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 vectorthe 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 vectorsis
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 ):
For in and weights we have
In particular, we have
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 .
where we just applied several elementary substitutions. ∎
Another important fact is, that the periodic -discrepancy can be expressed in terms of exponential sums.
where and where for we set
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  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 :
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  called the point sets , and the -sets. We have
For Korobov’s -sets we have
For the proof of Theorem 1 need the following lemma.
Let be a prime number and let . Then for all such that for at least one we have
Proof of Theorem 1.
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
Inserting this bound into the formula given in Proposition 2 gives
This yields the desired result for .
This yields the desired result for .
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
Hence, Theorem 1 shows that the periodic -discrepancy of Korobov’s -set is almost of average order.
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.
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.
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., ). 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.
For and we have
Moreover, for any there exists such that
for all and .
The lower bound for follows directly from the corresponding lower bound for the inverse of the non-periodic -discrepancy, see  or (9.17) in , together with Proposition 1. Similarly, a lower bound
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
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
Let be the smallest prime number larger or equal to
Then it follows from Theorem 1 that for Korobov’s -set with we have
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 .
-  J. Dick: Numerical integration of Hölder continuous absolutely convergent Fourier-, Fourier cosine-, and Walsh series. J. Approx. Theory 184: 111–145, 2014.
-  J. Dick and F. Pillichshammer: The weighted star discrepancy of Korobov’s -sets. Proc. Amer. Math. Soc. 143: 5043–5057, 2015.
-  A. Hinrichs: Discrepancy, integration and tractability. In: Monte Carlo and Quasi-Monte Carlo Methods 2012, pp. 123-163, Springer Proc. Math. Stat., 65, Springer, Berline, 2016.
-  A. Hinrichs and J. Oettershagen: Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives. In: Monte Carlo and Quasi-Monte Carlo Methods 2014, pp. 385-405, Springer Proc. Math. Stat., 163, Springer, Cham, 2013.
-  A. Hinrichs and H. Weyhausen: Asymptotic behavior of average -discrepancies. J. Complexity 28: 425–439, 2012.
-  L.K. Hua and Y. Wang: Applications of Number Theory to Numerical Analysis. Springer, Berlin, 1981.
-  N.M. Korobov: Approximate calculation of repeated integrals by number-theoretical methods. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 115: 1062–1065, 1957.
-  N.M. Korobov: On number-theoretic methods in approximate analysis (Russian). Probl. Numer. Math. Comp. Techn., Gosudarstv. Naučno-Tehn. Izdat. Mašinostr. Lit., Moscow, 1963, pp. 36–44.
-  V.F. Lev: On two versions of -discrepancy and geometrical interpretation of diaphony. Acta Math. Hungar. 69(4): 281–300, 1995.
-  R. Lidl and H. Niederreiter: Finite fields. Encyclopedia of Mathematics and its Applications, 20. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983; 2nd ed., Cambridge University Press, 1997.
-  E. Novak and H. Woźniakowski: Intractability results for integration and discrepancy, J. Complexity 17: 388–441, 2001.
-  E. Novak and H. Woźniakowski: Tractability of Multivariate Problems, Volume II: Standard Information for Functionals. European Mathematical Society, Zürich, 2010.
-  I. H. Sloan and H. Woźniakowski: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, J. Complexity 14, 1–33, 1998.
-  A. Weil: On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34: 204–207, 1948.
-  P. Zinterhof: Über einige Abschätzungen bei derApproximation von Funktionen mit Gleichverteilungsmethoden (German). Österr. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185: 121–132, 1976.