THE CONSTRUCTION OF GOOD EXTENSIBLE RANK-1 LATTICES. 1. Introduction We are interested in approximating a high dimensional integral [0,1]

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1 MATHEMATICS OF COMPUTATION Volue 00, Nuber 0, Pages S (XX) THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Abstract. It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield sall integration errors for n = p, p 2,... for soe prie p. This paper provides algoriths for the construction of generating vectors which are finitely extensible. The proofs which show that our algoriths yield good extensible rank- lattices are based on a sieve principle. Particular fast algoriths are obtained by using the fast coponent-by-coponent construction of Nuyens and Cools. Analogous results are presented for generating vectors with sall weighted star discrepancy.. Introduction We are interested in approxiating a high diensional integral [0,] f(x) dx s by an equal weight quadrature rule n n k=0 f(t k), where the quadrature points t 0,..., t n are deterinistically chosen fro [0, ) s. Such rules are called quasi- Monte Carlo rules. After putting soe restrictions on the integrand concerning its soothness the question of how to choose the quadrature points becoes iportant. Several branches of quasi-monte Carlo rules, which are nowadays quite interwoven, are known. Those are lattice rules [20, 25], Kronecker type sequences [7, 8, 9] and digital nets and sequences, see [9, 20, 22] and the references therein. In this paper we study lattice rules, where the quadrature points are given by t k = {kz/n + }, k = 0,..., n. Here {x} = x x denotes the fractional of each coponent of a real vector x, z is an integer vector and [0, ) s. The shift is either chosen 0 (for periodic functions) or i.i.d. (for non-periodic functions). We shall refer to the set of n such points as the point set P n,s (z, ) = {t k } n k=0 or P n,s (z) when = 0 or when the shift is averaged out (see below). An arbitrary point set will be denoted by P n,s. The quality of the lattice rule thus depends on the choice of the integer vector z. Until now no explicit constructions of z are known (except for diension s = 2), hence one has to resort to coputer search. Several construction algoriths for the generating vector z have been introduced and analysed [3, 5, 26, 28]. For certain Sobolev spaces H s of integrands one can 2000 Matheatics Subject Classification. Priary K45, 65C05, 65D30. Key words and phrases. Quasi-Monte Carlo, nuerical integration, extensible lattice rule, coponent-by-coponent construction. Supported by the Austrian Research Foundation (FWF), Project S 9609, that is part of the Austrian Research Network Analytic Cobinatorics and Probabilistic Nuber Theory. The support of the Australian Research Council under its Centre of Excellence progra is gratefully acknowledged. c 997 Aerican Matheatical Society

2 2 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE obtain an explicit for of the root ean square worst-case error ( ) /2 () e n,s (P n,s (z); K s ) := e 2 n,s(p n,s (z, ); K s )d, [0,] s where e n,s (P n,s (z, ); K s ) = sup I s (f) Q n,s (f), f H s f n k=0 f(t k). Such a forula can then be used for a coputer with Q n,s (f) = n search of the generating vector z of a lattice rule. The proofs of the upper bounds on the integration error for certain classes of integrands depend on soe averaging arguent where one obtains a bound on the average of the worst-case error over all (or a suitable subset of) generating vectors (note that we can restrict z {,..., n } s ). As there is always at least one generating vector which is better than average one can introduce a suitable algorith to find a generating vector better than average. This approach has been used successfully to show that search algoriths (Korobov algorith [28] or coponent-by-coponent algorith [3, 5, 26]) yield good generating vectors. In the case of coponent-by-coponent algoriths the generating vector can even be extended in the nuber of diensions. Though a desirable property [9], an extension in the nuber of points on the other hand has until now not been shown to be possible with these types of algoriths. That an extension of lattice rules in the nuber of points is possible at least theoretically has been shown in [0]. Such lattice rules are nowadays called extensible lattice rules. In this case the quadrature points are given by t k = {φ (k)z + }, where for k = k 0 + k p + + k p the radical inverse function φ is defined by φ (k) = k 0 p + + k p. Here z is a vector of p-adic nubers (see [2] for a definition of p-adic nubers; for our purposes here it is enough to assue that z is a vector of natural nubers hence we will not introduce p-adic nubers here). The existence of good extensible lattice rules has been proven in [2] (see Section 7 about a discussion of the precise eaning of extensible ). Therein it was shown that there exists a generating vector z which yields a lattice rule which is good for all oduli p, p 2, p 3,.... The prove is based on a ore sophisticated averaging arguent. After the existence had been established it reained a challenge to provide soe construction algorith which yields generating vectors for extensible lattice rules. Several successful nuerical investigations have been carried out [2, ], but a proof that those algoriths yield good extensible lattice rules was not provided. In this paper we provide such an algorith together with a proof (see also [4] for an analogue for polynoial lattice rules). The arguent for the proof is indeed siilar to the one used in [2] (and also [2]). It uses a cobination of Markov s inequality, Jensen s inequality and an extension of the following siple fact: let A, B be two subsets of a finite set N and let N denote the nuber of eleents in N. Then A, B > N /2 iplies that A B. (We use an extension of this to an arbitrary nuber of subsets of N.) Using these principles we can obtain both, an algorith and a proof, thereby providing first construction algoriths for extensible lattice rules. To speed up the algorith we show that we can also use a coponent-by-coponent approach [26] together with the fast coputation ethod introduced in [23, 24]. This way we obtain a practically feasible construction of

3 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 3 extensible lattice rules in (for any applications) sufficiently large diensions and range of oduli. Unfortunately our construction algoriths are not extensible in both n and s siultaneously. The first sieve algorith (see Section 3), though slow, is in principle extensible in n, but is not extensible in the diension. The CBC sieve algorith and the fast CBC sieve algorith (see Section 4) construct generating vectors for a range of oduli and is extensible in the diension, but once the vector is constructed it is not possible to extend the vector to work well also for other oduli. Hence the CBC sieve constructions provide ebedded rather then extensible lattice rules [2]. Obtaining an algorith which is extensible in both n and s siultaneously reains an interesting open question. See also Section 7 at the end of the paper for a discussion of this topic. The rest of the paper is arranged as follows. In Section 2 we discuss the function space setting of weighted reproducing kernel Hilbert spaces. In Section 3 we introduce a sieve algorith which outlines the ideas used for the algorith and the proof. In Section 4 we extend the results fro Section 3 by showing the sieve algorith can be cobined with a coponent-by-coponent approach allowing us to efficiently construct generating vectors of high diension. Further we also show that the fast CBC construction of [23] can be incorporated in the algorith as well. Section 5 contains soe brief nuerical experients, whereas in Section 6 we develop siilar results to those in Sections 3 which are based on iniising the quantity R n,s,γ (z), rather than the worst-case error. Using these results we are able to construct extensible lattice point sets with sall weighted star discrepancy. Finally, in Section 7 we state several rearks and ention soe open probles. 2. Reproducing kernel Hilbert spaces In this section we introduce classes of integrands for which we consider nuerical integration. Reproducing kernel Hilbert spaces are nowadays widely used in nuerical analysis and other areas and are also used here to define function classes of integrands. The theory of reproducing kernels was developed in [], see also [8] where reproducing kernel Hilbert spaces where used to investigate nuerical integration. A reproducing kernel Hilbert space over [0, ] is a Hilbert space H aditting a function K : [0, ] [0, ] R such that K(, y) H for all y [0, ] and f, K(, y) H = f(y) for all y [0, ] and f H. A kernel function K with these properties is unique and it can be shown that K is also syetric and positive definite. For diensions s > we consider tensor products of one-diensional spaces. It can be shown that the reproducing kernel for those spaces is just the product of the one-diensional kernels, i.e., K(x, y) = s K(x j, y j ), where x = (x,..., x s ) and y = (y,..., y s ). In the following we introduce the particular reproducing kernel Hilbert spaces in which nuerical integration is frequently considered [3, 5, 6, 8, 2, 3, 5, 6, 25, 26, 27, 28]. 2.. Weighted Sobolev spaces. We consider a tensor product Sobolev space H s,γ of absolutely continues functions whose partial ixed derivatives of order one in each variable are square integrable. The nor in the unanchored weighted

4 4 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Sobolev space H s,γ [5] is given by f Hs,γ = γ j u {,...,s} j u [0,] u ( ) 2 u f(x) dx {,...,s}\u dx u [0,] x s u u where u / x u f denotes the partial ixed derivative with respect to all variables j u. Here and in the rest of the paper the quantities γ j are non-negative real nubers called weights which are introduced to odify the iportance of different coordinate directions [27]. The reproducing kernel of the s-diensional unanchored weighted Sobolev space [5] is given by ( [ ( (2) K s,γ (x, y) = + γ j 2 B 2( x j y j ) + x j ) ( y j )]), 2 2 where B 2 ( ) denotes the Bernoulli polynoial of degree 2, given by B 2 (x) = x 2 x + /6, which can also be written as (3) B 2 (x) = x 2 x + 6 = 2π 2 h= e 2πihx h 2 x [0, ]. Here and throughout this paper the notation indicates a suation with the zero ter excluded. We can associate a shift invariant kernel [8] with K s,γ by setting Ks,γ(x, sh y) = K s,γ ({x + }, {y + }) d. [0,] s The shift-invariant kernel associated with K s,γ is given by (4) Ks,γ(x, sh y) = ( + γ j B 2 ( x j y j )). Using these definitions it follows that the ean square worst-case error () for the weighted Sobolev space is given by [8] e 2 n,s,γ (P n,s; K s,γ ) = Ks,γ sh (x, y) dxdy 2 n (5) Ks,γ sh [0,] n (x, t k) dx 2s [0,] s + n n n 2 Ks,γ(t sh i, t k ), k=0 i=0 which for a randoly shifted (extensible) lattice rule can be siplified to (see for exaple [26]) (6) e 2 n,s,γ(p n,s ; K s,γ ) = + n K sh n s,γ(t k, 0) = + n ( + γ j B 2 (t k,j )). n k=0 k=0 k=0 Note that the above forula can easily be evaluated using (3) for a given point set P n,s. Note that the shift-invariant kernel Ks,γ sh is related to the reproducing kernel of a certain weighted Sobolev space of periodic functions which we introduce in the following. /2,

5 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES Weighted Korobov spaces. The s-diensional weighted Korobov space H per,s,α,γ has a reproducing kernel of the for [8] ( ) e 2πih(xj yj) (7) K per,s,α,γ (x, y) = + γ j h α where for h = (h,..., h s ), r α (h, γ) = h= = h Z s e 2πih (x y) r α (h, γ) r α (h j, γ j ) and r α (h j, γ j ) = { if h j = 0, γ j h j α if h j 0. The paraeter α restricts the convergence of the Fourier coefficients of the functions in the Korobov space. Throughout the paper we will assue that α >. Equation (5) can again be used to obtain a forula for the worst-case error in the Korobov space H per,s,α,γ, (8) (9) (0) e 2 per,n,s,α,γ (P n,s(z); K per,s,α,γ ) = + n It follows fro (3), (6) and (8) that = + n = n k=0 n k=0 h Z s \{0} h z 0 (od n) ( + γ j h= e2πikh z/n r α (h, γ) h Z s r α (h, γ). () e n,s,2π2 γ(p n,s (z); K s,γ ) = e per,n,s,2,γ (P n,s (z); K per,s,α,γ ), e 2πikhzj/n where 2π 2 γ denotes the sequence of weights (2π 2 γ j ) j. Thus the results shown in the following are valid for the root ean square error for nuerical integration in the Sobolev space as well as for the worst-case error for nuerical integration in the Korobov space. Hence it is enough to state the only for e per,n,s,α,γ (equation () can be used to obtain results also for e n,s,γ ). In the following section we introduce the arguents used for obtaining an algorith and a proof for the construction of good extensible lattice rules. 3. The sieve algorith In [2] the authors used p-adic nubers to show the existence of good extensible lattices. Basically we could use p-adic nubers too, but as we focus on the construction of extensible lattices by coputer search, which in practice (though theoretically possible if one lets the coputer search infinitely long) can only be finite, it is enough in our case to assue that the generating vector is in a subset of N s. Throughout the paper let p be an arbitrary but fixed prie nuber. Then we restrict the set of adissible generating vectors to (2) Z s p = {z = (z,..., z s ) N s : gcd(z j, p) =, j =,..., s}. h α )

6 6 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Clearly, there is an infinite nuber of eleents in this set. Since e 2 per,p,s,α,γ (ẑ) = e 2 per,p,s,α,γ(z) if z ẑ (od p ), we exploit the structure inherent in lattice rules by defining the set (3) Z s p, = {z Z s p : z j < p, j =,..., s}. Where s = for the above definitions, we will typically oit the superscript. For any positive integer, a vector z Z s p has soe corresponding vector z Z s p, such that z z (od p ). Note there are φ(p ) s eleents in the set Z s p,, where φ is Euler s totient function and therefore φ(p ) = p p. 3.. Bounds on the worst-case error. In this section we prove soe essential results which will shed light on how we intend to construct good extensible lattice rules. In the following let ζ(α) = i= i α denote the Rieann zeta function. Theore. Let p be a prie and let, s be positive integers. Then we have φ(p ) s e 2 per,p,s,α,γ (z) E2 p,s,α,γ where z Z s p, En,s,α,γ 2 = ( + 2γ j ζ(α)). n Further there exists a vector z Zp, s such that for any λ (/α, ] where E 2 n,s,α,γ (λ) = n /λ e 2 per,p,s,α,γ (z) E2 p,s,α,γ (λ) ( + 2γ λ j ζ(αλ) ) Proof. The proof of the first part is siilar to the proof of [6, Theore 2.2]. We see that φ(p ) s e 2 per,p,s,α,γ (z) z Z s p, = + p p k=0 + γ j φ(p ) By [6, Lea 2.] we see that φ(p ) z j Z p, h= e 2πikhzj/p h α z j Z p, h= /λ. e 2πikhzj/p h α. = 2ζ(α) gcd(p, k) α φ(p ) p ( p) < 0 for any prie p and k p. The result follows iediately fro this. The proof of the second part is siilar to the proof of a result in [6] which akes use of Jensen s inequality, naely that for a sequence of positive nubers {a k } (4) ( /λ a k ak) λ for all 0 < λ. k k

7 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 7 Note that in the theore we ake the restriction λ > /α so that the function ζ(αλ) is well defined. We will ake this restriction throughout the paper. Cobining the worst-case error in (0) with Jensen s inequality we see that for any λ (/α, ] we have (5) e 2 per,p,s,α,γ (z) (e 2 per,p,s,αλ,γ λ(z) ) /λ. Here and in the following γ λ denotes the sequence of weights γ λ = (γ λ, γλ 2,...). Fro part of this theore, we know that φ(p ) s e 2 per,p,s,αλ,γ λ(z) E2 p,s,αλ,γ λ z Z s p, and hence it follows that for any λ (/α, ] there exists a vector z λ Zp, s such that e 2 per,p,s,αλ,γ (z λ λ ) Ep 2,s,αλ,γ. Putting these two results together, we find λ that for any λ (/α, ] there exists a vector z λ Zp, s such that ( e 2 per,p,s,α,γ(z λ ) ) λ E 2 p,s,αλ,γ λ. Let now z Zp, s such that e2 per,p,s,α,γ (z) = in z Zp, s e2 per,p,s,α,γ(z). Then we obtain ( ) /λ e 2 per,p,s,α,γ (z) e2 per,p,s,α,γ (z λ) Ep 2,s,αλ,γ = E 2 λ p,s,α,γ (λ) for any λ (/α, ]. This gives the desired result. We wish to define a probability easure over the set of all generating vectors Zp s [2, 2]. We would like to do so such that the easure of corresponding vectors in s is equiprobable. For N let µ s, be the equiprobable easure on the set Zp,. s We say a subset A of Zp s is of finite type, if there exists an integer = (A) N and a subset A of Zp, s such that A = {z Z s p : (z (od p )) A }. The easure of the finite type subset A is then defined as Thus, µ s (A) := µ s,(a) (A ). (6) µ s (A) = #A φ(p ) s. (Of course the nuber = (A) is not uniquely defined by A since if works, then also any nuber larger than will work in the definition of a finite type subset. It is easy to see that (6) does not depend on the specific choice of.) We now define the following set. For a real c define the set (7) C n,s,α,γ (c) = {z Z s p : e2 per,n,s,α,γ (z) ce2 n,s,α,γ }. This set has the following property. Theore 2. Let, s be positive integers. For any c we have µ s (C p,s,α,γ(c)) > c. Proof. This follows iediately fro applying Markov s inequality to the first part of Theore.

8 8 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE We now ake a sall adjustent to this set which allows us to incorporate Jensen s inequality, see [6] where a siilar arguent was used. For a real c define the set (8) C n,s,α,γ (c) = {z Zp s : e2 per,n,s,α,γ (z) c/λ En,s,α,γ 2 (λ) for all /α < λ }. We obtain the following theore. Theore 3. Let, s be positive integers. For any c we have µ s ( C p,s,α,γ(c)) > c. Proof. Let c be given and choose /α < λ such that c /λ E 2 n,s,α,γ(λ ) c /λ E 2 n,s,α,γ(λ) for all /α < λ. Fro Theore 2 we see that (9) µ s (C p,s,αλ,γ λ (c)) > c. Now, if z C p,s,αλ,γ λ (c), then e 2 per,p,s,αλ,γ λ (z) ce2 p,s,αλ,γ λ. By (5) this iplies that ( e 2 per,p,s,α,γ(z) ) λ ce 2 p,s,αλ,γ λ, which can be re-written as e 2 per,p,s,α,γ (z) c/λ ( E 2 p,s,αλ,γ λ ) /λ = c /λ E p,s,α,γ(λ ), which in turn iplies that z C p,s,α,γ(c). This eans that C p,s,αλ,γ λ (c) C p,s,α,γ(c). Using (9) as a lower bound, we find that µ s ( C p,s,α,γ(c)) µ s (C p,s,αλ,γ λ (c)) > c. In the following we will use the above theore to construct lattices for a range of oduli The sieve principle. We now want to construct lattice rules which work well for several choices of. Let p be the lowest nuber of points and p 2 be the highest nuber of points in which we are interested in, i.e., 2. Then for each =,..., 2 we can define a set C p,s,α,γ(c ) as in (8). In order to obtain a generating vector which works well for all choices of =,..., 2 we need to show that the intersection 2 Cp =,s,α,γ(c ) is not epty, or equivalently, has easure greater than 0. To this end choose c such that (20) 2 = c, then the easure of the intersection of the sets above can be shown to be strictly positive. In the following we will write C c p,s,α,γ (c ) to denote the copleent of the set C p,s,α,γ(c ) in Z s p.

9 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 9 Theore 4. Let s be a positive integer and let 0 < 2. Let c for all =,..., 2 such that 2 = c. Then there exists a vector z Zp s such that e 2 per,p,s,α,γ(z) c /λ Ep 2,s,α,γ(λ) for all /α < λ and =,..., 2. ( 2 ) Proof. We need to show that µ s Cp =,s,α,γ(c ) > 0. This is a siple calculation, ( 2 ) ( 2 ) µ s Cp,s,α,γ(c ) = µ s Cc p,s,α,γ (c ) = = 2 µ s ( C p c,s,α,γ(c )) = 2 > 0. = c The arguents used to prove Theore 4 are very siilar to the arguents used in [2]. (Using p-adic nubers we could indeed also allow 2 to be infinity. As in [2], using the above arguents, it is also possible to show the existence of a large nuber of good generating vectors.) An advantage of our presentation is aybe that it is ore apparent of how an algorith for the construction of good generating vectors can be obtained fro the arguents in the proof. This is done in the following section The sieve algorith. In this subsection we introduce the idea of how a good generating vector can be found by describing a sieve algorith for the construction of a generating vector z Zp s which works well for =,..., 2. This algorith is quite slow, but in later sections we will give soe odifications which speed up the sieve algorith. We wish to find a vector z Zp s which satisfies e 2 per,n,s,α,γ(z ) c /λ E 2 n,s,α,γ(λ) for all /α < λ and =,..., 2. That is, we wish to find a vector z Zp s that lies in 2 = Cp,s,α,γ(c ). For = we use coputer search to find ( c )φ(p ) s + of the φ(p ) s vectors in Zp, s vectors, which satisfy e 2 per,p,s,α,γ(z) c /λ Ep 2,s,α,γ(λ) for all /α < λ and label this set T. By Theore 3 we know that at least such a nuber of the exist. We then construct the set S + of all vectors z Zp, s +, such that there exists soe z T with z z (od p ). Fro the set S + we only keep ( (c c + + ))φ(p+ ) s + vectors which satisfy the inequality e 2 per,p +,s,α,γ (z) c /λ + E2 p +,s,α,γ (λ) for all /α < λ and label this set T +. Again by Theore 3 we know there ust be at least ( c + )φ(p+ ) s + vectors in Zp, s which satisfy + e2 per,p +,s,α,γ (z ) c /λ + E2 (λ) for all p +,s,α,γ /α < λ. Therefore, there ust be at least ( (c + c + ))φ(p+ ) s +

10 0 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE vectors which satisfy e 2 per,p,s,α,γ(z ) c /λ Ep 2,s,α,γ(λ) for all /α < λ as well. In the sae way, we construct the sets S +2, T +2,..., S 2 and T 2. Finally, by Theore 4 above, T 2 is guaranteed not to be epty. We ay select z to be any vector fro T 2 (see Section 4.2 for soe coents on how to choose a vector fro T 2 ). Reark. In principle we can allow 2 to be infinite, i.e., we can choose c such that = c. Then we can stop the coputer search at soe finite >. If one stores all the necessary values fro the initial search it is then also possible to resue the coputer search at a later point in tie to obtain an extensible lattice rules also for oduli larger than p. Hence the construction is truly extensible in the odulus (see also Section 7 for ore inforation on extensibility). As we will show in the next section, the vector can also be extended in the diension using a CBC approach, but once this is done, it becoes ebedded (see [2]) rather than extensible in the odulus, since the values of and 2 ay not be altered once chosen. Further, as can be seen fro the arguents above, one need not choose successive values of, i.e., one could choose an arbitrary subset K N and construct a good lattice rule with p points for all K. See also [4] for further coents. Reark 2. The constants c for =,..., 2 ay be chosen to be any positive sequence of reals such that (20) is satisfied. If 2 is finite, one possible choice of c to satisfy (20) is c = 2 +. This corresponds to the lattice rule having in soe sense the sae quality for each value of. This choice will be used later in Section 5. If 2 is chosen to be infinite, we cannot choose c to be independent of as we did above. Instead, the constants c ust grow with sufficiently fast so that the su in (20) converges. One possible choice is c = C(log( + )) +ɛ for any ɛ > 0 where C is chosen to be larger than = (log( + )) (+ɛ). This is the choice used in [2]. A siilar choice would be c = ζ( + ɛ) +ɛ again for any ɛ > 0. The following theore now applies to generating vectors constructed by the sieve algorith. Theore 5. Let s be a positive integer and let 0 < 2. Let c for all =,..., 2 such that 2 = c. Then the sieve algorith constructs a vector z Zp s such that e 2 per,n,s,α,γ (z ) c /λ E2 n,s,α,γ (λ) for all /α < λ and =,..., 2. Note that it is always possible to choose c of order +ε for soe ε > 0, hence the factor c in the bound in the above theore contributes at ost another factor of (+ε)/λ = (log n) (+ε)/λ, where n is the nuber of points. It can be shown that for every 0 < δ < there is a constant D δ > 0 such that (log n) c n D δ n δ, hence for every /α < λ there is a constant C λ > 0 such that e 2 per,p,s,α,γ (z ) C λ p /λ ( + 2γ λ j ζ(αλ) ) /λ

11 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES for all =,..., 2. (Here the constant C λ ay depend on the particular choice of c ; on the other hand there is also a constant C λ even if 2 =, see [2]). So using the sieve algorith we can construct generating vectors for lattice rules which achieve the optial rate of convergence for a range of oduli. 4. The coponent-by-coponent sieve algorith In the previous section we gave the idea of how to construct extensible lattice rules. In this section we show that the sieve algorith can be cobined with a coponent-by-coponent (CBC) approach [26] to obtain a faster construction algorith which will allow us to construct good lattice rules for a practically relevant range of oduli and diensions. This also gives the added benefit of obtaining a construction which is also extensible in the diension, but unfortunately the range of oduli in this case has to be chosen in advance and cannot be extended anyore. In this sense our lattice rules are ebedded rather then extensible, see also Section 7 and [2]. 4.. The CBC sieve algorith. We ay reduce the construction cost by constructing the vector z coponent-by-coponent. This approach has been shown to be very useful and effective in constructing lattice rules for fixed n where one has φ(n) s choices of z. In short, the CBC algorith works the following way: choose the first coponent of the generating vector z =. Then, for z s = (z,..., zs) already chosen, we will choose a coponent zs+ such that the worst-case error e 2 per,p,s+,α,γ(z s, zs+) satisfies a certain bound. This way we can obtain a good generating vector inductively [3, 5, 26]. We will now establish a siilar sequence of theores to those of Theores 4 which now include the coponent-by-coponent approach. Since we construct a vector z s+ = (z s, z s+) with z s fixed, we are concerned only with the increental ipact of the choice of z s+ on the worst-case error. We shall require the following technical lea. Lea. For any positive integer we have p φ(p ) h α k=0 z Z p, h= e 2πikhz/p 4ζ(α). Proof. This follows directly fro [6, Lea 2. and Lea 2.3]. We define the quantity θ p,s+,α,γ(z s, z s+ ) = e 2 per,p,s+,α,γ(z s, z s+ ) e 2 per,p,s,α,γ(z s) which will be needed in the following. We odify Theore as follows. Theore 6. Let and s be positive integers. Then we have φ(p θ p,s+,α,γ(z ) s, z s+ ) θ p,s+,α,γ z s+ Z p, where θ p,s+,α,γ = 4 p γ s+ζ(α) ( + 2γ j ζ(α)).

12 2 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Proof. Note that by (0), θ p,s+,α,γ(z s, z s+) can be written in the for θ p,s+,α,γ(z s, z s+) = r α (γ, (h, h s+ )) (2) = (h,h s+) Z s+ \{0} (h,h s+) (z s,zs+) 0 (od p ) h Z s \{0} h z s 0 (od p ) r α (γ, h) (h,h s+) Z s+,h s+ 0 (h,h s+) (z s,zs+) 0 (od p ) and so each ter θ p,s+,α,γ(z s, z s+ ) is non-negative. Fro (9) we see that φ(p θ p,s+,α,γ(z ) s, z s+ ) z s+ Z p, ( p ) e 2πikhz j /p = p + γ j h α k=0 h= γ s+ φ(p ) h α p p k=0 p p k=0 z s+ Z p, h= e 2πikhzs+/p + 2γ cos(2πkhzj /p ) j h α h= γ s+ φ(p ) h α z s+ Z p, h= ( + 2γ j ζ(α)) γ s+ φ(p ) z s+ Z p, ( + 2γ j ζ(α)). 4 p γ s+ζ(α) which uses Lea in the final step. e 2πikhzs+/p h= r α (γ, h) e 2πikhzs+/p h α We now define a set which is analogous to (7). For a real c and z s Zp s let (22) C p,s+,α,γ(c; z s) = {z s+ Z p : θ p,s+,α,γ(z s, z s+ ) c θ p,s+,α,γ}. The following theore follows iediately fro Markov s inequality. Recall that each ter θ p,s+,α,γ(z s, z s+ ) is non-negative as seen in (2) and hence Markov s inequality can be applied. As in this section we only deal with sets of onediensional vectors we siply write µ for the easure µ. Theore 7. Let and s be positive integers. Then for any c we have µ(c p,s+,α,γ(c ; z s)) > c.

13 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 3 Proof. This follows iediately fro applying Markov s inequality to Theore 6. We will be able to achieve stronger convergence results for the worst-case error if we use Jensen s inequality. We define the set C p,s+,α,γ(c; z s ) = { z s+ Z p : θ p,s+,α,γ(z s, z s+) c /λ ( θ p,s+,αλ,γ λ ) /λ (23) This new set has the following property. for all /α < λ }. Theore 8. Let and s be positive integers. Then for any c we have Proof. Fro Theore 7 we can say Now, if z s+ C p,s+,αλ,γ λ(c; z s) then µ( C p,s+,α,γ(c; z s )) > c. µ(c p,s+,αλ,γ λ(c; z s )) > c. θ p,s+,αλ,γ λ(z s, z s+) c θ p,s+,αλ,γ λ. Applying Jensen s inequality to (2) we see that (θ p,s+,α,γ(z s, z s+ )) λ θ p,s+,αλ,γ λ(z s, z s+ ). Cobining the last two inequalities iplies θ p,s+,α,γ(z s, z s+ ) c /λ ( θ p,s+,αλ,γ λ ) /λ which iplies that z s+ C p,s+,α,γ(c; z s ). This eans that C p,s+,αλ,γ λ(c; z s ) C p,s+,α,γ(c; z s), which by using Theore 7 as a lower bound iplies that µ( C p,s+,α,γ(c; z s)) µ(c p,s+,αλ,γ λ(c; z s)) > c. In the sae vein as Theore 4, we show in the following theore that there exists a coponent z s+ Z p such that the worst-case error e 2 per,p,s,α,γ (z s, z s+ ) is sall for all =,..., 2. Theore 9. Let and s be positive integers. Let z s Zs p. Let c for all =,..., 2 such that 2 = c. Then there exists a zs+ Z p such that θ p,s+,α,γ(z s, z s+ ) ( ) /λ c/λ θp,s+,αλ,γ λ for all /α < λ and =,..., 2. ( 2 ) Proof. We need to show that µ Cp =,s+,α,γ(c ; z s ) > 0. This is siple calculation, µ ( 2 = Cp,s+,α,γ(c ; z s) ) ( 2 = µ Cc p,s+(c ; z s) = 2 µ( C p c,s+ (c ; z s )) = > 2 = c 0. )

14 4 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE We can put the existing vector z s together with the new coponent z s+ to show that the vector z s+ = (z s, z s+) has the following properties. Theore 0. Let and s be positive integers. Let z s be chosen so that and z s+ be chosen so that for all /α < λ. Then e 2 per,p,s,α,γ (z s ) c/λ E2 p,s,α,γ (λ) θ p,s+,α,γ(z s, z s+) c /λ ( θp,s+,αλ,γ λ ) /λ e 2 per,p,s+,α,γ (z s+ ) c/λ E2 p,s+,α,γ (λ) for all /α < λ, where z s+ = (z s, z s+) and Proof. We have E 2 n,s,α,γ(λ) = n /λ ( + 4γ λ j ζ(αλ) ) e 2 per,p,s+,α,γ(z s, zs+) = e 2 per,p,s,α,γ(z s) + θ p,s+,α,γ(z s, zs+) ( + 4γ λ j ζ(αλ) ) /λ c/λ p /λ + c/λ p /λ c/λ p /λ = c/λ p /λ 4γs+ λ ζ(αλ) ( + 4γ λ j ζ(αλ) ) /λ /λ ( + 4γ λ j ζ(αλ) ) + 4γs+ λ ζ(αλ) ( + 4γ λ j ζ(αλ) ) s+ ( + 4γ λ j ζ(αλ) ) /λ = c /λ E 2 p,s+,α,γ(λ). /λ where the second inequality uses another application of Jensen s inequality. We ay now construct the extensible generating vector z using the CBC ethod. The algorith to do this is stated forally in Algorith. Theore. Let s be a positive integer and 0 < 2. Let c for all =,..., 2 such that 2 = c. Then Algorith constructs a vector z Zp s such that e 2 per,p,s,α,γ (z ) c /λ E2 p,s,α,γ (λ) for all /α < λ and =,..., 2.

15 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 5 Algorith CBC construction of z with sall e 2 per,p,s,α,γ(z ) for =,..., 2 Require: 2 N 0, α >, a positive sequence of weights γ, p prie and a sequence c,..., c 2 such that 2 = c : Set z = 2: for s = to s ax do 3: Find ( c )φ(p ) + coponents z s+ Z p, to populate the set T,s+ {z s+ Z p, : e 2 per,p,s+,α,γ (z s, z s+) c /λ E2 p,s+,α,γ (λ) for all /α < λ }. 4: for = + to 2 do 5: Define the set S,s+ = {z s+ Z p,, z T,s+ such that z s+ z (od p )} 6: Find ( i= c i )φ(p ) + vectors to populate the set T,s+ {z s+ S,s+ : e 2 per,p,s+,α,γ(z s, z s+ ) c /λ E 2 p,s+,α,γ(λ) for all /α < λ }. 7: end for 8: Select z s+ T 2,s+ 9: Set z s+ = (z s, z s+) 0: end for : Set z = z s ax 4.2. Optiising the CBC sieve algorith. The classical CBC algorith constructs one coponent of the generating vector at a tie. For each diension, it takes the coponent which iniises the worst-case error. The requireent that this coponent is the iniu is iportant in using Jensen s inequality to gain the optial rate of convergence (see [5]). The sieve algorith does not have this requireent. Rather than finding the iniiser at each step, we require a certain nuber of adissible vectors, that is, vectors whose worst-case error is lower than soe bound. Therefore, Algorith will find an extensible lattice rule without the need for any optiisation. However, it is instinctive that we should attept to go beyond looking for siply a set of adissible vectors and attept to find the best (in soe sense) generating vectors at each step. This can be done by odifying the choice of the set T,s+ for =,..., 2 and s =,..., s ax in Algorith. Rather than just constructing T,s+ with the first ( i= c i )φ(p ) + coponents z s+ such that e 2 per,p,s+,α,γ (z s, z s+) c /λ E2 p,s+,α,γ (λ) for all /α < λ, we construct the set T,s+ to contain all coponents that satisfy the bound. We then truncate the set T,s+ to contain exactly those ( i= c i )φ(p ) + eleents which have the sallest worst-case error e 2 per,p,s+,α,γ(z s, z s+ ), for the given z s. As we will see in the nuerical section, the bound is significantly larger than the actual errors and hence using such an optiisation step ensures that we do not choose a coponent which barely satisfies the bound but rather is aongst the best ones.

16 6 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE 4.3. The fast CBC sieve algorith. In the previous section we constructed the coponents of the generating vector by first choosing the first digits and then extending those up to 2 digits step-by-step for a set of good coponents. Though this algorith is feasible for practical values, it does not allow us to use the fast coponent-by-coponent algorith introduced by Nuyens and Cools [23, 24]. Their construction algorith reduces the usual construction cost of the CBC algorith fro O(sn 2 ) to O(sn log n) (which is a rearkable speed-up for large n) by exploiting the structure of the calculation. In order to ake use of the fast CBC algorith we odify the previous construction algoriths. In this case it is necessary to search over all possible choices of the new coponent z s+, rather than just those which have been shown to be good for earlier values of. Here we siply store all the good coponents z s+ for the generating vector (z s, z s+ ) for each value of. The construction is then perfored by iniising a new error easure, which, for given z s Zs p, 2, is defined by (24) F, 2,s+,α,γ(z s+ ) = 2 ax /α<λ = e 2 per,p,s+,α,γ ((z s, z s+)) c /λ E 2. p,s+,α,γ(λ) Using this easure we now construct a generating vector coponent-by-coponent, in each step choosing z s+ which iniises the quantity F, 2,s+,α,γ. Algorith 2 Fast CBC sieve construction of a good generating vector z with sall e 2 per,p,s,α,γ (z ) Require: 2 N 0, α >, a positive sequence of weights γ, p prie and the positive sequence c,..., c 2 such that 2 = c : Set z = 2: for s = to s ax do 3: for = to 2 do 4: Copute λ (/α, ] which iniises N,c (λ) = c /λ as a function of λ. 5: For each z s+, Z p, copute 6: end for 7: Set T s+ = { z s+ Z p,2 : e 2 per,p,s+,α,γ ((z s, z s+)) N,c (λ ). 8: Select z s+ T s+ which iniises E 2 p,s+,α,γ (λ) } e 2 per,p,s+,α,γ((z s, z ax s+ )) 2 N,c (λ. ) 2 e2 per,p,s+,α,γ ((z s, z s+)) N =,c (λ ). 9: Set z s+ = (z s, zs+ ) 0: end for : Set z = z s ax

17 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 7 We can now use Theore 9 to show that there ust be at least one choice of z s+ Z p which is good for all =,..., 2. Theore 2. Let s be a positive integer and 0 < 2. Let c for all =,..., 2 such that 2 = c. Then Algorith 2 constructs a vector z Zp s such that e 2 per,p,s,α,γ (z ) c /λ E2 p,s,α,γ (λ) for all /α < λ and =,..., 2. Reark 3. In Algorith 2 note that instead of choosing zs+ T s+ which iniises the quantity 2 e2 per,p,s+,α,γ((z s, z s+ )) N =,c (λ ) it would be enough in theory to pick any zs+ T s+ (which ust be non-epty by Theores 9 and 0). But as searching for the iniu increases the construction cost only arginally it is advisable to include this step since the bound is typically very loose (see Section 5). Note that another legitiate choice in line 8 would be to select the zs+ T s+ which iniises e 2 per,p ax,s+,α,γ ((z s, z s+)) 2 N,c (λ ). In this instance we have chosen the forer because it gives saller worst case errors in the nuerical experients. The iniu of N,c (λ) can be found with sufficient accuracy using any standard one-diensional constrained optiisation software. Coputing the noralised worst-case error in lines 3 and 6 in Algorith 2 can be done in order n log n operations using the fast CBC algorith of [24] Theoretical bounds on the algorith of Cools et al. Algorith 2 is very siilar in nature to the algorith suggested in [2]. Their algorith is different in that given z s they choose zs+ to iniise the error easure defined by (25) V p,, 2,s+,α,γ((z s, z s+)) := ax 2 e 2 per,p,s+,α,γ((z s, z s+ )) e 2 per,p,s+,α,γ (z() ) for s =,..., s ax where z () is the generating vector with the CBC algorith for n = p for =,..., 2. In [2] there was no foral proof with any bound on the size of the error easure V p,, 2,s,α,γ(z), although the nuerical experients suggested that it reained sall. Here we show how our algorith ay be odified to give an upper bound on the error easure V p,, 2,s,α,γ(z). Theore 3. Let s be a positive integer and 0 < 2. Let z () be the generating vector constructed by the CBC algorith with n = p for =,..., 2. Then there exists a choice of c for =,..., 2 with 2 = c such that Algoriths and 2 construct a vector z Zp s where V p,, 2,s,α,γ(z ) for =,..., 2. in /α<λ ( 2 = 2 ) /λ Ep,s,α,γ(λ) ( e 2 per,p,s,α,γ(z () ) ) λ

18 8 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Proof. Let c for all =,... 2 such that 2 be constructed using Algoriths or 2 so that = c e 2 per,p,s,α,γ (z ) c /λ E2 p,s,α,γ (λ). Then z ay for all /α < λ and =,..., 2. Let the quantities c for =,... 2 be chosen such that (26) c /λ E 2 p,s,α,γ(λ) = X(λ)e 2 per,p,s,α,γ(z () ) for soe /α < λ and constant X(λ). In choosing X(λ), we require that 2 = c. Rearranging (26) we see that this condition is satisfied if we take X(λ) = ( 2 = 2 ) /λ Ep,s,α,γ(λ) ( e 2 per,p,s,α,γ(z () ) ). λ Putting Theore 2 and (26) together we see that e 2 per,p,s,α,γ(z ) X(λ)e 2 per,p,s,α,γ(z () ) for all /α < λ and =,..., 2, which iplies that which gives the desired result. V p,, 2,s,α,γ(z ) in X(λ) /α<λ 5. Nuerical testing We have shown that it is possible to construct a generating vector z such that e 2 per,p,s,α,γ (z ) c /λ E2 p,s,α,γ (λ) for all /α < λ where c for =,..., 2 such that 2. The testing was all perfored using the fast CBC algorith since it is the fastest coputationally. There are several paraeters for each calculation which we ust choose. In each exaple we take p = α = 2. We also assue that the constants c for =,..., 2 are equal for each, that is c = 2 +. For these experients we take = 0, 2 = 20 and s ax = 360. There are two conclusions which can be drawn fro our nuerical experients. The first conclusion we ay draw is that the worst-case error for the extensible lattice rule is uch saller than the bound in Theore 2 suggests. To deonstrate this we define the quantity e per,p,s,α,γ(z (,2) ) (27) U p,, 2,s ax,α,γ = ax s s ax c /2λ E p,s,α,γ(λ ) = c where c /λ E 2 p,s,α,γ(λ ) c /λ E 2 p,s,α,γ(λ) for all /α < λ and z (,2) is an extensible lattice rule constructed with Algorith 2. Theore 2 shows that U p,, 2,s ax,α,γ is bounded by. In Tables 3 we copare the values of U 2,0,20,360,2,γ for different choices of γ. We see that in each case U 2,0,20,360,2,γ is orders of agnitude less than. In fact, the nuerical tests do not find any exaples where U 2,0,20,360,2,γ is greater than The second conclusion we ay draw is that the worst-case error for the extensible lattice rule is not significantly greater than the worst-case error for the near

19 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 9 e per,2,360,2,γ(z ) c /λ E 2,360,2,γ(λ ) U 2,0,20,360,2,γ = e-02.44e e-02 = 5.33e-02.0e e-02 = 2 3.4e e e-02 = 3 2.2e e e-02 = 4.44e e e-02 = 5 9.4e e-0 3.7e-02 = 6 5.8e-03.79e e-02 = e-03.27e e-02 = e e e-02 = 9.53e e e-02 = e e e-02 Table. Worst-case error of the extensible lattice rule where γ j = /j 2 e per,2,360,2,γ(z ) c /λ E 2,360,2,γ(λ ) U 2,0,20,360,2,γ = e e e-02 = 2.83e e e-02 = e+02.75e e-02 = 3.4e+02.24e e-02 = e e e-02 = e+0 6.8e e-02 = e e e-02 = e e e-02 = e+0 2.9e e-02 = 9.77e+0.55e e-02 = 20.25e+0.09e e-02 Table 2. Worst-case error of the extensible lattice rule where γ j = 0.9 j e per,2,360,2,γ(z ) c /λ E 2,360,2,γ(λ ) U 2,0,20,360,2,γ = 0 2.5e+0.80e+2 6.9e-02 =.77e+0.27e e-02 = 2.25e+0 9.0e e-02 = e e e-02 = e e e-02 = e e e-02 = 6 3.4e e e-02 = e+09.59e e-02 = 8.57e+09.3e e-02 = 9.e e e-02 = e e e-02 Table 3. Worst-case error of the extensible lattice rule where γ j = 0.05

20 20 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE optial lattice rule as constructed by the CBC algorith. To deonstrate this we exaine the error easure V p,, 2,s,α,γ(z ) defined above V 2,0,20,s,2,γ.4.2 γ j = / j 2 γ j = 0.05 γ j = 0.9 j s Figure. Graph of V 2,0,20,s,2,γ for 3 choices of γ and s =,..., 360 In Figure we see when = 0, 2 = 20 and p = α = 2 the greatest ratio of the worst-case error of the extensible lattice z and the worst-case error of the corresponding near optial choice z () (as constructed by the CBC algorith) is always less than 2 for these particular choices of γ. This is siilar to the results in [2, Table 6.]. 6. Extensible lattice rules with sall star discrepancy In the sections above, we have developed three algoriths to construct an extensible lattice rule with sall worst-case error. Another easure of the quality of a lattice rule (or any quasi-monte Carlo rule for that atter) is the weighted star discrepancy of the underlying nodes. The weighted star discrepancy for a point set P n consisting of n points in the s-diensional unit cube is defined by (28) Dn,s,γ(P n ) := sup u {,...,s} where γ u sup disc((x u, ), P n ), x u [0,] u (29) disc(x, P n ) := #(P n [0, x)) Vol([0, x)), n γ u = j u γ j and for x = (x,..., x s ) the vector (x u, ) denotes the vector where the j-th coponent is x j if j u and otherwise.

21 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 2 If P n is the node set of a lattice rule with generating vector z we will in the following write Dn,s,γ(z) instead of Dn,s,γ(P n ). The quantity Dn,s,γ (z) is difficult to copute. However, fro [4] we easily deduce that the weighted star discrepancy is bounded by Dn,s,γ(z) < 2 R ( n,s,γ(z) + γ u ( ) ) u, n u {,...,s} where the quantity R n,s,γ (z) is defined by (30) R n,s,γ (z) = with and r(h, γ) = h B n,s h z 0 (od n) r(h j, γ j ) and r(h j, γ j ) = r(h, γ), { ( + γ j ) if h j = 0, γj h j if h j 0, B n,s = Z s ( n/2, n/2] s and B n,s = B n,s \ {0}. In the case where s =, we will usually drop the subscript s. It was proved by Joe [4] that if the vector of weights γ = (γ, γ 2,...) satisfies γ j <, then we have u {,...,s} where Γ := i= γ i +γ i. ( γ u ( ) ) u ax(, Γ)eP i= γi n n s, We therefore ai to construct extensible lattice rules with sall weighted star discrepancy by iniising the quantity R n,s,γ (z), for n = p and =,..., 2. In this section we will provide theores analogous to theores above which are based upon the worst-case error. These theores can be used to derive algoriths which are alost identical to the ones above. Given the siilarity between the theores, soe of the results below are stated without proof. 6.. The sieve algorith. The sieve algorith can be perfored in the sae way as described in Section 3 where one replaces the worst-case error with the quantity R p,s,γ and the bound E p,s,α,γ with R p,s,γ given in the following. The following theores are the natural analogs of Theores 7 and Theores 9 2. Theore 4. Let and s be positive integers. Then we have φ(p ) s R p,s,γ(z) R p,s,γ, where R p,s,γ = p z Z s p, ( + γ j + γ j L(p )) ( + γ j ) and L(n) = h B n h.

22 22 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Proof. The proof is siilar to the proof of [20, Theore 5.0]. We have (3) φ(p ) s z Z s p, R p,s,γ(z) = = φ(p ) s φ(p ) s z Z s p, h B p,s h B p,s h z 0 (od p ) A(h) r(h, γ), r(h, γ) where A(h) is the nuber of vectors z Z s p, such that h z 0 (od p ). Note that A(0) = φ(p ) s and r(0, γ) = s ( + γ j). Since we can write (3) as (32) (33) where φ(p ) s = = z Z s p, A(h) = z Z s p, R p,s,γ(z) p p k=0 p ( + γ j ) + φ(p ) s p k=0 p ( + γ j ) + φ(p ) s p T (k, p, γ j ) = k=0 e 2πikh z/p h B p,s z Zp, s T (k, p, γ j ), e2πikh z/p r(h, γ h B p j ) z Z p, e 2πikh z/p r(h, γ) Clearly, T (0, p, γ j ) = φ(p ) ( + γ j + γ j L(p )). For k p we note that gcd(p, k) = gcd(p, k) which by following the proof of [20, Eq. (5.7)] gives. (34) T (k, p, γ j ) = φ(p )( + γ j ) + γ j gcd(p, k) ( L(gcd(p, k)) L(p gcd(p, k)) ). Since L(gcd(p, k)) L(p gcd(p, k)) < 0, then T (k, p, γ j ) < φ(p )( + γ j ). This eans we can write (33) as φ(p ) s = p z Z s p, ( + γ j ) + R p,s,γ(z) p ( + γ j + γ j L(p )) ( + γ j + γ j L(p )) + p p k= ( + γ j ) = R p,s,γ ( + γ j ) which finishes the proof.

23 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 23 Reark 4. While the quantity L(n) ay be coputed exactly, for large n, it ay be ore practical to approxiate L(n) by the the following function [20]. L(n) = 2 log n + 2γ log 4 + ɛ(n) with ɛ(n) < 4n 2. where γ = li N ( N = log N ) = is the Euler-Mascheroni constant. For reals b define (35) B n,s,γ (b) = {z Z s p : R n,s,γ(z) br n,s,γ }. Then we have Theore 5. Let and s be positive integers. Then we have µ s (B p,s,γ(b)) > b. Proof. The result follows fro Theore 4 together with an application of Markov s inequality and fact that R p,s,γ(z) is non-negative. Theore 6. Let and s be positive integers. Let b for =,..., 2 such that 2 = b. Then there exists a vector z Zp s such that R p,s,γ(z) b R p,s,γ for =,..., 2. Proof. The proof is analogous to that of Theore 4. Theore 7. Let s be a positive integer and 0 < 2. Let b for =,..., 2 such that 2 = b. Then the sieve algorith based on R p,s,γ constructs a vector z Zp s such that R p,s,γ(z ) b R p,s,γ for =,..., The CBC sieve algorith. The CBC sieve algorith is Algorith where one replaces the worst-case error with the quantity R p,s,γ and E p,s,α,γ with R p,s,γ. The quantity Υ p,s+,γ(z s, z s+ ) = R p,s+,γ(z s, z s+ ) R p,s,γ(z s). will be needed subsequently. The construction of z coponent-by-coponent is justified by the following theore. Theore 8. Let and s be positive integers. Then φ(p Υ p,s+,γ(z s ), z s+) Υ p,s+,γ, z s+ Z p, where Proof. Note first that Υ p,s+,γ(z s, z s+ ) Υ p,s+,γ = 2γ s+ p L(p ) ( + γ j + γ j L(p )). (h.h s+) (z s,zs+) 0 (od p ) h B p,s, h s+ 0 r(h, γ) r(h s+, γ s+ ).

24 24 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Now, following a siilar arguent as in the proof of Theore 6 we see where φ(p ) = Υ p,s+,γ(z s, z s+ ) z s+ Z p, φ(p ) φ(p ) z s+ Z p, h B p,s (h,h s+) (z s,zs+) 0 (od p ) h B p,s, h s+ 0 A(h, h s+, z s ) r(h, γ) h s+ B p r(h, γ) r(h s+, γ s+ ) r(h s+, γ s+ ), A(h, h s+, z s) = #{z s+ Z p, : (h, h s+ ) (z s, z s+ ) 0 (od p )} = p p e 2πik(h,hs+) (z s,zs+)/p. z s+ Z p, k=0 We can therefore find an upper bound with φ(p ) p φ(p ) p φ(p ) = φ(p ) z s+ Z p, Υ p,s+,γ(z s, z s+ ) p p k=0 p k=0 p k=0 e2πikhs+ zs+/p r(h h s+ Bp s+, γ s+ ) z s+ Z p, e2πikhs+ zs+/p r(h h s+ Bp s+, γ s+ ) z s+ Z p, T (k, p, γ s+ ) F (k, p, z, γ j ), h j B p h j B p e 2πikhj z j /p r(h j, γ j ) e 2πikhj z j /p r(h j, γ j ) where and T (k, p, γ s+ ) = e2πikhs+ zs+/p r(h h s+ Bp s+, γ s+ ) z s+ Z p, =T (k, p, γ s+ ) ( + γ s+ )φ(p ) F (k, p, z, γ j ) = h j B p e 2πikhj z j /p. r(h j, γ j ) Clearly F (k, p, z, γ j ) +γ j +γ j L(p ) and T (0, p, γ s+ ) = φ(p )γ s+ L(p ). Fro (34) we see that for k 0 we have T (k, p, γ s+ ) = γ s+ gcd(p, k) (L(p gcd(p, k)) L(gcd(p, k))).

25 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES 25 Since F (k, p, z, γ j ) is bounded independently of k, it reains to calculate φ(p ) = = p p k= T (k, p, γ s+ ) p γ s+ φ(p ) p gcd(p, k) (L(p gcd(p, k)) L(gcd(p, k))) γ s+ φ(p ) p = γ s+ p L(p ). k= φ(p l ) ( L(p l+ ) L(p l ) ) l= Cobining this with the k = 0 ter, we see that φ(p ) z s+ Z p, Υ p,s+,γ(z s, z s+ ) 2γ s+ p L(p ) ( + γ j + γ j L(p )). For b and z s Zs p, define (36) B p,s+,γ(b; z s ) = { z s+ Z p : Υ p,s,γ(z s, z s+) b Υ p,s+,γ}. As before we again write siply µ for the easure µ. Then we have Theore 9. Let and s be positive integers. Then for any b we have µ(b p,s+,γ(b ; z s )) > b. Theore 20. Let s be a positive integer and let 0 < 2. Let b for =,..., 2 such that 2 = b. Then there exists a z s+ Zs p such that Υ p,s+,γ((z s, z s+)) b Υ p,s+,γ for =,..., 2. Proof. The proof is analogous to that of Theore 9. Theore 2. Let and s be positive integers. Let z s be chosen so that R p,s,γ(z s ) b p ( + γ j + 2γ j L(p )) ( + γ j ) and let zs+ be chosen so that Then Υ p,s+,γ(z s, z s+) b 2γ s+ p L(p ) R p,s+,γ(z s+ ) b p where z s+ = (z s, z s+ ). s+ ( + γ j + γ j L(p )). s+ ( + γ j + 2γ j L(p )) ( + γ j ),

26 26 JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Proof. We have R p,s+,γ(z s+) = R p,s,γ(z s) + Υ p,s+,γ(z s, zs+) b ( p + γ s+ + 2γ s+ L(p )) ( + γ j + 2γ j L(p )) b p γ s+ s+ ( + γ j + 2γ j L(p )) s+ ( + γ j + 2γ j L(p )) ( + γ j ) ( + γ j ). Theore 22. Let s be a positive integer and let 0 < 2. Let b for =,..., 2 such that 2 = b. Then an algorith equivalent to Algorith constructs a vector z Zp s such that for =,..., 2 R p,s,γ(z ) b s+ s+ p ( + γ j + 2γ j L(p )) ( + γ j ) The fast CBC sieve algorith. Finally, the fast CBC sieve construction ay be used to find a generating vector z with sall R p,s,γ(z ) for =,..., 2 again by aking the necessary adjustents, i.e., replacing the worstcase error with the quantity R p,s,γ and replacing the bound E p,s,α,γ with R p,s,γ. The following theore now applies to generating vectors constructed using the fast CBC algorith based on R p,s,γ. Theore 23. Let s be a positive integer and let 0 < 2. Let b for =,..., 2 such that 2 = b. Then an algorith equivalent to Algorith 2 constructs a vector z Zp s such that for =,..., 2 R p,s,γ(z ) b s+ p ( + γ j + 2γ j L(p )) ( + γ j ). 7. Concluding rearks Though we provide soe useful constructions here there are still soe open questions. First let us address the eaning of extensible. In the introduction we wrote that the existence of good extensible lattice rules was shown in [2]. Here the eaning of extensible is fro a practical point of view, naely that there exists a lattice rule whose generating vector z is such that one can obtain good lattice rules for all oduli p, p 2,.... What would be an interesting result in this direction, but was not shown in [2], is the following: For any generating vector of a good lattice rule in diension s with nuber of points p, there exists an extension of this generating vector such that one obtains a good lattice rule for soe other nuber of points p with. (Copare this stateent with a probabilistic version in Reark. Further, an analogue result for the diension is known if s > s, see [3, 5].) Such a result would