θr ρ j k 1+ρ k 1+ρ j k 1 q P − , P − < q P − , and P − < q P . ≤ ≤ ≤ ≤ 10 M. G. MADRITSCH
We will, similar to above, treat the first and last range by Proposi- tion 4.3 and Proposition 4.1, respectively. For the middle range we will apply Proposition 4.7. Since 2ρ < θr, we note that the middle range is empty if k = 1. Since the size of j represents the position of the digit in the expansion (cf. (3.2)), we will deal in the following subsection with the “most significant digits”, the “least significant digits” and the “digits in the middle”, respectively.
4.1. Most significant digits. We start our series of estimates for the exponential sum S(P, j, ν) for j being in the highest range. In particular, we want to show the following Proposition 4.1. Suppose that for some k 1 we have f (k)(x) Λ for any x on [a, b] with Λ > 0. Then ≥ ≥
1 1 P S(P, j, ν) Λ− k + . ≪ log P (log P )G The main idea of the proof is to use Riemann-Stieltjes integration together with Lemma 4.2 ( [9, Lemma 8.10]). Let F :[a, b] R and suppose that for some k 1 we have F (k)(x) Λ for any →x on [a, b] with Λ > 0. Then ≥ ≥ b k 1/k e(F (x))dx k2 Λ− . ≤ Za
Proof of Proposition 4.1. We rewrite the sum into a Riemann-Stieltjes integral: ν P ν S(P, j, ν)= e f(p) = e f(t) dπ(t)+ (1). qj qj O p P 2 X≤ Z Then we apply the prime number theorem in the form (2.2) to gain the usual integral back. Thus P ν dt P S(P, j, ν)= e j f(t) + G . −G q log t O (log P ) ZP (log P ) Now we use the second mean-value theorem to get 1 ξ ν P (4.1) S(P, j, ν) sup e j f(t) dt + G . ≪ log P −G q (log P ) ξ ZP (log P )
Finally an application of Lemma 4.2 proves the lemma.
CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES11
4.2. Least significant digits. Now we turn our attention to the low- est range of j. In particular, the goal is the proof of the following Proposition 4.3. Let P and ρ be positive reals and f be a pseudo- polynomial as in (3.1). If j is such that
j θr ρ (4.2) 1 q P − ≤ ≤ holds, then for 1 ν P γ there exists η > 0 (depending only on f and ρ) such that ≤ ≤ 8 1 η S(P, j, ν) = (log P ) P − . Before we launch into the proof we collect some tools that will be necessary in the sequel. A standard idea for estimating exponential sums over the primes is to rewrite them into ordinary exponential sums over the integers having von Mangoldt’s function as weights and then to apply Vaughan’s identity. We denote by log p, if n = pk for some prime p and an integer k 1; Λ(n)= ≥ (0, otherwise. von Mangoldt’s function. For the rewriting process we use the following Lemma 4.4. Let g be a function such that g(n) 1 for all integers n. Then | | ≤ 1 g(p) max Λ(n)g(n) + (√P ). ≪ log P t P O p P ≤ n t ≤ X≤ X
Proof. This is Lemma 11 of [15]. However, the proof is short and we need some piece later. We start with a summation by parts yielding 1 P dt g(p)= log(p)g(p)+ log(p)g(p) . log P t(log t)2 p P p x 2 p t ! X≤ X≤ Z X≤ Now we cut the integral at √P and use Chebyshev’s inequality (cf.
[21, Th´eor`eme 1.3]) in the form p t log(p) log(t)π(t) t for the lower part. Thus ≤ ≤ ≪ P P 1 dt √ g(p) + 2 max log(p)g(p) + ( P ) ≤ log P √P t(log t) √P 12 M. G. MADRITSCH Finally we again use Chebyshev’s inequality π(t) t/ log(t) to ob- tain ≪ (4.3) log(t) ⌊ log(p) ⌋ Λ(n)g(n) log(p)g(p) log(p) 1 π(√t) log(t) √t. − ≤ ≤ ≪ n t p t √ a=2 ≤ ≤ p t X X X≤ X In the next step we use Vaughan’s identity to subdivide this weighted exponential sum into several sums of Type I and II. Lemma 4.5 ( [1, Lemma 2.3]). Assume F (x) to be any function defined on the real line, supported on [P/2,P ] and bounded by F0. Let further U,V,Z be any parameters satisfying 3 U Λ(n)F (n) K log P + F + L(log P )8, ≪ 0 P/2 ∞ K = max d3(m) F (mn) , M m=1 Z m=1 U CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES13 Now we have the necessary tools to state the Proof of Proposition 4.3. An application of Lemma 4.4 yields 1 ν 1 S(P, j, ν) max Λ(n)e (g(n)+ h(n)) + P 2 . ≪ log P qj n P ≤ X We split the inner sum into log P sub sums of the form ≤ ν Λ(n)e (g(n)+ h(n)) qj X ν S Xε e (g(xy)+ h(xy)) . | 1|≪ qj x 2X X For estimating the inner sum we fix x and denote Y = X . Since θ Z x r and θ >k 0, we have that 6∈ r ≥ ℓ ∂ g(xy) θr ℓ X Y − . ∂yℓ ≍ qj P θr ρ νq jXθr Xρ Now on the one hand, since − , we have − . On the other hand for ℓ 5( θ +1)≤ we get ≫ ≥ ⌊ r⌋ 2 1 ν θr ℓ γ θr ℓ X Y − P X − 5 X− 2 . qj ≤ ≪ 14 M. G. MADRITSCH Thus an application of Lemma 4.6 yields the following estimate: 1 1 ε 1 k ρ S X Y Y − K + (log Y ) X− K + X− 2 4K·8L5−2K | 1|≪ x 2X/Z (4.5) ≤X h i 1 1 K 1+ε ρ 5 X (log X) X− + X− 64L −4 , ≪ k 1 where we have used that K < 1 and ρ< 3 . X 2X For the second sum S2 we start by splitting the interval ( V , U ] into log X subintervals of the form (X , 2X ]. Thus ≤ 1 1 ν S (log X)Xε b(y)e (g(xy)+ h(xy)) 2 j | | ≤ q X1 ε Now an application of Cauchy’s inequality together with b(y) X yields | |≪ 2 2 2 2ε ν S2 (log X) X X1 b(y)e j (g(xy)+ h(xy)) | | ≤ q X1 S 2 (log X)2X4εX | 2| ≪ 1 ν X + e (g(xy ) g(xy )+ h(xy ) h(xy )) × qj 1 − 2 1 − 2 A As above we want to apply Lemma 4.6. To this end we fix y1 and y = y . Similarly to above we get that 2 6 1 ℓ ∂ (g(xy1) g(xy2)+ h(xy1) h(xy2)) y1 y2 θr ℓ − − | − |X X− . ∂xℓ ≍ y 1 1 CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES15 ν y1 y2 θr ρ Now, on the one hand we have j | − | X X and on the other q y1 ≫ hand ℓ − 2 1 ν y1 y2 θr ℓ γ+θr X γ+θr ℓ | − |X X− X X − 3 X− 2 qj y 1 ≪ V ≪ ≪ 1 if ℓ 2 θ + 3. Thus again an application of Lemma 4.6 yields ≥ ⌊ r⌋ (4.6) 1 ρ 1 1 2 2 4ε K S (log X) X X X + X X− + X− K + X− 2 4K·2L2−2K | 2| ≪ 1 1 1 A θr ρ j k 1+ρ (4.7) P − < q P − ≤ holds, then for 1 ν P γ we have ≤ ≤ νf(p) 1 ρ S(P, j, ν)= e P − 4k . qj ≪ p P X≤ The main idea in this range is to use that the dominant part of f comes from the polynomial h. Therefore after getting rid of the function g we will estimate the sum over the polynomial by the following Lemma 4.8. Let h R[X] be a polynomial of degree k 2. Suppose α is the leading coefficient∈ of h and that there are integers≥ a, q such that 1 qα a < with (a, q)=1. | − | q Then we have for any ε> 0 and H X ≤ 41−k 1+ε 1 1 q log(p)e(h(p)) H + 1 + k . ≪ q H 2 H X Now we have enough tools to state the Proof of Proposition 4.7. As in the Proof of Proposition 4.3 we start by an application of Lemma 4.4 yielding 1 ν 1 S(P, j, ν) max Λ(n)e (g(n)+ h(n)) + P 2 . ≪ log P qj n P ≤ X We split the inner sum into log P sub sums of the form ≤ ν S := Λ(n)e (g(n)+ h(n)) qj X We distinguish three cases according to the size of b. Case 1. Hρ < b. In this case we may apply Lemma 4.8 together with (4.3) to get 1 ρ +ε T (h) H − 4k−1 . ≪ CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES17 Case 2. 2 b j θr ρ Since 2 ρ < θ r, this contradicts our lower bound q P − . Case 3. b = 1. This case requires a further distinction≥ accord- ing to whether a = 0 or not. Case 3.1. ναk 1 . It follows that qj ≥ 2 j q 2 ναk ≤ | | j θr ρ again contradicting our lower bound q P − . ≥ ναk 1 Case 3.2. qj < 2 . This implies that a = 0 which yields j k ρ (4.9) q ναk H − . ≥| | 1 θr 1 j We distinguish two further cases according to whether P − ν − q X or not. | | ≤ 1 θr 1 j Case 3.2.1 P − ν − q X. This implies that qj P θr ν and | | ≤ ≤ | | 1 θr 1 j 1 ρ 1 1 2ρ H = P − ν − q P − ν − P − . | | ≥ | | ≥ Plugging these estimates into (4.9) gives θr (1 2ρ)(k ρ) P α P − − . ≥| k| However, since 4(k + 1)ρ< 1, we have (1 2ρ)(k ρ) >k 1+2ρ θ − − − ≥ r yielding a contradiction. 1 θr 1 j Case 3.2.2 P − ν − q > X. Then H = X 1 2ρ | | ≥ P − and (4.9) becomes k 1+ρ (1 2ρ)(k ρ) P − να P − − ≥| k| yielding a similar contradiction as in Case 3.2.1. Therefore Case 1 is the only possible and we may always apply Lemma 4.8 together with (4.3). Plugging this into (4.8) yields ν 1 ρ +ε Λ(n)e (g(n)+ h(n)) H − 4k−1 1+ ϕ(n) ϕ(n + 1) qj ≪ | − | X Thus ν 1 ρ +ε Λ(n)e (g(n)+ h(n)) H − 4k−1 . qj ≪ X n X+H ≤ X≤ 5. Proof of Theorem 2.1, Part II Now we use all the tools from the section above in order to estimate (5.1) J H J j π(P ) 1 1 (q− f(p)) π(P )H− J + ν− S(P, j, ν). I − qℓ ≪ j=ℓ p P ν=1 j=ℓ X X≤ X X As indicated in the section above, we split the sum over j into two or three parts according to whether θr > k or not. In any case an application of Proposition 4.3 yields for the least significant digits that 1 9 1 η (5.2) ν− S(P, j, ν) (log P ) JP − . ≪ 1 ν P γ 1 qj P θr−ρ ≤X≤ ≤ X≤ Now let us suppose that θr >k. Then an application of Proposition 4.1 yields (5.3) 1 ν− S(P, j, ν) 1 ν P γ P θr−ρ J j π(P ) P (q− f(p)) , I − qℓ ≪ log P j=ℓ p P ≤ X X which together with (3.3) proves Theorem 2.1 in the case that θr >k. On the other side if θr θr ρ j k 1+ρ k 1+ρ j k P − < q P − and P − < q P . ≤ ≤ CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES19 For the “digits in the middle” an application of Proposition 4.7 yields (5.4) ρ 1 1 1 ν− S(P, j, ν) ν− P − 4k ≪ 1 ν P γ P θr−ρ J j π(P ) P (q− f(p)) , I − qℓ ≪ log P j=ℓ p P ≤ X X which together with (3.3) proves Theorem 2.1 in the case that θ References [1] V. Bergelson, G. Kolesnik, M. 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Vinogradov, The method of trigonometrical sums in the theory of num- bers, Dover Publications Inc., Mineola, NY, 2004, Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation. CONSTRUCTION OF NORMAL NUMBERS VIA PSEUDO POLYNOMIAL PRIME SEQUENCES21 1. Universite´ de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-les-Nancy,` F-54506, France; 2. CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre- les-Nancy,` F-54506, France E-mail address: [email protected]