On Weyl products and uniform distribution modulo one.

In the present paper we study the asymptotic behavior of trigonometric products of the form ∏ k = 1 N 2 sin ( π x k ) for N → ∞ , where the numbers ω = ( x k ) k = 1 N 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 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 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

Let f be a function and be a sequence of numbers in the unit interval. Much work was done on analyzing so-called Weyl sums of the form , and on the convergence behavior of to . See for example [8, 17, 36, 41]. It is the aim of this paper to propagate the analysis of corresponding “Weyl products”in particular with respect to their asymptotic behavior for .

Note that, formally, studying products in fact is just a special case of studying , sinceunless for some . Thus we will concentrate on functions f for which (and possibly also .

Assuming an even distribution of the sequence , one expects to tend to the integral if this exists. That means, very roughly, that we expectwhich we can rewrite asHence it makes sense to study the asymptotic behavior of the normalized productA 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 formfor were analyzed. (Note that , hence the normalization factor 2 in this case.)

It was shown there that for almost all and all we havefor all sufficiently large N andfor infinitely many N.

In the present paper we restrict ourselves to and we will extend the analysis of such products to other types of sequences . In particular we will consider two well-known types of uniformly distributed sequences, namely the van der Corput sequence and the Kronecker sequence with irrational . Furthermore, we will determine the typical behavior ofthat is, the almost sure order of this product for “random” sequences in a suitable probabilistic model.

Such sine-products and estimates for such products play an important role in many different fields of mathematics. We just mention a few of them: interpolation theory (see [18, 19]), partition theory (see [42, 48]), Padé approximation (see [33]), KAM theory and q-series (see [2, 15, 24, 26, 29]), analytic continuation of Dirichlet series (see [25, 45]), and many more.

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 be numbers in [0, 1]. Their star-discrepancy is defined aswhere . An infinite sequence in [0, 1] is called uniformly distributed modulo one (u.d. mod 1) if for all we haveor, equivalently,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 in terms of the star-discrepancy of . A similar result in a weaker form was obtained by Hlawka [18] (see also [19]).

Theorem 1

Let be a sequence of real numbers from [0, 1] which is u.d. mod 1. Then for all sufficiently large N we havewhere .

Concerning the quality of Theorem 1, consider the case when is a low-discrepancy sequence such as the van der Corput sequence (which is treated in Theorem 5 below). Then , and Theorem 1 givesfor some 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 for . This point set has star-discrepancy , and hence the general estimate (3) givesTo 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 which is designed such that for infinitely many N we have for .

On the other hand, the product on the left-hand side of (5) is well known to be exactly (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 formwhere are points on the unit circle. The main results in [18, 19] are lower and upper bounds of in terms of the star-discrepancy of the sequence 1 It should also be mentioned that Wagner [45] proved the general lower boundfor infinitely N, where 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 in terms of the star-discrepancy of the sequence . Let be numbers in [0, 1] and let . Let denote the star-discrepancy of . Furthermore, let 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 inwhere the supremum is taken over all with . We will show

Theorem 2

Let be an arbitrary sequence of reals of the form with M(N) positive integers, and . Then we have:

For all there exist and such that for all we have

For all sufficiently large N we have

Let us now focus on products of the formwhere is a given irrational number, i.e., we consider the special case when is the Kronecker sequence . 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 is everywhere dense modulo 1; i.e., the sequence of fractional parts is dense in [0, 1]. At the beginning of the 20th century various authors considered this sequence (and generalizations such as , 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 in the unit interval (as well as in the unit cube ). 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 , where is the sequence of best approximation denominators of ,Moreover we conjecture that alwaysCompare these considerations also with the conjectures stated in [32]. To illustrate these two assertions see Figs. 1 and 2, where for we plot for (Fig. 1) and the normalized version for (Fig. 2). Note that the first best approximation denominators of are given by .

Fig. 1

for and

Fig. 2

for and

For the case for some best approximation denominator q the product 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 thatwhen 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 , 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 . Then

Next we consider general :

Theorem 4

Let be the continued fraction expansion of the irrational number . Let be given, and denote its Ostrowski expansion bywhere is the unique integer such that , where , and where are the best approximation denominators for . Then we have

Corollary 1

For all N with we havewhere and hence

The second part of Corollary 1 can also be obtained from [7, Lemma 4].

In the following we say that a real is of type if there is a constant such thatfor all with .

The next result essentially improves a result given in [25]. There a bound on for of type t of the form instead of our much sharper bound was given. Note that our result only holds for , so we cannot obtain the sharp result of Lubinsky [32] in the case of with bounded continued fraction coefficients.

Corollary 2

Assume that is of type . Then for some constant C and all N large enough .

Now we will deal with , where is the van der Corput-sequence. The van der Corput sequence (in base 2) is defined as follows: for with binary expansion with digits (of course the expansion is finite) the 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:

Theorem 5

Let be the van der Corput sequence in base 2. Thenand

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 studywhere is a sequence of independent, identically distributed (i.i.d.) random variables in [0, 1]. The second probabilistic model are random subsequences of the Kronecker sequences , where the elements of are selected from independently and with probability 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].

Theorem 6

Let be a sequence of i.i.d. random variables having uniform distribution on [0, 1], and letThen for all we have, almost surely,for all sufficiently large N, andfor infinitely many N.

Theorem 7

Let be an irrational number with bounded continued fraction coefficients. Let be a sequence of i.i.d. -valued random variables with mean 1 / 2, defined on some probability space , which induce a random sequence as the sequence of all numbers , sorted in increasing order. SetThen for all we have, -almost surely,for all sufficiently large N, andfor infinitely many N

Remark 1

The conclusion of Theorem 7 remains valid if is only assumed to be of finite approximation type (see [28, Chapter 2, Section 3] for details on this notion).

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 in Theorem 1 is exactly the same as in results obtained by Fukuyama [13] for products and under the “super-lacunary” gap condition .

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 . 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 of bounded variation V(g), any N and numbers we havewhere is the star-discrepancy of . Let andFor letNote, that , henceBy partial integration we obtain(with a positive -constant for small enough). Furthermore, we haveAltogether we have, using the Koksma-Hlawka inequality and since Hencefor some constant . We choose and obtainFor some . Note that can be chosen such that if for .

Next we come to the proof of Theorem 2. We will need several auxiliary lemmas, before proving the theorem.

Lemma 1

For with let . Consider the following point set : If N is even, the is given by the pointstogether with 2M times the point .

If N is odd, the is given bytogether with times the point .

Then has star-discrepancy

If any of the points of is moved nearer to , 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 for as it is plotted in Fig. 3.

Fig. 3

Discrepancy function of

Lemma 2

For as in Lemma 1 we have .

Let denote the points of . Assume, there is another N-point set different from with points such that and . Let be minimal such that , and assume that is chosen such that this is maximal. If i is such that , then , otherwise (see Fig. 3) we had .

By translating to we obtain a new point-set with , and , a contradiction.

In the analogous way we can argue if i is such that , or such that . Hence, such an cannot exist.

Lemma 3

For all and all we have

, and

The proof of Equation (ii) is based on noting that and are the zeros of . Then, the polynomial has 2N zeros and these areHence, we getTaking and , the last equation is written asThis is a standard formula that can be found in [14, Formula 1.392].

Putting , the proof of assertion (ii) is complete. Equation (i) follows immediately from Equation (ii) by noting thatLetting and using l’Hospital’s rule, we conclude thatAnother nice proof of Equation (i) can be found for example in [35].

Lemma 4

There is an such that for all we have

This follows immediately from the Taylor expansion

Lemma 5

There is an such that for all we have

This follows from

Proof of Theorem 2

Let with (for the result is easily checked by following the considerations below) and as in Lemmas 1 and 2. Note that depends on N. We assume M(N) even. For M(N) odd the calculations are carried out quite analogously. We have, using also equation (i) of Lemma 3,Note that the function is of the form as presented in Fig. 4. Hence for we haveBy Lemma 4 for all M with for the integral above we haveand hence, using also Lemma 5,This givesand consequentlyThis proves assertion (b) of Theorem 2.

Fig. 4

The function

On the other hand we haveand henceThis givesand consequentlyIt remains to show that for all there are and such that for all the right hand side of (9) is at most .

To this end let be large enough such that for all we have . Furthermore, let be large enough such that for all the value is so small such thatThen for all and all we haveIf , then the penultimate expression can be estimated bywhereThis implies the desired result.

Proofs of the results for Kronecker sequences

Proof of Theorem 3

Let with , where is the best approximation denominator following q. The case of negative can be handled quite analogously. There is exactly one of the points for in each interval for . Note that the point in the interval is the point , where is the best approximation denominator preceding q. We haveHence, on the one hand (by equation (i) of Lemma 3),On the other hand

Proof of Theorem 4

Let for and . ThenWe considerLet with, say, . (The case of negative is handled quite analogously.)

Let for some and , then, with and we havefor some . Since , for given d there is always exactly one point in the interval for each (the interval taken modulo one).

We replace now the points by new points, namely:Using the new points instead of the by construction we obtain an upper bound for . ThenHenceBy equation (ii) of Lemma 3 we haveand henceNote that and therefore also always. HenceHenceWe havesince for . Henceand thereforeas desired.

if with then in the representation (10) of we replace by 0, unless .

if with then in the representation (10) of we replace by

if , where is such that then

for the d such that in the representation (10) of we replace by 0,

for the d such that in the representation (10) of we replace by ,

for the single such that we replace by .

if with , then

for the h such that in the representation (10) of we replace  by ,

for the h such that in the representation (10) of we replace by 0,

for the single such that we replace in the representation (10) of the by 0 if and by otherwise, where here and in the following we use the notation . Let the second be the case, the other case is handled quite analogously.

Proof of Corollary 1

By Theorem 4 we haveWe haveBy iteration we obtainif l is even andif l is odd. With these estimates we getNote that and hence , where .

Hence

Proof of Corollary 2

Since is of type we haveand hence . Especially we have the following: Let , then, because ofwe haveHence the bound from Theorem 4 can be estimated byfor N large enough.

Proof of the result on the van der Corput sequence

Letwhere is the element of the van der Corput sequence.

Lemma 6

Let (in dyadic representation)andThen .

We haveSince withwe obtain from equation (ii) of Lemma 3 Furthermore, withand hence, again by equation (ii) of Lemma 3,Note that .

In the same way we have withand hence by equation (ii) of Lemma 3 SoWe have to show thatSince and it follows thatLet and . Then we have andLet . Then we have andHere we used that for is minimal for and for is minimal for .

Lemma 7

We have:

We only prove (ii), which is the most elaborate part of the lemma. The other assertions can be handled in the same way but even simpler. In (ii) we havewith . HenceHere and . Some tedious but elementary analysis of the functionfor and shows that in this region. Hence .

Proof of Theorem 5

Consider n with . From Lemma 6 and Lemma 7 it follows that for the product has its largest values forand for the product has its largest values forBy equation (i) of Lemma 3 we havehence for s to infinity. Furthermore,and hence for s to infinity. FinallyLet now be arbitrary. Thenand the last term tends toHence for all s large enough we have for all .

We still have to consider n with . With equation (ii) of Lemma 3 we haveThe product of the last two factors tends to for s to infinity.

Furthermore, it is easily checked that and are smaller than . Hence for all n with we havewhich tends to for s to infinity. So altogether we have shown thatFrom Lemma 6 and from equation (i) of Lemma 3 it also follows that for all s we havewhich tends to 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 productswhere 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 definewhere 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.

Lemma 8

Let be a sequence of independent random variables with expectations and finite variances , and let . Assume there are positive numbers such thatThen satisfies a law of the iterated logarithm

In the case of centered i.i.d. random variables with finite variance, we have with . Thus in this caseIn order to apply Lemma 8 to the sum (12), we note thatand compute the varianceThis proves Theorem 6.

For the proof of Theorem 7 we split the corresponding logarithmic sum into two partswhere denotes the Rademacher function on [0, 1] and the space of subsequences of the positive integers corresponds to [0, 1] equipped with the Lebesgue measure. For irrationals with bounded continued fraction expansion, by Corollary 1 we haveFor the second sum in (16) we set and apply Lemma 8. The random variables are clearly independent and thus we have to compute the quantities and check condition (13). Obviously, and . Using the fact thatwith some positive constant , we obtainwith some . Using Koksma’s inequality and discrepancy estimates for it can easily been shown thatThus, the conditions of Lemma 8 are satisfied and we haveConsequently, from (16) and (17) we obtainFinally, note that by the strong law of large numbers we have, -almost surely, thatConsequently, from (18) we can deduce thatThis proves Theorem 7.