A note on the periodic L_2-discrepancy of Korobov's p-sets

01/07/2020
by   Josef Dick, et al.
0

We study the periodic L_2-discrepancy of point sets in the d-dimensional torus. This discrepancy is intimately connected with the root-mean-square L_2-discrepancy of shifted point sets, with the notion of diaphony, and with the worst case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory many results are based on averaging arguments. In order to make such results relevant for applications one requires explicit constructions of point sets with “average” discrepancy. In our main result we study Korobov's p-sets and show that this point sets have periodic L_2-discrepancy of average order. This result is related to an open question of Novak and Woźniakowski.

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1 Introduction

In this note we study the periodic -discrepancy which is a quantitative measure for the irregularity of distribution of a point set, but which is also closely related to the worst-case integration error of quasi-Monte Carlo integration rules (see, for example, [3, 4, 5, 9]). In order to state its definition we first explain the “test-sets” that are considered in this specific notion of discrepancy.

For we define the periodic “interval” as

For dimension and and in the periodic “boxes” are given by

The local discrepancy of a point set consisting of elements in the -dimensional unit cube with respect to a periodic box is given by

Then the periodic -discrepancy of is the -norm of the local discrepancy taken over all periodic boxes , i.e.,

The usual -discrepancy of a point set is defined as

In the closely connected context of worst case errors for cubature formulas it is natural to extend these notions to weighted point sets. If, additionally to the point set we also have a set of associated real weights , the local discrepancy of the weighted point set is given as

Then is obtained for equal weights . The periodic -discrepancy of the weighted point set is

The usual -discrepancy of the weighted point set is

For a (weighted) point set

and a real vector

the shifted point set is defined as

where means that the fractional-part-function for non-negative real numbers , is applied component-wise to the vector . The root-mean-square -discrepancy of a shifted (and weighted) point set

with respect to all uniformly distributed shift vectors

is

Note that

If the weights are the standard equal weights , we drop from the notation.

The following relation between periodic -discrepancy and root-mean-square -discrepancy of a shifted point set holds (see also [9]):

Proposition 1.

For in and weights we have

In particular, we have

Proof.

We write down the proof for dimension and the equal weighted case only. The general case follows by similar arguments. We have

We consider two cases:

  • If , then iff or and this holds iff . Hence

    Note that .

  • If , then iff or and this holds iff . Hence

    Note that .

Hence,

where we just applied several elementary substitutions. ∎

Another important fact is, that the periodic -discrepancy can be expressed in terms of exponential sums.

Proposition 2.

We have

where and where for we set

Proof.

See [4, p. 390]. ∎

The above formula shows that the periodic -discrepancy is—up to a multiplicative factor—exactly the diaphony which is a well-known measure for the irregularity of distribution of point sets and which was introduced by Zinterhof [15] in the year 1976.

In Section 2

we will estimate the periodic

-discrepancy of certain types of point sets that recently gained some attention in the context of error bounds with favourable weak dependence on the dimension (see [1, 2]). On the other hand, we also show in Section 3 that even the weighted version of the periodic

-discrepancy suffers from the curse of dimensionality.

2 The periodic -discrepancy of Korobov’s -sets

Let be a prime number. We consider the following point sets in :

  • Let with

    The point set was introduced by Korobov [8] (see also [6, Section 4.3]).

  • Let with

    The point set was introduced by Korobov [7] (see also [6, Section 4.3]).

  • Let with

    Note that is the multi-set union of all Korobov lattice point sets with modulus . The point set was introduced by Hua and Wang (see [6, Section 4.3]).

Hua and Wang [6] called the point sets , and the -sets. We have

Theorem 1.

For Korobov’s -sets we have

For the proof of Theorem 1 need the following lemma.

Lemma 1.

Let be a prime number and let . Then for all such that for at least one we have

(1)
(2)
(3)
Proof.

Eq. (1) follows from a bound from A. Weil [14] on exponential sums which is widely known as Weil bound (see also [10, Theorem 5.38]). For details we refer to [1]. For a proof of Eq. (2) we refer to [6, Lemma 4.6]. A proof of Eq. (3) can be found in [2]. ∎

Proof of Theorem 1.

We use the formula from Proposition 2 for the periodic -discrepancy and estimate the exponential sum with the help of Lemma 1. We provide the details only for and . The proof for is analoguous.

Before we start we mention the following easy results that will be used later in the proof. For we write to indicate that for all . Likewise, indicates that there is at least one index with . We have

and

First we study . Then . Using Eq. (1) from Lemma 1 we have

Inserting this bound into the formula given in Proposition 2 gives

This yields the desired result for .

Now we turn to . Here . Using Eq. (2) from Lemma 1 we have

Hence

This yields the desired result for .

In order to prove the bound on use Eq. (3) from Lemma 1. ∎

Remark 1.

Note that the dependence on the dimension of our bound on the periodic -discrepancy is only , which looks very promising regarding tractability properties at first sight. However, it can be easily checked that already the periodic -discrepancy of the empty set is only (see the forthcoming Lemma 2) which is much smaller than . We will see in the next section that the periodic -discrepancy suffers from the curse of dimensionality.

In discrepancy theory many results are based on averaging arguments. In order to make such results relevant for applications one requires explicit constructions of point sets with “averge” discrepancy. It is easily checked that the average squared periodic -discrepancy

equals

Hence, Theorem 1 shows that the periodic -discrepancy of Korobov’s -set is almost of average order.

Remark 2.

There is some relation to Open Problem 40 in [12, p. 57]. There the authors ask for a construction of an -element point set in dimension in time polynomial in and for which the -discrepancy is less than the average -discrepancy taken over all -element point sets in dimension . Here we have—up to a linear factor —the answer to this question for the periodic -discrepancy.

Remark 3.

Our result also gives some information about the usual -discrepancy: The average -discrepancy is

and this is well-known to be

Using Proposition 1 we obtain that the root mean square -discrepancy of the shifted -sets is at most and hence almost of average order. This also implies that there must exist a shift for the -sets such that

Whether the -discrepancy of the -sets itself satisfies the bound is an interesting open problem.

3 The curse of dimensionality for the periodic -discrepancy

In this section we are interested in the inverse of periodic -discrepancy which is the minimal number for which there exists an -element point set in whose periodic -discrepancy is less then times the initial periodic -discrepancy. Since lower bounds become stronger if proved also for the weighted version, we consider here also the inverse of the weighted periodic -discrepancy and compare the results to the results for the unweighted case and to the case where only positive weights are allowed.

Lemma 2.

The initial periodic -discrepancy is

We omit the easy proof of this lemma.

Hence, for and

Analogously, we define and if arbitrary or only positive weights are allowed, respectively.

In information based complexity one is particularly interested in the dependence of the inverse discrepancy on and . In particular, a polynomial dependence on would be favourable (see, e.g., [12]). This however is not achievable in the case of the periodic -discrepancy. Indeed, the following theorem shows that the periodic -discrepancy suffers from the curse of dimensionality.

Theorem 2.

For and we have

Moreover, for any there exists such that

for all and .

Proof.

The lower bound for follows directly from the corresponding lower bound for the inverse of the non-periodic -discrepancy, see [11] or (9.17) in [12], together with Proposition 1. Similarly, a lower bound

follows from the corresponding lower bound for the inverse of the non-periodic -discrepancy, see [13] or (9.16) in [12], together with Proposition 1.

To prove the better lower bound for we observe that according to [4, p. 389-390] we have

where is the second Bernoulli polynomial, and is the coordinate of the point . Note that and for we have . Now we have

Hence,

implies

and therefore

The slightly worse lower bound for follows similarly using

If the weights are nonnegative, estimating the double sum by the diagonal terms, we have

Now it is easily seen that

implies

Remark 4.

Let be the smallest prime number larger or equal to

Then it follows from Theorem 1 that for Korobov’s -set with we have

and hence

where we used Bertrand’s postulate, which tells us that .

This means, that the term in the lower bounds in Theorem 2 is the exact basis for the exponential dependence of the information complexity in .

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