On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod _{k=1}^N 2 \sin (\pi x_k)$$\end{document}∏k=1N2sin(πxk) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}N→∞, where the numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =(x_k)_{k=1}^N$$\end{document}ω=(xk)k=1N are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}ω, thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97–118, 1969). Furthermore, we consider the special cases when the points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}ω are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.


Introduction and statement of the results
. It is the aim of this paper to propagate the analysis of corresponding "Weyl products" in particular with respect to their asymptotic behavior for N → ∞.
Note that, formally, studying products P N in fact is just a special case of studying S N , since log P N = Hence it makes sense to study the asymptotic behavior of the normalized product N k=1 S f f (x k ) rather than N k=1 f (x k ).
A special example of such products played an important role in [1] in the context of pseudorandomness properties of the Thue-Morse sequence, where lacunary trigonometric products of the form N k=1 2 sin(π 2 k α) for α ∈ R were analyzed. (Note that 1 0 log sin(π x)dx = − log 2, hence the normalization factor 2 in this case.) It was shown there that for almost all α and all ε > 0 we have N k=1 |2 sin(π 2 k α)| ≤ exp (π + ε) N log log N for all sufficiently large N and for infinitely many N .
In the present paper we restrict ourselves to f (x) = sin(π x) and we will extend the analysis of such products to other types of sequences (x k ) k≥1 . In particular we will consider two well-known types of uniformly distributed sequences, namely the van der Corput sequence (x k ) k≥1 and the Kronecker sequence ({kα}) k≥1 with irrational α ∈ [0, 1]. Furthermore, we will determine the typical behavior of N k=1 2 sin(π x k ), that is, the almost sure order of this product for "random" sequences (x k ) k≥1 in a suitable probabilistic model.
All our results use methods from uniform distribution theory and discrepancy theory, so we will introduce some of the basic notions from these subjects. Let x 1 , . . . , x N be numbers in [0, 1]. Their star-discrepancy is defined as For more basic information on uniform distribution theory and discrepancy, we refer to [10,28]. Now we come to our new results. First we will give general estimates for products N k=1 2 sin(π x k ) in terms of the star-discrepancy D * N of (x k ) 1≤k≤N . A similar result in a weaker form was obtained by Hlawka [18] (see also [19] where N := N D * N . Concerning the quality of Theorem 1, consider the case when (x k ) k≥1 is a lowdiscrepancy sequence such as the van der Corput sequence (which is treated in Theorem 5 below). Then N = O (log N ), and Theorem 1 gives for some γ ∈ R + and all sufficiently large N . Stronger asymptotic bounds are provided by Theorem 5 below; thus, Theorem 1 does not provide a sharp upper bound in this case.
As another example, let x k = k/(N + 1) for k = 1, 2, . . . , N . This point set has star-discrepancy D * N = 1/(N + 1), and hence the general estimate (3) gives To be precise we can obtain this estimate directly from Theorem 1 only for "infinitely many N " instead of "for arbitrary N ".
Theorem 1 is stated for sequences, hence the "sufficiently large N " may depend on the sequence. But we can apply the Theorem 1 to a sequence (x k ) k≥1 which is designed such that for infinitely many N we have x k = k/(N + 1) for k = 1, 2, . . . , N .
On the other hand, the product on the left-hand side of (5) is well known to be exactly N + 1 (see also Lemma 3 below). Thus, the general estimate from Theorem 1 has an additional factor N in comparison with the correct order in this case, which is quite close to optimality.
As already mentioned above, Hlawka [18,19] studied similar questions in connection with interpolation of analytic functions on the complex unit disc. There he considered products of the form where ξ k are points on the unit circle. The main results in [18,19] are lower and upper bounds of |ω N (z)| in terms of the star-discrepancy D * N of the sequence (arg 1 2π ξ k ), k = 1, . . . , N . 1 It should also be mentioned that Wagner [45] proved the general lower bound for infinitely N , where c > 0 is some explicitly given constant. This solved a problem stated by Erdős.
In the sequel we will give a second, essentially optimal theorem which estimates products N k=1 2 sin(π x k ) in terms of the star-discrepancy of the sequence (x k ) k≥1 . Let ω = {x 1 , . . . , x N } be numbers in [0, 1] and let P N (ω) = N k=1 2 sin(π x k ). Let D * N (ω) denote the star-discrepancy of ω. Furthermore, let d N be a real number from the interval [1/(2N ), 1], which is the possible range of the star-discrepancy of N -element point sets. We are interested in where the supremum is taken over all ω with D * N (ω) ≤ d N . We will show (b) For all sufficiently large N we have Let us now focus on products of the form where α is a given irrational number, i.e., we consider the special case when (x n ) n≥1 is the Kronecker sequence ({nα}) n≥1 . Such products play an essential role in many fields and are the best studied such Weyl products in the literature. See for example [7,9,16,21,25,32,39,44]. Before discussing these products in detail, let us recall some historical facts. By Kronecker's approximation theorem, the sequence (nα) n≥1 is everywhere dense modulo 1; i.e., the sequence of fractional parts ({nα}) n≥1 is dense in [0, 1]. At the beginning of the 20th century various authors considered this sequence (and generalizations such as ({αn d }) n≥1 , etc.) from different points of view; see for instance Bohl [5], Weyl [46] and Sierpińksi [40]. An important impetus came from celestial mechanics. It was Hermann Weyl in his seminal paper [47] who opened new and much more general features of this subject by introducing the concept of uniform distribution for arbitrary sequences (x k ) k≥1 in the unit interval (as well as in the unit cube [0, 1] s ). This paper heavily influenced the development of uniform distribution theory, discrepancy theory and the theory of quasi-Monte Carlo integration throughout the last 100 years. For the early history of the subject we refer to Hlawka and Binder [20]. Numerical experiments suggest that for integers N with q l ≤ N < q l+1 , where (q l ) l≥0 is the sequence of best approximation denominators of α, the product attains its maximal value for N = q l+1 − 1.
For the case N = q − 1 for some best approximation denominator q the product q−1 n=1 |2 sin(π nα)| already was considered in [9,39], and in much more general form in [3] (see also [37]). In particular, it follows from the results given there that  |2 sin(π nα)| = lim when q runs through the sequence of best approximation denominators. Indeed, we are neither able to prove assertion (6) nor assertion (7). Nevertheless we want to give a quantitative estimate for the case N = q − 1, i.e., also a quantitative version of (8), before we will deal with the general case.
Theorem 3 Let q be a best approximation denominator for α. Then Next we consider general N ∈ N: . .] be the continued fraction expansion of the irrational number α ∈ [0, 1]. Let N ∈ N be given, and denote its Ostrowski expansion by The second part of Corollary 1 can also be obtained from [7,Lemma 4].
In the following we say that a real α is of type t ≥ 1 if there is a constant c > 0 such that The next result essentially improves a result given in [25]. There a bound on N n=1 |2 sin (π nα)| for α of type t of the form N cN 1−1/t log N instead of our much sharper bound 2 C N 1−1/t was given. Note that our result only holds for t > 1, so we cannot obtain the sharp result of Lubinsky [32] in the case of α with bounded continued fraction coefficients.

Corollary 2 Assume that α is of type t > 1. Then for some constant C and all N large enough
is the van der Corputsequence. The van der Corput sequence (in base 2) is defined as follows: for n ∈ N with binary expansion n = a 0 + a 1 2 + a 2 2 3 + · · · with digits a 0 , a 1 , a 2 , . . . ∈ {0, 1} (of course the expansion is finite) the n th element is given as (see the recent survey [11] for detailed information about the van der Corput sequence). For this sequence, in contrast to the Kronecker sequence, we can give very precise results. We show: Finally, we study probabilistic analogues of Weyl products, in order to be able to quantify the typical order of such products for "random" sequences and to have a basis for comparison for the results obtained for deterministic sequences in Theorems 3-5. We will consider two probabilistic models. First we study where (X k ) k≥1 is a sequence of independent, identically distributed (i.i.d.) random variables in [0, 1]. The second probabilistic model are random subsequences (n k α) k≥1 of the Kronecker sequences (nα), where the elements of n k are selected from N independently and with probability 1 2 for each number. This model is frequently used in the theory of random series (see for example the monograph of Kahane [23]) and was introduced to the theory of uniform distribution by Petersen and McGregor [38] and later extensively studied by Tichy [43], Losert [30], and Losert and Tichy [31].
Then for all ε > 0 we have, almost surely, for all sufficiently large N , and for infinitely many N .
, 1}-valued random variables with mean 1/2, defined on some probability space ( , A, P), which induce a random sequence (n k ) k≥1 = (n k (ω)) k≥1 as the sequence of all numbers {n ≥ 1 : ξ n = 1}, sorted in increasing order. Set Then for all ε > 0 we have, P-almost surely, Remark 2 It is interesting to compare the conclusions of Theorems 6 (for purely random sequences) and 7 (for randomized subsequences of linear sequences) to the results in equations (1) and (2), which hold for lacunary trigonometric products. The results coincide almost exactly, except for the constants in the exponential term (which can be seen as the standard deviations in a related random system; see the proofs). The larger constant in the lacunary setting comes from an interference phenomenon, which appears frequently in the theory of lacunary functions systems (see for example Kac [22] and Maruyama [34]). On the other hand, the smaller constant in Theorem 7 represents a "loss of mass" phenomenon, which can be observed in the theory of slowly growing (randomized) trigonometric systems; it appears in a very similar form for example in Berkes [4] and Bobkov-Götze [6]. It is also interesting that the constant π/ √ 6 in Theorem 1 is exactly the same as in results obtained by Fukuyama [13] for products |2 sin(π n k α)| and |2 cos(π n k α)| under the "super-lacunary" gap condition n k+1 /n k → ∞.
The outline of the remaining part of this paper is as follows. In Sect. 2 we will prove Theorems 1 and 2, which give estimates of Weyl products in terms of the discrepancy of the numbers (x k ) 1≤k≤N . In Sect. 3 we prove the results for Kronecker sequences (Theorems 3 and 4), and in Sect. 4 the results for the van der Corput sequence (Theorem 5). Finally, in Sect. 5 we prove the results about probabilistic sequences (Theorems 6 and 7).

Proofs of Theorems 1 and 2
Proof of Theorem 1 The Koksma-Hlawka-inequality (see e.g. [28]) states that for any function g : By partial integration we obtain (with a positive O-constant for ε small enough). Furthermore, we have Altogether we have, using the Koksma-Hlawka inequality and since log sin(π ε) = log(π ε) − π 2 ε 2 6 − O(ε 4 ), for some constant c > 0. We choose ε = D * N and obtain Next we come to the proof of Theorem 2. We will need several auxiliary lemmas, before proving the theorem. (ii) If any of the points of ω is moved nearer to 1 2 , then the star-discrepancy of the new point set is larger than D.
Proof We give the proof for N even only (the proof for N odd runs quite analogously). The parts (i) and (ii) immediately follow from the form of the graph of the discrepancy function a → A N (a) a − a for a ∈ [0, 1] as it is plotted in Fig. 3.

Lemma 2 For ω as in Lemma
In the analogous way we can argue if i is such that x i = 1 2 , or such that x i > 1 2 . Hence, such an ω cannot exist. Proof The proof of Equation (ii) is based on noting that e ia N and e −ia N are the zeros of X 2 − 2 cos(a N )X + 1. Then, the polynomial X 2N − 2 cos(a N )X N + 1 has 2N zeros and these are 0, 1, . . . , N − 1) .
Hence, we get Taking X = 1 and a = 2b, the last equation is written as This is a standard formula that can be found in [14, Formula 1.392].
Putting b = π x N , the proof of assertion (ii) is complete. Equation (i) follows immediately from Equation (ii) by noting that Letting b → 0 and using l'Hospital's rule, we conclude that Another nice proof of Equation (i) can be found for example in [35].

Fig. 4 The function log sin(π x)
Lemma 5 There is an ε 0 > 0 such that for all ε < ε 0 we have Proof This follows from

Proof of Theorem 2 Let
Note that the function x → log sin(π x) is of the form as presented in Fig. 4. Hence for M < N 2 we have log sin π By Lemma 4 for all M with M N < ε 0 for the integral above we have and hence, using also Lemma 5, This proves assertion (b) of Theorem 2.
On the other hand we have This gives , and consequently It remains to show that for all ε > 0 there are c(ε) and N (ε) such that for all N ≥ N (ε) the right hand side of (9)

Proofs of the results for Kronecker sequences
Proof of Theorem 3 Let α = p q + θ with 0 < θ < 1 q·q + , where q + is the best approximation denominator following q. The case of negative θ can be handled quite analogously. There is exactly one of the points {kα} for k = 1, . . . , q − 1 in each interval [ m q , m+1 q ) for m = 1, . . . , q − 1. Note that the point in the interval [ q−1 q , 1) is the point q − α , where q − is the best approximation denominator preceding q. We have Hence, on the one hand (by equation (i) of Lemma 3), On the other hand Proof of Theorem 4 Let N i := b l q l + b l−1 q l−1 + · · · + b i+1 q i+1 for i = 0, . . . , l − 1 and N l := 0. Then |2 sin(π nα)| .
• if {nα} ∈ I l with κ + l+1 q i < 1 2 then in the representation (10) of {nα} we replace kθ i byθ i . • if {nα} ∈ I l 0 , where l 0 is such that κ + l 0 q i < 1 2 ≤ κ + l 0 +1 q i then -for the d such that κ + l 0 q i + dθ i ≥ 1 2 in the representation (10) of {nα} we replace kθ i by 0, -for the d such that κ + l 0 q i + (d + 1)θ i < 1 2 in the representation (10) of {nα} we replace kθ i byθ i , -for the single d 0 such that κ -for the h such that κ + q i −1 q i + hθ i ≥ 1 in the representation (10) of {nα} we replace kθ i byθ i , -for the h such that κ + q i −1 q i + (h + 1)θ i ≤ 1 in the representation (10) of {nα} we replace kθ i by 0, -for the single h 0 such that κ + q i −1 q i + h 0θi < 1 < κ + q i −1 q i + (h 0 + 1)θ i we replace in the representation (10) of {nα} the kθ i by 0 if g(κ + q i −1 q i + h 0θi ) ≥ g(κ + q i −1 q + (h 0 + 1)θ i ) and byθ i otherwise, where here and in the following we use the notation g(x) := |2 sin π x|. Let the second be the case, the other case is handled quite analogously.
Using the new points instead of the {nα} by construction we obtain an upper bound i for i . Then
We have q l ≥ a l q l−1 + q l−2 ≥ a l a l−1 q l−2 + a l q l−3 + q l−2 ≥ (a l a l−1 + 1) q l−2 .
By iteration we obtain q l ≥ (a l a l−1 + 1) (a l−2 a l−3 + 1) · · · (a 2 a 1 + 1) ≥ 2 if l is even and if l is odd. With these estimates we get Note that q l ≥ φ l−1 and hence l ≤ log q l Proof of Corollary 2 Since α is of type t > 1 we have Hence the bound from Theorem 4 can be estimated by for N large enough.

Proof of the result on the van der Corput sequence
Let P N := Then P n > 2P n .
Proof of Theorem 5 Consider n with 2 s ≤ n < 2 s+1 . From Lemma 6 and Lemma 7 it follows that for 2 s + 2 s−1 ≤ n < 2 s+1 the product P n has its largest values for and for 2 s ≤ n < 2 s + 2 s−1 the product P n has its largest values for
Let now 2 s + 2 s−1 ≤ n ≤ n 3 be arbitrary. Then 1 n 2 P n ≤ 1 2 s + 2 s−1 2 P n 3 , and the last term tends to 2 9π sin π Hence for all s large enough we have 1 n 2 P n < 1 2π for all 2 s + 2 s−1 ≤ n < n 3 . We still have to consider n with 2 s ≤ n < 2 s + 2 s−1 . With equation (ii) of Lemma 3 we have P n 4 = P n 1  for s to infinity. Furthermore, it is easily checked that P n 5 and P n 6 are smaller than P n 4 . Hence for all n with 2 s ≤ n < 2 s + 2 s−1 we have P n n 2 ≤ P n 4  which tends to π for s to infinity. This gives the lower bound in Theorem 5.

Proof of the probabilistic results
In the first part of this section we consider products where (X k ) k≥1 is a sequence of i.i.d. random variables on [0, 1]. We want to determine the almost sure asymptotic behavior of (11). We take logarithms and define where Y k = log(2 sin(π X k )) is again an i.i.d. sequence. Thus we can apply Kolmogorov's law of the iterated logarithm [27] (see also Feller [12]) in the i.i.d. case. However, for later use we state this LIL in a more general form below.
Thus, the conditions of Lemma 8 are satisfied and we have lim sup , P-almost surely.
Finally, note that by the strong law of large numbers we have, P-almost surely, that Consequently, from (18) we can deduce that lim sup N →∞ N k=1 log(2 sin(π n k α)) √ N log log N = π √ 12 , P-almost surely.