Fibonacci sets and symmetrization in discrepancy theory

Transcription

1 Fibonacci sets and symmetrization in discrepancy theory Dmitriy Bilyk, V.N. Temlyakov, Rui Yu Department of Mathematics, University of South Carolina, Columbia, SC, 29208, U.S.A. Abstract We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let { } n=0 be the sequence of Fibonacci numbers. The -point Fibonacci set F n [0, ] 2 is defined as F n := {(µ/, {µ / })} µ=, bn where {x} is the fractional part of a number x R. It is known that cubature formulas based on Fibonacci set F n give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F n = F n {(p, p 2 ) : (p, p 2 ) F n } has asymptotically minimal L 2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L 2 discrepancy among two-dimensional point sets. We also introduce quartered L p discrepancy which is a modification of the L p discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set F n has minimal in the sense of order quartered L p discrepancy for all p (, ). This in turn implies that certain twofold symmetrizations of the Fibonacci set F n are optimal with respect to the standard L p discrepancy. Keywords: Discrepancy, Fibonacci numbers, cubature formulas, numerical integration, Fourier coefficients. 200 MSC: K38, 42A6, 65D30 Corresponding author. address: bilyk@math.sc.edu (Dmitriy Bilyk ) Preprint submitted to Journal of Complexity May 4, 20

2 . Introduction.. Discrepancy Let P N be a set of N points in the unit cube [0, ] d in dimension d. The extent of uniform distribution of P N can be measured by the discrepancy function: D(P N, x) := # { P N [0, x) } N [0, x), (.) where x = (x,..., x d ), [0, x) = d [0, x j ), and denotes the Lebesgue j= measure. The L p norm of the above discrepancy function, usually called the L p discrepancy, is a benchmark that one uses to evaluate the quality of a particular set of N points. The fundamental problem of the discrepancy theory is to construct sets with small L p discrepancy. The main principle of discrepancy theory, or theory of irregularities of distribution, states that the quantity D(N, d) p := inf P N D(P N, x) p must necessarily go to infinity with N when d 2. We refer to Kuipers and Niederreiter [22], Beck and Chen [], Matoušek [25], and Chazelle [5] for detailed surveys. The principal lower estimates for D(N, d) p are: K. Roth s Theorem.([28], 954) In all dimensions d 2, we have D(N, d) 2 C(d)(log N) d 2, (.2) where C(d) is a positive constant that may depend on d. W. Schmidt s Theorem. ([3], 972) In dimension d = 2, where C is a positive absolute constant. D(N, 2) C log N, (.3) Both bounds (.2) and(.3) are known to be sharp in the sense of order, see e.g. van der Corput [0], Davenport [], Roth [29] and Frolov [5] 2

3 for more details. One of the most famous (and relevant to our discussion) examples demonstrating sharpness of (.3) is the irrational lattice: {( µ )} N A N (α) := N, {µα} µ=, (.4) where α is an irrational number and {x} is the fractional part of the number x. If the partial quotients of the continued fraction of α are bounded, then the L discrepancy of this set is of the order log N (see, e.g. [25], [20]). The idea of this example goes back to Lerch, 904 [24]. In the present paper we study the distributional properties of the closely related Fibonacci sets. These sets are known in the theory of Quasi-Monte Carlo methods under the names Fibonacci lattice points sets or Fibonacci lattice rules, but we shall adhere to the abbreviated name. Let { } n=0 be the sequence of Fibonacci numbers: b 0 = b =, = + 2, for n 2. (.5) The -point Fibonacci set F n [0, ] 2 is defined as F n := {(µ/, {µ / })} bn µ=. (.6) Obviously, for large n, the set F n is close to the irrational lattice A N (α) with 5 N = and α =, i.e., the reciprocal of the golden section. It is well 2 known (see [26]) that D(F n, x) C log, (.7) hence, according to Schmidt s bound (.3), Fibonacci sets also have optimal L discrepancy. Finally, we mention another important example of a low-discrepancy construction: the van der Corput (or Hammersley) digit-reversing set, introduced in [0], whose L discrepancy is of the order log N (see [25] for a geometric proof). While this set is not directly related to our discussion, we shall often use it as a point of comparison..2. Numerical integration It is well known (see, for instance, [36]) that the L discrepancy (as well as other notions of discrepancy) of a finite set is closely related to the error 3

4 of numerical integration with knots at the given points. We shall discuss this topic in more detail here. The quality of a set of N points for numerical integration can be measured in the following standard way. For a certain function class W compare the error of numerical integration with knots from the given set with optimal error for cubature formulas with N knots. We give a precise formulation of the problem. Numerical integration seeks good ways of approximating an integral f(x)dµ by an expression of the form Ω Λ N (f, ξ) := N λ j f(ξ j ), ξ = (ξ,..., ξ N ), ξ j Ω, j =,..., N. (.8) j= It is clear that f has to be integrable and defined at the points ξ,..., ξ N. The expression (.8) is called a cubature formula (Λ, ξ) (in our case Ω R 2 ) with knots ξ = (ξ,..., ξ N ) and weights Λ = (λ,..., λ N ). For a function class W the error of the cubature formula Λ N (, ξ) is defined by Λ N (W, ξ) := sup fdµ Λ N (f, ξ). (.9) f W In the case of equal weights λ j = /N we denote this error by Λ e N (W, ξ). Set δ N (W ) := Ω inf Λ N (W, ξ); δ λ,...,λ N N(W e ) := inf Λ e N(W, ξ) ξ,...,ξ N ξ,...,ξ N to be the best errors achieved by cubature formulas with N knots. With these definitions at hand, the relation between the L discrepancy of a set P N [0, ] 2 and the error of numerical integration with knots at P N is straightforward. Define the following class of functions χ d := {χ [0,x] (y) := d χ [0,xj ](y j ), x j [0, ], j =,..., d}, j= where χ [0,u] (v) is a characteristic function of the interval [0, u]. Then it is clear that Λ e N(χ d, P N ) = N D(P N, x). (.0) We now define classes of (periodic) functions with bounded mixed derivative, which arise naturally in numerical integration. For r > 0, let F r (t) := + 2 k r cos(2πkt rπ/2). (.) k= 4

5 For x = (x, x 2 ) denote F r (x) := F r (x )F r (x 2 ) and MW r p := {f : f = ϕ F r : ϕ p }, where means convolution and p is the standard L p norm. It is known (see, for instance, survey [36]) that the Fibonacci sets F n are also good for numerical integration of functions from the classes MW r p. The following known result gives the order of Λ e (MW r p, F n ) for all parameters p, r > /p. In our paper, stands for of the same order of magnitude as and stands for less than a constant multiple of. Theorem.. We have Λ e (MWp r, F n ) b r n (log ) /2, < p, r > max ( ), ; p 2 b r n log, p =, r > ; b r n (log ) r, 2 < p, < r < ; p 2 b r n ((log )(log log )) 2, 2 < p, r = /2. (.2) The following theorem gives the lower bounds for optimal rates of numerical integration (again, see survey [36]). Theorem.2. The following lower bound is valid for any cubature formula (Λ, ξ) with N knots (r > /p) Λ N (MW r p, ξ) C(r, p)n r (log N) 2, p <. The lower bounds provided by Theorem.2 and the upper bounds from Theorem. show that the Fibonacci cubature formulas Λ e (, F n ) are optimal (in the sense of order) among all cubature formulas in the case < p <, r > max(/p, /2): δ bn (MW r p ) Λ e (MW r p, F n ) b r n (log ) /2. We shall also make a remark in Section 2 which shows that the sets F n are much better than their siblings A N (α) from the point of view of numerical integration of smooth functions. It is well known (see, e.g., [36], Proposition.2) that the L discrepancy governs integration errors for the class MW : c (d)λ e N(χ d, ξ) Λ e N(MW, ξ) c 2 (d)λ e N(χ d, ξ). (.3) 5

6 This, together with inequality (.7), yields the relation Λ e (MW, F n ) b n log, (.4) that was not covered by Theorem.. All these results motivate us to conduct a thorough study of the Fibonacci sets..3. Optimal L 2 vs. L discrepancies At this point we would like to demonstrate that the issue of constructing sets with low L 2 discrepancy is even more subtle than in the case of L. This situation is in natural contrast with the lower discrepancy estimates, where L 2 bounds are generally much simpler than L. One may be tempted to think that the optimality of the Fibonacci set F n with respect to L 2 discrepancy may be implied by Theorems. and.2. However, this is not the case! While there is a direct relation between L p discrepancy for < p < and the error of cubature formulas for the (non-periodic) function classes MẆ p (Ω d) (see [36], formula (.5)), there is no such connection for the (periodic) classes MWp treated in Theorem.. Only the general equivalence δ N (MWp ) δ N(MẆ p (Ω d)) between the rates of decay of errors of optimal cubature formulas for these classes is available (see Theorem. in [36] and the remark thereafter), which is not enough to derive that F n has optimal L 2 discrepancy. Unfortunately, the L 2 discrepancy of the classical examples either fails to be of optimal order (the L 2 discrepancy of the N-point van der Corput set is of order log N, not log N, [6]), or requires much more delicate arguments than L (as in the case of the Fibonacci set F n, [33]), or is even unknown (lattices A N (α) for general α). However, discrepancy theory provides several standard ways to modify these sets in order to achieve the smallest possible order of the L 2 discrepancy and/or simplify the calculations:. Cyclic shifts. The translation idea, originated in K. Roth s papers [29], [30], was applied probabilistically to the van der Corput set. A deterministic example of such a shift was recently constructed by Bilyk [2]. 2. Digit scrambling (digit shifts). This approach is introduced in [6] and one may refer to [25] for a comprehensive discussion and interesting constructive examples. In the past decade substantial work in this direction has been done in the context of two-dimensional low discrepancy sets, see [2], [9], [2], [3], [3]. 6

7 3. Davenport s Reflection Principle. This idea in various guises is explored in the current paper. Roughly speaking, it states that if a finite set P N has low L discrepancy, then symmetrizing this set produces a new set of low L 2 discrepancy. This approach was initiated by Davenport [, 956] in the case of irrational lattice. Symmetrization was subsequently used by Proinov [27], Chaix and Faure [4] for the generalized van der Corput sequences, Chen and Skriganov [8] for the van der Corput set, Larcher and Pillichshammer [23] for (0, m, 2)-nets and (0, )-sequences in base 2, and by other authors. The original Davenport s construction historically was the first example demonstrating the sharpness of (.2) (in dimension d = 2). His construction involved an irrational lattice A N (α), where α is an irrational number with bounded partial quotients, symmetrized with respect to the vertical line x =. For a long time it was not clear whether this symmetrization is really 2 necessary. The first partial answer appeared more than 20 years later. In 979, Sós and Zaremba [33] proved that when all the partial quotients of the (finite or infinite) continued fraction of α are equal, then the set A(α) has optimal L 2 discrepancy. In particular, this result covers the Fibonacci set F n and the irrational lattice A N (( 5 )/2) in these cases all the partial quotients are equal to : D(F n, x) 2 D(A bn (( 5 )/2), x) 2 log. (.5) It is also suggested in the same paper that perhaps the L 2 discrepancy is not optimal for some other values of α. This means that the L 2 discrepancy depends on much finer properties of α than simply the boundedness of its partial quotients. The situation with L p discrepancy is even less clear. These issues will be further explored in our upcoming work. To further convince the reader of the difficulty of L 2 constructions we should mention that in higher dimensions (d 3) explicit examples of sets with optimal order of L 2 discrepancy have been constructed only in the last few years by Chen and Skriganov [7] (simplified in [9] and extended to L p for p 2 by Skriganov [32]). However, the constant in the leading term of their estimate is rather large. In the two-dimensional case, Faure, Pillichshammer, Pirsic, and Schmid [3] find an effective value of this constant by considering the L 2 discrepancy of the so-called generalized Hammersley point sets..4. Main results In the present paper, we apply Davenport s symmetrization idea to the Fibonacci set. In Section 2 we prove that the symmetrized Fibonacci set F n 7

8 has minimal in the sense of order L 2 discrepancy, i.e. (see Theorem 2.8) D(F n, x) 2 C log. (.6) This is achieved by a meticulous examination of the Fourier coefficients of the function D(F n, x). This result may seem superfluous in view of the aforementioned result (.5) of Sós and Zaremba. Nevertheless, both the result and the method present several advantages. First of all, we are able to provide an exact formula allowing one to compute the precise value of L 2 norm of the discrepancy function (Theorem 2.). This formula enabled us to computationally evaluate the constant C in the upper bound (.6). We show that the constant we get is around , which is better than the best previously known constant in the L 2 discrepancy upper bounds, , provided in [3]. Unfortunately, at present we cannot compute this constant for the nonsymmetrized Fibonacci set F n, since an analog of formulas (2.58)-(2.59) from Theorem 2. is not available. Technically speaking, in the non-symmetrized case certain difficulties arise in the computation of the coefficient D(F n, 0) (cf. Lemma 2.2) as well as D(F n, k) with k = (k, 0) (cf. Lemma 2.5). This is perhaps not surprising: Davenport introduced his technique precisely to take care of the zero-order Fourier coefficient. In addition, in the case of the van der Corput set it is exactly this coefficient D(V n, 0) = D(V n, x) dx that is responsible for the large L 2 norm, see [6], [3], [2]. Finally, the proof of Sós and Zaremba was quite complicated and involved numerous ideas from number theory and probability. At the same time, our proof, which only relies on computing the Fourier coefficients of the discrepancy function, is much more transparent and opens the door to investigating more general lattices, which is the theme of our ongoing work. In Section 3 we further develop the symmetrization idea and introduce quartered L p discrepancy: a version of the L p discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set F n has minimal in the sense of order quartered L p discrepancy for all p (, ). While these result by itself may seem artificial, it leads to the construction of a two-fold symmetrization of the Fibonacci set Fn sym, which has optimal standard L p discrepancy D(F sym n, x) p C(p) log (.7) for all p (, ). We note that constructions of sets with optimal L p discrepancy for p 2 are even more scarce than for p = 2. In particular, we 8

9 do not know if the standard Fibonacci set F n satisfies (.7). The methods of Fourier analysis, including Littlewood-Paley theory, are applied to prove these results. This research was supported by the National Science Foundation Grants DMS and DMS The L 2 discrepancy of the symmetrized Fibonacci set We shall start by briefly discussing the L discrepancy of the Fibonacci set F n = {(µ/, {µ / })} bn µ= and its similarities to the irrational lattice, as well as their differences, from the point of view of discrepancy and numerical integration. As we stated in the introduction, it is a classical and over a century old result [24] that the irrational lattice {( µ )} N A N (α) := N, {µα} has sharp L norm if the partial quotients of the continued fraction of α are 5 bounded. In the special case when N = and α = (the reciprocal 2 of the golden section), the set A N (α) is closely related to the set F n and satisfies the estimate µ= D(A n (α), x) log. (2.8) The sets F n and A N (α) are close to each other in the following sense. For µ, the x-coordinates of the µth points of F n and A n (α) are the same and the differences between the y-coordinates of these points are small. This follows from the well-known inequality α. (2.9) For completeness we give a simple proof of the above inequality. Consider P (x) = x 2 + x. Then P (α) = 0 and P ( / ) = b 2 n. We have 2b 2 n P ( / ) P (α) = P (ξ) / α, 9

10 ξ ( bn Therefore, ), α. It is easy to see that and 2 α 2 3. This implies (2.9). Using (2.9) we obtain 2 P (ξ) 7 3. (2.20) {µ / } {µα} = µ / µα µ 2b 2 n 2. (2.2) (The identity above may be violated only when µ =, but a single point bears no significance on the results.) As mentioned earlier, it is well known [26] that Fibbonaci sets have optimal L discrepancy: D(F n, x) log, n 2 (2.22) Inequality (2.2) and the following simple known lemma show that this bound can also be derived as a perturbation of (2.8). Lemma 2.. Let P N = {p k } N k= [0, ]d and Q N = {q k } N k= [0, ]d be such that p k q k δ, k =,..., N. Then D(PN, x) D(Q N, x) Nδd. The bounds (2.8) and (2.22) show that the sets F n and A n (α) are equally good from the point of view of the L discrepancy. Theorem. from the introduction shows that the sets F n are good for numerical integration. We now demonstrate by a simple example that sets A n (α) are not good for numerical integration of functions with high smoothness. Indeed, consider a function f(x, x 2 ) := e 2πix 2. It is easy to check that f MW r p for all r and p. The error of numerical integration of f using A n (α) with equal weights is µ= e 2πiµα = e 2πibnα e 2πiα. 0

11 Using (2.20) we get This implies for n b 2 n α. 2b 2 n e 2πibnα sin 2π{ α} 2 π 2π 3 7 b 2 n = 2 7. Therefore, the error of numerical integration of f is bounded from below by cb 2 n, i.e. the error estimates do not improve when the smoothness r > 2. It means that the cubature formula Q n,α (g) := g(q) q A n(α) has a saturation property for r > 2. We note that this example resonates with ideas explored in [7] and [37]. We now turn our attention to the estimates for the L 2 discrepancy. Inspired by the Davenport s Reflection Principle [], described in the introduction, and the similarities between the Fibonacci and irrational lattices, we symmetrize F n to a 2 -point set Its discrepancy function is F n := {(p, p 2 ) (p, p 2 ) : (p, p 2 ) F n }. (2.23) D(F n, x) := #{F n [0, x ) [0, x 2 )} 2 x x 2, where x = (x, x 2 ) (0, ] 2. Rewriting it to [ D(F n, x) = χ[p,) [p 2,)(x) + χ [p,) [ p 2,)(x) ] 2 x x 2, p=(p,p 2 ) F n and computing the Fourier coefficients of the D(F n, x) yields D(F n, k) = [ χ[p,) [p 2,)(k) + χ [p,) [ p 2,)(k) ] 2 x x 2 = p=(p,p 2 ) F n [ p F 0 0 n χ [p,) [p 2,)(x, x 2 )e 2πik x dx dx 2

12 = χ [p,) [ p 2,)(x, x 2 )e 2πik x dx dx 2 ] [ x x 2 e 2πik x dx dx 2 e 2πik x dx e 2πik 2x 2 dx 2 p F p p 2 n ] + e 2πik x dx e 2πik 2x 2 dx 2 p p 2 2 x e 2πik x dx x 2 e 2πik 2x 2 dx (2.24) Note that µ= e 2πilµ/bn = { bn, l 0 (mod ), 0, l 0 (mod ). Let L(n) := {k = (k, k 2 ) Z 2 : k + k 2 0 (mod )}, then µ= e 2πi(k + k 2 )µ/ = Now let us consider different cases: { bn, (k, k 2 ) L(n), 0, (k, k 2 ) L(n). (2.25) (2.26) Case. k = 0, k 2 = 0. We have the following lemma: Lemma 2.2. D(F n, 0) = 2. Proof. From (2.24) we get D(F n, 0) = [ ] ( p )( p 2 ) + ( p )p 2 2 p Fn = p F n ( p ) 2 = µ= ( µ/ ) 2 2

13 = ( + ) 2 2 = 2. (2.27) Case 2. k 0, k 2 0. In this case D(F n, [ k) = ( e 2πikp )( e 2πik2p2 ) 4π 2 k k 2 p F n = Then we have the following lemma: Lemma 2.3. If k 0, k 2 0, then +( e 2πik p )( e 2πik 2( p 2 ) ) ] + 2π 2 k k 2 [ ( e 2πikp )( e 2πik2p2 ) 4π 2 k k 2 p F n +( e 2πik p )( e 2πik 2p 2 ) ] + 2π 2 k k 2. (2.28) D(F n, k) = provided that at least one of k and k 2 is 0 modulo. 2π 2 k k 2 (2.29) Proof. Without loss of generality assume k 0 (mod ), then So from (2.28) we get e 2πik p = e 2πik µ bn =. D(F n, k) = 2π 2 k k 2. (2.30) Lemma 2.4. Assume k 0 (mod ) and k 2 0 (mod ), then, k 2π 2 + k 2 0, k k 2 0, k k 2 D(F n,, k k) = 4π 2 + k 2 0, k k 2 0, k k 2, k 4π 2 + k 2 0, k k 2 0, k k 2 0, k + k 2 0, k k 2 0, where all congruences are taken modulo. 3 (2.3)

14 Proof. Since by (2.25) p F n e ±2πxik jp j = 0 for j =, 2, we can rewrite (2.28) as D(F n, k) = = = [ 2 + e 2πi(kp+k2p2) + e ] 2πi(kp k2p2) + 4π 2 k k 2 2π 2 k k 2 p F n [ e 2πi(kp+k2p2) + e ] 2πi(kp k2p2) 4π 2 k k 2 p F n 4π 2 k k 2 µ= [ 2πiµ(k +k 2 ) e bn + e 2πiµ(k k 2 ) ] bn. (2.32) If both k + k 2 0 (mod ) and k k 2 0 (mod ) hold, i.e. (k, k 2 ) L(n) and (k, k 2 ) L(n), we get D(F n, k) = 2π 2 k k 2. (2.33) Note that for odd the congruences k +k 2 0 (mod ), k k 2 0 (mod ) imply k 0 (mod ) that violates the assumptions of Lemma 3.3. Thus this case is possible only for even. If only one of k + k 2 0 (mod ), k k 2 0 (mod ) holds, or in other words only one of (k, k 2 ), (k, k 2 ) is in L(n), then D(F n, k) = 4π 2 k k 2. (2.34) If k + k 2 0 (mod ) and k k 2 0 (mod ), i.e. both (k, k 2 ) and (k, k 2 ) are not in L(n), then we get D(F n, k) = 0. (2.35) Case 3. k 0, k 2 = 0. We have the following lemma: Lemma 2.5. If k 0, k 2 = 0, D(F n,, k k) = 0 (mod ), 2πik 0, k 0 (mod ). (2.36) 4

15 Proof. We obtain from (2.24), D(F n, k) = 2πik = 2πik [ ] ( e 2πikp )( p 2 ) + ( e 2πikp )p 2 + 2πik p F n [ ] e 2πikp +. (2.37) 2πik p F n If k 0 (mod ), then e 2πik p =, thus D(F n, k) = If k 0 (mod ), then p F n e 2πik p = 0, hence 2πik. (2.38) D(F n, k) = 0. (2.39) Case 4. k = 0, k 2 0. We have the following lemma: Lemma 2.6. If k = 0, k 2 0, D(F n,, k k) = 2 0 (mod ), 2πik 2 0, k 2 0 (mod ). (2.40) Proof. From (2.24) we obtain D(F n, k) = 2πik 2 p F n [ ( p )( e 2πik2p2 ) + ( p )( e 2πik2p2 ) ] + 2πik 2 = [ ( p )(2 e 2πik2p2 e 2πik2p2 ) ] +. 2πik 2 2πik 2 p F n If k 2 0 (mod ), then e ±2πik 2p 2 =, and D(F n, k) = 5 2πik 2. (2.4)

16 If k 2 0 (mod ), then p F n e ±2πik 2p 2 = 0, and we get D(F n, k) = 2πik 2 Let us set = 2πik 2 = 2πik 2 = On one hand, and thus on the other hand [ 2 2p + p e 2πik2p2 + p e ] 2πik2p2 + 2πik 2 p F n [ 2 2 µ + µ e 2πik 2 µ bn µ= [ 2πik πik 2 f(x) = [ bn µ=0 b n µ=0 f (x) = f (k 2 ) = µ= ( µ e 2πik 2 µ bn ( µ e 2πik 2 µ bn e 2πiµx b n µ=0 f (x) = 2πie2πix (e 2πix bn b n µ=0 bn = e2πix bn. e 2πix + µ ] e 2πik 2 µ bn + 2πik 2 + µ ) ] e 2πik 2 µ bn + µ ) ] e 2πik 2 µ bn + 2. (2.42) 2πiµ e 2πiµx bn, (2.43) 2πiµ e 2πiµk2bn bn ; (2.44) ) (e 2πix ) 2πi bn ( 2πix e bn ). (2.45) 2 e 2πix Note that e 2πik 2 = and thus f (k 2 ) = = 2πi(e 2πik2bn bn ) ( e 2πik 2 bn ) 2 2πi. e 2πik 2 bn 6 (2.46)

17 Comparing (2.44) and (2.46) we find b n µ=0 In the same way we get b n µ=0 µ e 2πik2µbn bn = µ e 2πik2µbn bn =. e 2πik 2 bn. e 2πik 2 bn Therefore, b n µ=0 [ µ e 2πik2µbn bn + µ ] e 2πik 2 µ bn = = + bn e 2πik 2 ( e 2πik 2 bn ( e 2πik 2 bn = e 2πik2bn bn =. 2 e 2πik 2 bn e 2πik 2 bn ) + ( e 2πik 2 bn ) )( e 2πik 2 bn ) + e 2πik 2 bn 2 e 2πik 2 bn Hence from (2.42) Remark 2.7. We define the sets D(F n, k) = + ( + 2) 2πik 2 2πik 2 = 0. (2.47) S = {(k, k 2 ) : k, k 2 0, k 0 (mod )}, S 2 = {(k, k 2 ) : k, k 2 0, k 2 0 (mod )}, S 3 = {(k, 0) : k 0 (mod ), k 0}, S 4 = {(0, k 2 ) : k 2 0 (mod ), k 2 0}, S 5 = {(k, k 2 ) : (k, k 2 ) L(n) \ {0}, k, k 2 0 (mod )}, S 6 = {(k, k 2 ) : (k, k 2 ) L(n) \ {0}, k, k 2 0 (mod )}. 7

18 Based on previous lemmas, we have the following observations. The results of lemmas 2.3, 2.4, 2.5, and 2.6 imply that for k S... S 6 we have D(F n, k). (2.48) 2 max( k j, ) j= In all other cases, the corresponding Fourier coefficients are equal to zero, see (2.35), (2.39) and (2.47). For k S, we write k = l, where l Z \ {0}. Then D(F n, k) = 2π 2 k l. We deal with S 2, S 3, and S 4 similarly. We are now ready to proceed to the main theorem. Theorem 2.8. For the symmetrized Fibonacci set F n [0, ] 2, we have Proof. By Parseval s theorem, D(F n, x) 2 log. (2.49) D(F n, x) 2 2 = D(F n, k) 2 2 D(F n, 0) 2 + k L(n)\{0} + 6 i= b 2 n 2 max(kj 2, ) j= (k, k 2 ) L(n)\{0} +2 l 0 k 0 k S i D(F n, k) 2 b 2 n 2 max(kj 2, ) j= (kl) l 0 It is easy to see that the last two sums converge to some constants and the l 2. 8

19 first two are completely similar to each other. We can thus estimate D(F n, x) 2 2 k L(n)\{0} b 2 n. (2.50) 2 max(kj 2, ) We now use the following lemma, see Lemma 2. from Chapter 4 of [35]. Lemma 2.9. Denote Γ(N) := { d k = (k,, k d ) Z d : max( k j, ) N } and Z l := ( Γ(2 l+ γ ) \ Γ(2 l γ ) ) L(n), l = 0,, 2,..., then there exists an absolute constant γ > 0 such that for any n > 2 Γ(γ ) ( L(n) \ 0 ) =, j= j= and Z l 2 l (l + ) log, l = 0,, 2,.... (2.5) Therefore, the summation in (2.50) can be estimated as D(F n, x) 2 2 l 0 and using the cardinality estimate of Z l in (2.5), we get, D(F n, x) 2 2 l 0 = log log. k Z l 2 l 2, (2.52) 2 l (l + ) log (2 l ) 2 l 0 l + 2 l Hence D(F n, x) 2 log. 9

20 Remark 2.0. In this section we symmetrize the original Fibonacci set to obtain a 2 point set F n = {(p, p 2 ) {(p, p 2 ) : (p, p 2 ) F n }. Obviously, the L discrepancy of F n satisfies the same upper bound as F n in the order of magnitude and thus is optimal. Theorem 2.8 verifies the sharpness of its L 2 discrepancy. In fact, we can also demonstrate that a 4 point set F n = {(p, p 2 ) ( p, p 2 ) {(p, p 2 ) {( p, p 2 ) : (p, p 2 ) F n } achieves the minimal L 2 discrepancy as well. The computation is completely analogous, and, in Case 4 (Lemma 2.6), it is much more straightforward. Next, we derive a formula which provides the exact value of D(F n, x) 2. For simplicity, we shall first assume that is odd, and thus S 5 S 6 =. We start with the contribution of k S 5, using the notation introduced in Remark 2.7. In this case, D(F n, k) = bn 4π 2 k k 2. We shall make use of the well-known identity (see e.g. [34], page 65, ex. 5): n Z (n + x) 2 = π 2 sin 2 (πx). (2.53) Denote k +k 2 = l, for l Z and toward the end of the computation write k 2 = m + r, where m Z and r =,...,. We have, by Lemma 2.4 D(F n, k) 2 = b2 n 6π 4 k S 5 = = = 6π 2 b n 6π 2 r= 6b 2 n k 2 k 2 0 mod 2 l Z b 2 n ( ) k 2 0 mod k2 2 sin 2 πbn k 2 b n r= ( ) sin 2 πbn r ( ) 2 l k 2 b 2 m Z n ( ) 2 m + r ( ) ( ), (2.54) sin 2 πbn r sin 2 πr where we have used identity (2.53) in the second and the last equalities above. It is obvious that the contribution of k S 6 is identical. If is even, a correction term arises due to the fact that S 8b 2 5 S 6 (we leave the n computation to the reader). 20

21 Using the inclusion-exclusion principle and the identity we obtain by Lemma 2.3 k S S 2 D(F n, k) 2 = 4 4 = 8 l N l N,k 2 N l N,l 2 N l 2 = π2 6, (2.55) b 2 n 4π 4 l 2 b 2 n k 2 2 b 2 n 4π 4 b 4 nl 2 l π 4 π2 6 π b 2 n k N,l 2 N = 36 b 2 n 4π 4 k 2 l 2 2b 2 n ( 2 ). (2.56) b 2 n (The multiplication by 4 above accounts for all possible choices of signs). Finally, Lemmas 2.5 and 2.6 yield k S 3 S 4 D(F n, k) 2 = 2 b 2 n 4π 2 l Z\{0} b 2 nl 2 = 6. (2.57) Putting together equations (2.54), (2.56), and (2.57), and the relation D(F n, 0) = 2 (Lemma 2.2) we obtain Theorem 2.. For n 2 we have D(F n, x) 2 2 = b n 8b 2 n r= D(F n, x) 2 2 = b n 8b 2 n r= ( ) ( sin 2 πbn r sin 2 πr ( ) ( sin 2 πbn r sin 2 πr 7 )+ 7 ) b 2 n b 2 n when is odd, (2.58) when is even. (2.59) We should recall that the L 2 discrepancy of an arbitrary N-point set can be computed precisely using Warnock s formula [38]. However, the fastest known way to perform this computation requires O(N log N) steps [8], [4] 2

22 Figure : The values of S n for n 35. (see also the discussion in 2.4 of [25]). The formulas of Theorem 2. require only of the order of N steps to compute the discrepancy of the symmetrized Fibonacci set F n. It can be shown directly that the main term in equations (2.58) and (2.59) is of the order log n. Besides, numerical experiments indicate that S n = b 2 n b n r= ( ) ( ) n. (2.60) sin 2 πbn r sin 2 πr We have performed these computations using MATLAB and Maple up to n = 35, which corresponds to N = 2b 35 = 29, 860, 704. The differences between successive values of S n stabilize very quickly (up to the sixth decimal digit starting with n = 6, see Table and Figure ). Straightforward computations become unstable and too slow beyond this value; in particular, very time-consuming computations for 36 n 40 yielded consecutive differences between and We plan to conduct more sophisticated and precise calculations in the future. We are extremely grateful and indebted to Douglas Meade for his help with the numerical experiments. 22

23 Since the symmetrized Fibonacci set F n has N = 2 points, we have ( ) log N 5 + lim = log n n 2 Here log stands for the natural logarithm in order to compare our results with the upper bound in [3]. Assuming that the results of the numerical experiments are indeed true, we obtain D(F n, x) 2 2 = b n 8b 2 n r= ( ) ( ) + O() (2.6) sin 2 πbn r sin 2 πr = (0.25) ( ) n + O() = log N + o(log N). This (numerically obtained) constant above is smaller than the analogous best constant, , found in [3] for the scrambled generalized Hammersley point sets. Hence, numerical computations indicate that among all two-dimensional point sets, the symmetrized Fibonacci lattice has the smallest known L 2 discrepancy: Corollary 2.2. The symmetrized Fibonacci sets F n with N = 2 points satisfy: D(F lim n, x) (2.62) n log N The previously best known constant, obtained in [3] is slightly larger, However, our corollary, strictly speaking, is not a mathematical fact, but rather a result of experiments. The actual values of the L 2 discrepancy provided by (2.58) and (2.59) for moderate values of n are somewhat larger. For example, for n = 35, i.e. N = 29, 860, 704, we have D(F n,x) 2 log N (see Table for a full list of values). It is worth mentioning that the best currently known constant in the lower estimates was found by Hinrichs and Markhasin [9]. They prove that, in our notation, D(N, 2) 2 log N log N. 2 6 log 2 23

24 Table : The results of numerical computations n N = 2 S n D(F n, x) 2 2 D(F n,x) 2 log N Quartered L p discrepancy and two-fold symmetrization We shall consider a modification of the classical L p discrepancy function. For a parameter a [0, /2] define the following univariate characteristic function for t [0, ). S(a, t) := χ [/2 a,/2+a] (t), and for the multivariate case x [0, /2] d, y [0, ] d S(x, y) := d S(x j, y j ). j= 24

25 For a set ξ := {ξ µ } N µ= [0, ] d, define the quartered L p discrepancy as follows D q N (ξ, N, d) p := S(x, ξ µ ) N S(x, y)dy [0,] d µ= L p([0,/2] d,x). (3.63) The expression inside the norm is simply the discrepancy of ξ with respect to the box centered at /2 = (/2,..., /2) and opposite corners at /2±x. Let us note that this notion of discrepancy does not quite measure the uniformity of distribution of ξ as it doesn t change when we move all points to the same quadrant with respect to the center of the square. However, precisely these considerations relate the quartered L p discrepancy and standard L p discrepancy. We have S(a, t) = χ [0, 2 +a](t) χ [0, 2 a](t). This allows us to obtain the following inequality D q (ξ, N, d) p 2 d D(ξ, x) p. The quartered L p discrepancy can be bounded from below by the L p discrepancy of a symmetrized set ξ sym, that we define momentarily. We describe it in the case d = 2. Let R and R 2 be reflection operators that act as follows: for u = (u, u 2 ) [0, ] 2 R (u) := ( u, u 2 ), R 2 (u) := (u, u 2 ). For a set ξ = {ξ j } N j= [0, ] 2, define the symmetrized set ξ := ξ R (ξ) R 2 (ξ) R 2 (R (ξ)). This set contains 4N points, counting multiplicity. The sets [ G (x) := 2, ) [ 2 + x 2, ) [ 2 + x 2, G 2 (x) := 2, ) [ 2 x [ G 3 (x) := 2, ) [ 2 x 2, ) 2 x 2, G 4 (x) := [ 2, 2 + x ) ), 2, 2 + x 2 [ 2, ) 2 x 2, contain the same number of points of ξ since we split the points in set ξ on the boundary evenly. 25

26 We now define ξ sym the two-fold symmetrization of ξ in the following way: take all the points of ξ that lie in the same quadrant [/2, ] [/2, ], then shift and rescale them to the unit square [0, ] 2 : ξ sym := { ( v = 2 u ) 2 : u ξ ([ 2, ] [2, ] )}. (3.64) Then for the quartered L p discrepancy of ξ we have p D q ( ξ, 4N, 2) p 2 2 p = 4 χ G (x)(u) 4N x x 2 dx dx u ξ [ 2,] [ 2,] p = χ 0 0 [0,z] (v) N z z 2 dz dz 2 = D(ξ sym, z) p p, v ξ sym where z = 2x. On the other hand, obviously D q ( ξ, 4N, 2) p = 4D q (ξ, N, 2) p. Thus we have proved the following simple property that we formulate as a proposition. Proposition 3.. Let ξ sym be the two-fold symmetrization of ξ as defined by (3.64). Then D(ξ sym, x) p = 4D q (ξ, N, 2) p. Proposition 3. can be used in both directions. First, it allows us to get a lower bound for D q (ξ, N, 2) p. It is known that for all p > and any set P N of N points one has D(P N, x) p C log N, (3.65) where C is some positive absolute constant. Therefore, for any ξ D q (ξ, N, 2) p C log N. Second, it gives a way to build a set (in our case ξ sym ) with good L p discrepancy from a set (in our case ξ) with good quartered L p discrepancy. For instance, as we prove below, the Fibonacci sets F n have optimal quartered L p discrepancy for p (, ) in the sense of order. Therefore, by Proposition 3. the set Fn sym, obtained from the Fibonacci set F n by the symmetrization procedure described above, has optimal in the sense of order standard L p discrepancy for all p (, ). 26

27 We proceed to estimate D q (ξ, N, d) p, p <, from above in the case when ξ = F n is the Fibonacci set, i.e. d = 2, N = and ξ µ = (µ/, {µ / }), ξ = F n := {ξ µ } bn µ=. We apply the technique that is based on the Fourier representation of S(x, y) as a function on y. First, we find the Fourier coefficients of the univariate function Ŝ(a, k) = 0 S(a, t)e 2πikt dt = ( ) k (2πik) (e 2πika e 2πika ). It is clear that Ŝ(a, 0) = 2a. Second, it follows directly from the definition of S(x, y) and the above formulas that Denote d Ŝ(x, k) = Ŝ(x j, k j ) j= Φ(k) = µ= d max( k j, ). (3.66) j= e 2πi(k,ξµ). Then for a trigonometric polynomial f one has Φ n (f) := µ= f ( µ/, {µ / } ) = k It is known and easy to see that the following relation holds { bn, k L(n), Φ(k) = 0, k L(n). ˆf(k)Φ(k). (3.67) (3.68) Therefore, in the case p = 2, that we discuss first D q (F n,, 2) 2 Φ(k)Ŝ(x, k). (3.69) 2 k (0,0) Using the fact that functions Ŝ(x, k) and Ŝ(x, k ) are orthogonal on [0, ] 2 if ( k, k 2 ) ( k, k 2 ), the bounds (3.66), (3.69), and estimate (2.5) we obtain D q (F n,, 2) 2 2 b 2 n(2 l ) 2 Z l log l=0 27 l=0 2 l (l + ) 2 2l log. (3.70)

28 Thus, D q (F n,, 2) 2 log. (3.7) We now proceed to the case p [2, ). Let ψ l (x) := Ŝ(x, k). k Z l Then D q (F n,, 2) p l=0 ψ l p. (3.72) By the corollary of the Littlewood-Paley theorem we have for ψ l p ( ψ l p δs (ψ l ) /2 p) 2, (3.73) where for s = (s, s 2 ), s j are nonnegative integers δ s (f, x) := ˆf(k)e i(k,x). s [2 s j ] k j <2 sj, j=,2 It is not difficult to see that for ψ l only those δ s (ψ l ) can be nonzero for which s log 2 (2 l γ ) C. In addition by Lemma 2.9 the number of terms of δ s (ψ l ) is not greater than C2 l. Therefore, δ s (ψ l ) p δ s (ψ l ) 2/p δ s (ψ l ) 2/p 2 l/p b n (3.74) 2 and ψ l p (l + log ) /2 2 l/p b n. (3.75) The bounds (3.72), (3.74) and Proposition 3. imply Theorem 3.2. i) For all p (, ), the quartered L p discrepancy of the Fibonacci set F n satisfies D q (F n,, 2) p C(p) log. (3.76) ii) For all p (, ), the two-fold symmetrization Fn sym F n has optimal L p discrepancy: of the Fibonacci set D(F sym n, x) p C (p) log 4. (3.77) 28

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