MEASURES OF PSEUDORANDOMNESS FOR FINITE SEQUENCES: MINIMUM AND TYPICAL VALUES (EXTENDED ABSTRACT)

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1 MEASURES OF PSEUDORADOMESS FOR FIITE SEQUECES: MIIMUM AD TYPICAL VALUES (EXTEDED ABSTRACT) Y. KOHAYAKAWA, C. MAUDUIT, C. G. MOREIRA, AD V. RÖDL Dedicated to Professor Imre Simon on the occasion of his 60th birthday Abstract. Mauduit and Sárözy introduced and studied certain numerical parameters associated to finite binary sequences E { 1, 1} in order to measure their level of randomness. These parameters, the normality measure (E ), the well-distribution measure W (E ), and the correlation measure C (E ) of order, focus on different combinatorial aspects of E. In their wor, amongst others, Mauduit and Sárözy (i) investigated the relationship among these parameters and their minimal possible value, (ii) estimated (E ), W (E ), and C (E ) for certain explicitly constructed sequences E suggested to have a pseudorandom nature, and (iii) investigated the value of these parameters for genuinely random sequences E. In this paper, we continue the wor in the direction of (iii) above and determine the order of magnitude of (E ), W (E ), and C (E ) for typical E. We prove that, for most E { 1, 1}, both W (E ) and (E ) are of order, while C (E ) is of order log ( ) for any given 2 /4. We also prove a lower bound for the correlation measure C (E ) ( even) for arbitrary sequences E, which, in particular, gives that min E C (E ) c for some c > 0 for any constant even. 1. Introduction and statement of results In a series of papers, Mauduit and Sárözy studied finite pseudorandom binary sequences E = (e 1,..., e ) { 1, 1}. In particular, they investigated in [11] the measures of pseudorandomness to be defined shortly. The readers interested in detailed discussions concerning the definitions below Date: 15 August Mathematics Subject Classification. 68R15. Key words and phrases. Random sequences, pseudorandom sequences, finite words, normality, correlation, well-distribution, discrepancy. This research was partially supported by IM-AGIMB/IMPA. The first author was partially supported by MCT/CPq a ProEx project (Proc. CPq /1997 4) and by CPq (Proc /93 1 and /2000 0). The fourth author was partially supported by SF Grant The collaboration of the Brazilian authors and the US author is supported by a CPq/SF cooperative grant (910064/99 7, ).

2 2 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL and further related literature are referred to [10] (WORDS, Rouen 1999) and [11]. Let, M, X { 1, 1}, a Z, b, b > 0, and D = (d 1,..., d ) with 0 d 1 < < d < be given. Below, we write card S for the cardinality of a set S, and if S is a set of numbers, then we write S for the sum s S s. We let T (E, M, X) = card{n: 0 n < M, n +, and (e n+1, e n+2,..., e n+ ) = X}, (1) and U(E, M, a, b) = {e a+jb : 1 j M, 1 a + jb for all j}, (2) V (E, M, D) = {e n+d1 e n+d2... e n+d : 1 n M, n + d }. (3) In words, T (E, M, X) is the number of occurrences of the pattern X in E, counting only those occurrences whose first symbol is among the first M elements of E. The quantity U(E, M, a, b) is the discrepancy of E on an M-element arithmetic progression contained in {1,..., }. Finally, V (E, M, D) is the correlation among length M segments of E relatively positioned according to D = (d 1,..., d ). The normality measure of E is defined as (E ) = max max max X M T (E, M, X) M 2, (4) where the maxima are taen over all log 2, X { 1, 1}, and 0 < M + 1. The well-distribution measure of E is defined as W (E ) = max{ U(E, M, a, b) : a, b, and M such that 1 a + b < a + Mb }. (5) Finally, the correlation measure of order of E is defined as C (E ) = max{ V (E, M, D) : M and D such that M + d }. (6) In [3], Cassaigne, Mauduit, and Sárözy studied, amongst others, the value of W (E ) and C (E ) for random binary sequences E, with all the 2 sequences in { 1, 1} equiprobable, and the minimal possible values for W (E ) and C (E ). They proved the following theorems. (Below and elsewhere in this abstract, we write log for the natural logarithm.) Theorem A. For all ε > 0, there are numbers 0 = (ε) and δ = δ(ε) > 0 such that for 0 we have with probability at least 1 ε. δ < W (E ) < 6 log (7)

3 MEASURES OF PSEUDORADOMESS 3 Theorem B. For every integer 2 and real ε > 0, there are numbers 0 = 0 (ε, ) and δ = δ(ε, ) > 0 such that for all 0 we have with probability at least 1 ε. δ < C (E ) < 5 log (8) Theorem C. For all and with 2, we have (i) min{c (E ): E { 1, 1} } = 1 if is odd, (ii) min{c (E ): E { 1, 1} } log 2 (/) if is even. As it turns out, an improvement of the upper bound in Theorem A may be deduced from a proof of a closely related result of Erdős and Spencer. Indeed, an argument in [7, Chapter 8] tells us that one may drop the logarithmic factor in (7), at the expense of increasing the multiplicative constant. Let us also observe that it follows from the results of Roth [14] and Matouše and Spencer [9] that the order of magnitude of min { W (E ): E { 1, 1} } is 1/4. In [3], it is conjectured that for any even 2 there is a constant c > 0 such that for we have min { C (E ): E { 1, 1} } c, which would be a considerable strengthening of Theorem C(ii). In this extended abstract, we state stronger versions of Theorems A and B, we prove the conjecture above in a more general form, and we give a result concerning the typical value of (E ). Theorem 1. For any given ε > 0 there exist 0 and δ > 0 such that if 0, then with probability at least 1 ε. δ < W (E ) < 1 δ (9) Theorem 1 above is essentially proved in Erdős and Spencer [7, Chapter 8]. The reader is referred to [8] for an alternative proof, the main idea of which is also used in the proof of Theorem 4 below (see Section 2.2). We next state a result that establishes the typical order of magnitude of C (E ) for a wide range of, including values of proportional to. Theorem 2. Let 0 < ε 0 1 be fixed and let ε 1 = ε 1 () = (log log )/ log. There is a constant 0 = 0 (ε 0 ) such that if 0, then, with probability at least 1 ε 0, we have 2 5 ( ) ( ( )) log < C (E ) < (2 + ε 1 ) log ( ) < (3 + ε 0 ) log < 7 4 log ( ) (10)

4 4 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL for every integer with 2 /4. The proof of Theorem 2 is setched in Section 2.1. Our next result tells us that C (E ) is concentrated in the case in which is small. Theorem 3. For any fixed constant ε > 0 and any integer function = () with 2 log log log, there is a function Γ(, ) and a constant 0 for which the following holds. If 0, then the probability that 1 ε < C (E ) Γ(, ) < 1 + ε (11) holds is at least 1 ε. Theorem 3 follows from standard arguments involving martingales (see [8] for details). Clearly, Theorem 2 tells us that Γ(, ) is of order log ( ). We now turn to the normality measure (E ). Theorem 4. For any given ε > 0 there exist 0 and δ > 0 such that if 0, then with probability at least 1 ε. δ < (E ) < 1 δ (12) The proof of Theorem 4 is setched in Section 2.2. Finally, we state our result concerning the minimal possible value for the parameter C (E ). Theorem 5. If and are natural numbers with even and 2, then C (E ) > (13) 3( + 1) for any E { 1, 1}. The proof of Theorem 5 is given in Section 3. In Section 4, we remar that the upper bounds in Theorems 1 and 4 are in a certain sense best possible. 2. Estimates for C (E ) and (E ) for random sequences E We shall setch the proofs of Theorems 2 and 4 in this section. Recall that these results concern random elements E from the uniform space { 1, 1}. In this section, unless stated otherwise, E will always stand for such a random sequence Proof of Theorem 2. The upper estimate for C (E ) follows from standard estimates for the binomial distribution, and we refer the reader to [8] for the details. Here, we shall concentrate on the following result.

5 MEASURES OF PSEUDORADOMESS 5 Lemma 6. With probability tending to 1 as, we have C (E ) > 2 ( ) log 5 for every integer with 2 /4. We start by stating a technical result without proof. (14) Fact 7. Let m = /3. For every sufficiently large, the following hold. (i) If 2 log m, then ( ) ( ) /3 log 0.99 log. (15) (ii) If log m < /4, then ( ) / log 10 ( ) log. (16) 3 We now give the proof of the main result in this section. Proof of Lemma 6. Set m = /3, and recall that we write S(m, 1/2) for a random variable with binomial distribution Bi(m, 1/2). Fix 2 /4. We are interested in the largest integer r for which P (S(m, 1/2) 12 ) ( ) m (m + r) 2 (log ) (17) 1 holds. Indeed, we let r(m) = r (m) = max {r : inequality (17) holds}. (18) We need the following fact concerning r(m). Fact 8. For every sufficiently large, the following hold. (i) If 2 log m, then r(m) m log ( ) m + 1. (19) 1 (ii) If log m < /4, then ( ) r(m) ( m m + 1 ) log 2 log. (20) 1 (iii) If 2 /4, then r(m) 2 5 log ( ). (21)

6 6 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL We shall not prove Fact 8 here. Inequality (21) in Fact 8, which follows from (19) and (20) and the inequalities in Fact 7, will be used shortly. To prove Lemma 6, we shall show that, with probability 2/ 2 log, we have C (E ) r (m), (22) and then we shall sum over all 2 /4. This gives that (22) holds for some with 2 /4 with probability O(1/ log ) = o(1). Therefore, (22) fails for all 2 /4 with probability 1 o(1), and hence, taing into account (21), Lemma 6 will be proved. Let 2 /4 be fixed. Our strategy to estimate the probability that (22) should hold will be as follows. Recall E = (e 1,..., e ) and let u = (e 1,..., e m ). Let D be the set of ( 1)-tuples D = (d 1,..., d 1 ) with m d 1 < < d 1 2m. For each D D, let v D = (e 1+d1 e 1+d2... e 1+d 1,..., e m+d1 e m+d2... e m+d 1 ), (23) and let A(D) be the event that u, v D > r(m) = r (m) (24) holds. It suffices to show that some A(D) (D D ) holds with probability at least 1 2/ 2 log. For convenience, let X = X(E ) be the number of events A(D) (D D ) that hold for E. Let p = p(m) = P (S(m, 1/2) 12 ) (m + r(m)). (25) Because of (18), we have ( ) m + 1 E(X) = p D = p 2 log. (26) 1 We have now arrived at the ey claim: the events A(D) (D D ) are pairwise independent. Although this somewhat surprising fact is crucial, we shall omit its proof because of space constraints. (The reader is invited to amuse himself or herself proving this; for a one-page proof, see [8]). Claim 9. For all distinct D and D D, we have P(A(D) A(D )) = p 2. To complete the proof of Lemma 6, we mae use of the following result, which gives a lower bound for the probability of a union of pairwise independent events. We shall in fact state a stronger lemma, which has as hypothesis that the events should be asymptotically negatively correlated. Versions of this lemma may be found in [4] and [6]. More recently, Petrov used this result to generalize the Borel Cantelli lemma [12, 13]. Lemma 10. Let A 1,..., A M be events in a probability space, each with probability at least p. Let ε 0 be given, and suppose that P(A i A j ) p 2 (1 + ε) for all i j. Then ( ) P A Mp j 1 j M 1 + (M 1)p(1 + ε) > 1 ε 2 Mp.

7 MEASURES OF PSEUDORADOMESS 7 We conclude the proof of Lemma 6 combining Claim 9 and Lemma 10. It suffices to notice that we have M = ( ) m+1 1 pairwise independent events A(D) (that is, ε = 0), and pm 2 log (see (26)). Lemma 10 then tells us that, with probability greater than 1 2/ 2 log, the event A(D) happens for some D D. We conclude that (22) occurs with probability at most 2/ 2 log, and hence, as observed above, summing over all 2 /4, Lemma 6 follows The normality measure. Recall that the normality measure of E = (e 1,..., e ) { 1, 1} is defined as (E ) = max max max X M T (E, M, X) M 2, (27) where the maxima are taen over all log 2, X { 1, 1}, and 0 < M + 1, and T (E, M, X) is the number of occurrences of the pattern X in E, counting only those occurrences starting with some e j with j M (see (1)). In this section, we outline the proof of Theorem 4. Proof of Theorem 4 (Outline). We start with a setch of the proof of the lower bound in (12). We tae = 1 in (27); in fact, we consider T (E,, (1)), the number of occurrences of 1 in E. Then a simple application of the de Moivre Laplace theorem on the binomial distribution (see, e.g., [2]) tells us that for any ε > 0, there is δ > 0 so that the lower bound in (12) holds with probability at least 1 ε. It remains to prove the upper bound for (E ) for typical sequences E. The basic lemma that we shall use is Lemma 11 below. We shall consider intervals of integers of the form B m,r = (m2 r, (m + 1)2 r ] Z, where m and r are non-negative integers. Clearly, B m,r = 2 r. We refer to the B m,r as blocs. For an integer 1, X { 1, 1}, and B m,r a bloc with max B m,r = (m + 1)2 r + 1, (28) we shall write T (E, B m,r, X) for the number of occurrences of the pattern X in E, counting only those occurrences starting in B m,r, that is, T (E, B m,r, X) = card{n B m,r : E (n) = X}, where E(n) = (e j) n j<n+, and, as usual, E = (e 1,..., e ). Lemma 11. Let m and r be fixed non-negative integers with B m,r [1, ]. For all D > 0, the probability that there is X { 1, 1} with log 2 satisfying (28) such that T (E, B m,r, X) 2 r > D 2 r (29) is at most ( ) ( O e 2D2 /9 + 2(log 2 ) 2 exp 3D ) log 2 2r/2.

8 8 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL We shall not prove Lemma 11 here. Let us continue with the proof of Theorem 4. Let us apply Lemma 11 with D = C(K r) (30) for all blocs B m,r [1, ], where K = 1 + log 2, and C is a large constant. There are at most /2 r < 2 K r blocs B m,r contained in [1, ] and any such bloc is such that r < K. Let us call a bloc B m,r large if r 4 log 2 log 2 (31) and small otherwise. If C is a large enough constant, then, with the value of D given in (30), it follows from Lemma 11 that inequality (29) holds for some large bloc B m,r and some X { 1, 1} ( log 2 ) with probability ( ) O e 2C2 /9. (32) Since the bound in (32) tends to 0 as C, we may and shall suppose henceforth that (**) for all integers m, r, and with B m,r [1, ] satisfying (28) and (31), and every X { 1, 1} ( log 2 ), we have T (E, B m,r, X) 2 r C(K r) 2 r. ow observe that, for any 1 M + 1, we may write [1, M] as a disjoint union of blocs B m,r (r log 2 M < K) with at most one bloc of the form B m,r for each r. Indeed, such a decomposition of [1, M] may be read out from the binary expansion of M. Let us write I for the set of the pairs (m, r) for which B m,r occurs in this decomposition of [1, M]. Furthermore, let I = I + I be the partition of I with For later reference, observe that I + = {(m, r) I : r satisfies (31)}. (33) I < 4 log 2 log 2. (34) Observe also that if (m, r) I, then T (E, B m,r, X) 2 r 2 r < (log 2 ) 4 (35) for any X { 1, 1}. Using (**), (34), and (35), one may chec that, for any X { 1, 1} ( log 2 ), we have T (E, M, X) M2 T (E, B m,r, X) 2 r = O( ), (m,r) I as required (we omit the details).

9 MEASURES OF PSEUDORADOMESS Two remars and a problem on. We close this section with two observations about the normality measure and an open problem that we find quite attractive. Put (E ) = max X max M T (E, M, X) M 2, (36) where the maxima are taen over all X { 1, 1} and 0 < M + 1. ote that, then, we have (E ) = max{ (E ): log 2 }. Our first remar is that min E (E ) = 1 for any fixed : we consider powers of appropriate de Bruijn sequences. More precisely, we tae a circular sequence in which every member of { 1, 1} occurs exactly once, open it up (turning it into a linear sequence), and repeat it many times. Our second remar is that min E (E ) (1/2 o(1)) log 2 : if a sequence E { 1, 1} contains no segment of length = log 2 log 2 log 2 of repeated 1s (say), then (E ) log 2 = /2. If E contains such a segment, then 1 (E ) /2 = (1/2)(log 2 log 2 log 2 ). Problem 12. Is there an absolute constant α > 0 for which we have for all large enough? min (E ) > α E 3. The minimum of the correlation measure The proof of Theorem 5 that we give in this section is based on the following elementary lemma from linear algebra [5, Lemma 7], whose proof we include for completeness. Lemma 13. For any symmetric matrix M = (M ij ) 1 i,j n, we have ( 2 1 i n ii) M ran(m) (trace(m))2 trace(m 2 ) = 1 i,j n M2 ij. (37) Proof. Let r = ran(m). Then M has exactly r non-zero eigenvalues, say, λ 1,..., λ r. By the Cauchy Schwarz inequality, we have (trace(m)) 2 = (λ λ r ) 2 r(λ λ 2 r) = r trace(m 2 ). As M is symmetric, we have trace(m 2 ) = ( ) 1 i n 1 j n M ijm ji = 1 i,j n M2 ij, as required. Proof of Theorem 5. First we remar that C (E ) = 1, so that (13) is true for =, and we can suppose for a contradiction that there exist with 2 1 and E { 1, 1} such that C (E ) 3( + 1). (38)

10 10 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL Set = 2l, put M = + 1 and consider the 2M vectors v 0,..., v 2M 1 R M given by v i = e il+j, e il+j+1,..., 1 j l 1 j l, (39) 1 j l e il+j+m 1 for i {0,..., 2M 1}. Define the 2M 2M matrix M = (M ij ) 1 i,j 2M putting (40) M ij = v i, v j (41) for all (i, j) {1,..., 2M} 2. Then M has the following properties: (i) ran(m) M, (ii) M ii = M for any i {1,..., 2M}, (iii) M ij /3( + 1) for any (i, j) {1,..., 2M} 2 with i j. Indeed, (i) and (ii) are clear and (iii) follows from (38). It follows from Lemma 13 that ( 2M 2 ) 2 ran(m) 2M M 2 + 2M(2M 1)/3( + 1) 2M 3 = M 2 + (2M 1)/3( + 1), so that, by (39), we have ran(m) > 2M 3 M 2 + (2M 1)(M + 1)/3 = M 6M 2 5M 2 + M 1 > M, which contradicts property (i) above. This contradiction shows that for no with 2 1 there is E { 1, 1} such that (38) holds, and our result follows. 4. Concluding remars The upper bounds in Theorems 1 and 4 are best possible in the following sense. Let us consider W (E ). We claim that, for any C > 0, there is ε 0 > 0 such that ( P W (E ) < C ) 1 ε 0 (42) for all large enough. Therefore, the fact that the constant 1/δ in the upper bound in Theorem 1 depends on ε cannot be avoided. Inequality (42) follows simply from the de Moivre Laplace theorem on the binomial distribution (we omit the details). One may prove similar facts concerning the upper bound in Theorem 4 by considering T (E,, (1)), the number of occurrences of 1 in E.

11 MEASURES OF PSEUDORADOMESS 11 Problem 14. Investigate the existence of the limiting distributions of { } { } W (E ) (E ) and and Investigate these distributions. 1 C (E ) log ( ) 1. 1 It is most liely that all three sequences in Problem 14 have limiting distributions. ote that Theorems 2 and 3 tell us that there is a function Γ(, ), whose order of magnitude is log ( ), for which {C (E )/Γ(, )} 1 has a limiting distribution that is concentrated on a point, as long as = () log log log. Finally, we believe that Problem 12 merits investigation. ote added in the revision. Very recently, oga Alon has obtained an algebraic construction for a sequence E with (E ) of order at most 1/3 (log ) O(1). He has also observed that Theorem 9.3 from [1] may be used to prove further lower estimates for min E C (E ). We shall come bac to these results in the near future. References [1]. Alon. Problems and results in extremal combinatorics I. Discrete Math. To appear. [2] B. Bollobás. Random graphs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, [3] J. Cassaigne, C. Mauduit, and A. Sárözy. On finite pseudorandom binary sequences. VII. The measures of pseudorandomness. Acta Arith., 103(2):97 118, [4] K. L. Chung and P. Erdös. On the application of the Borel-Cantelli lemma. Trans. Amer. Math. Soc., 72: , [5] B. Codenotti, P. Pudlá, and G. Resta. Some structural properties of low-ran matrices related to computational complexity. Theoret. Comput. Sci., 235(1):89 107, Selected papers in honor of Manuel Blum (Hong Kong, 1998). [6] P. Erdős and A. Rényi. On Cantor s series with convergent 1/q n. Ann. Univ. Sci. Budapest. Eötvös. Sect. Math., 2:93 109, [7] P. Erdős and J. Spencer. Probabilistic methods in combinatorics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], ew Yor-London, Probability and Mathematical Statistics, Vol. 17. [8] Y. Kohayaawa, C. Mauduit, C. G. Moreira, and Rödl. Measures of pseudorandomness for finite sequences: minimum and typical values. In preparation, [9] J. Matouše and J. Spencer. Discrepancy in arithmetic progressions. J. Amer. Math. Soc., 9(1): , [10] C. Mauduit. Finite and infinite pseudorandom binary words. Theoret. Comput. Sci., 273(1-2): , WORDS (Rouen, 1999). [11] C. Mauduit and A. Sárözy. On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith., 82(4): , [12] V. V. Petrov. A generalisation of the Borel Cantelli lemma. manuscript, 2002, 8pp.

12 12 KOHAYAKAWA, MAUDUIT, MOREIRA, AD RÖDL [13] V. V. Petrov. A note on the Borel-Cantelli lemma. Statist. Probab. Lett., 58(3): , [14] K. F. Roth. Remar concerning integer sequences. Acta Arith., 9: , Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, Brazil address: yoshi@ime.usp.br Institut de Mathématiques de Luminy, CRS-UPR9016, 163 av. de Luminy, case 907, F-13288, Marseille Cedex 9, France address: mauduit@iml.univ-mrs.fr IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ, Brazil address: gugu@impa.br Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA address: rodl@mathcs.emory.edu