Distribution of short subsequences of inversive congruential pseudorandom numbers modulo 2 t
aa r X i v : . [ m a t h . N T ] J un DISTRIBUTION OF SHORT SUBSEQUENCES OFINVERSIVE CONGRUENTIAL PSEUDORANDOMNUMBERS MODULO t L ´ASZL ´O M´ERAI AND IGOR E. SHPARLINSKI
Abstract.
In this paper we study the distribution of very shortsequences of inversive congruential pseudorandom numbers mod-ulo 2 t . We derive a new bound on exponential sums with suchsequences and use it to estimate their discrepancy. The techniquewe use is based on the method of N. M. Korobov (1972) of estimat-ing double Weyl sums and a fully explicit form of the Vinogradovmean value theorem due to K. Ford (2002), which has never beenused in this area and is very likely to find further applications. Introduction
Background on the M¨obius tranformation.
Let t ě U t “ R ˚ t for the group of units of the residue ring R t “ Z { t Z modulo 2 t . Then U t “ t ´ . It is often be convenientto identify elements of R t with the corresponding elements of the leastresidue system modulo 2 t .We fix a matrix M “ ˆ m m m m ˙ P GL p R t q with(1.1) M ” ˆ ˙ or ˆ ˙ mod 2 . We then consider sequences generated by iterations of the
M¨obiustranformation (1.2) M : x ÞÑ m x ` m m x ` m which, under the condition (1.1), is always defined over U t , that is, M : U t Ñ U t . Mathematics Subject Classification.
Key words and phrases.
Inversive congruential pseudorandom numbers, primepowers, exponential sums, Vinogradov mean value theorem.
That is for u P R t we consider the trajectory(1.3) u n “ M p u n ´ q “ M n p u q , n “ , , . . . , generated by iterations of the M¨obius tranformation (1.2) associatedwith M .Assume that the characteristic polynomial of M has two distinct eigenvalues ϑ and ϑ from the algebraic closure Q of the field of 2-adic fractions Q .It is not difficult to prove by induction on n that there is an explicitrepresentation of the form(1.4) u n “ γ ϑ n ` γ ϑ n γ ϑ n ` γ ϑ n with some coefficients γ ij P Q , i, j “ , ϑ , ϑ P Z are2-adic integers, in which case, interpolating, we also have γ ij P Z , i, j “ , γ ” γ ” . Then, defining g P U t by the equation g “ ϑ { ϑ we have g P R t (recall that M is invertible in R ), thus the sequencegenerated by (1.3), the representation (1.4) has the form(1.5) u n “ ag n ´ b ` c with some coefficients a, b, c P R t . Furthermore, it is also easy to seethat b ” . Motivation.
The sequences (1.3) are interesting in their ownrights but they have also been used as a source of pseudorandom num-ber generation where this sequence is known as the inversive genera-tor , for example, see [4] for the period length and [10] for distributionalproperties.More precisely, let τ be the multiplicative order of g modulo 2 t . Then p u n q is a periodic sequence with period length τ , provided that a is odd.Niederreiter and Winterhof [10], extending the results of [9] from oddprime powers to powers of 2, obtained nontrivial results for segmentsof these sequences of length N satisfying(1.6) τ ě N ě p { ` η q t ISTRIBUTION OF INVERSIVE GENERATOR 3 for any fixed η ą t .Here using very different techniques we significantly reduce the range(1.6) and obtain results which are nontrivial for much shorter segments,namely, for(1.7) τ ě N ě ct { for some absolute constant c ą Our results.
Here we establish upper bounds for the exponentialsums S h p L, N q “ L ` N ´ ÿ n “ L e ` hu n { t ˘ , ď N ď τ, where, as usual, we denote e p z q “ exp p πiz q and, as before, τ is themultiplicative order of g modulo 2 t .Using the method of Korobov [8] together with the use of the Vino-gradov mean value theorem in the explicit form given by Ford [6], wecan estimate S h p L, N q for the values N in the range (1.7).Throughout the paper we always use the parameter(1.8) ρ “ log Nt which controls the size of N relative to the modulus 2 t on a logarithmicscale. Theorem 1.1.
Let gcd p g, q “ gcd p a, q “ and write g “ ` w β β , gcd p w β , q “ . Then for β ă N ď τ we have | S h p L, N q| ď cN ´ ηρ where ρ is given by (1.8) , for some absolute constants c, η ą uniformlyover all integers h with gcd p h, q “ . From a sequence p u n q defined by (1.5) we derive the inversive con-gruential pseudorandom numbers with modulus t : u L { t , u L ` { t , . . . , u L ` N ´ { t P r , q . The discrepancy D p L, N q of these numbers is defined by D p L, N q “ sup J Ăr , q ˇˇˇˇ A p J, N q N ´ | J | ˇˇˇˇ , L. M´ERAI AND I. E. SHPARLINSKI where the supremum is taken over all subintervals J of r , q , A p N, J q is the number of point u n { t in J for L ď n ă L ` N , and | J | isthe length of J . The Erd˝os–Tur´an inequality (see [5, Theorem 1.21])allows us to give an upper bound on the discrepancy D p L, N q in termsof S h p L, N q . Theorem 1.2.
Let p u n q be as in Theorem 1.1 and assume that β ă N ď τ . Then we have D p L, N q ď c N ´ η ρ where ρ is given by (1.8) , for some constants c , η ą . Writing N ´ ρ “ exp ˆ ´ p log N q t ˙ we see that Theorems 1.1 and 1.2 are nontrivial in the range (1.7).2. Preparation
Notation.
We recall that the notations U ! V , and V " U areequivalent to the statement that the inequality | U | ď cV holds withsome absolute constant c ą to the 2-adic valuation, that is, for non-zerointegers a P Z we let v p a q “ k if 2 k is the highest power of 2 whichdivides a , and v p a { b q “ v p a q ´ v p b q for a, b ‰ Multiplicative order of integers.
The following assertion de-scribes the order of elements modulo powers of 2.
Lemma 2.1.
Let g ‰ ˘ be an odd integer and write g “ ` w β β , gcd p w β , q “ . Then for s ě β the multiplicative order τ s of g modulo s is τ s “ s ´ β ` and (2.1) g τ s “ ` w s s , gcd p w s , q “ . Proof.
First we note that β ě
2. We prove (2.1) by induction of s .Clearly, we have (2.1) with s “ β . Furthermore, if (2.1) holds forsome s ě β , then by squaring it we get g τ s “ ` w s s ` ` w s s ` “ ` w s ` s ` , with w s ` “ ` w s s ´ ” s ` s . (cid:3) ISTRIBUTION OF INVERSIVE GENERATOR 5
Explicit form of the Vinogradov mean value theorem.
Let N k,n p M q be the number of integral solutions of the system of equations x j ` . . . ` x jk “ y j ` . . . ` y jk , j “ , . . . , n, ď x i , y i ď M, i “ , . . . , k. Our application of Lemma 2.3 below rests on a version of the Vino-gradov mean value theorem which gives a precise bound on N k,n p M q .We use its fully explicit version given by Ford [6, Theorem 3], whichwe present here in the following weakened and simplified form. Lemma 2.2.
For every integer n ě there exists an integer k Pr n , n s such that for any integer M ě we have N k,n p M q ď n n M k ´ . n . We note that the recent striking advances in the Vinogradov meanvalue theorem due to Bourgain, Demeter and Guth [3] and Wooley [11]are not suitable for our purposes here as they contain implicit constantsthat depend on k and n , while in our approach k and n grow togetherwith M .2.4. Double exponential sums with polynomials.
Our main toolto bound the exponential sum S h p L, N q is the following result of Ko-robov [8, Lemma 3]. Lemma 2.3.
Assume that ˇˇˇˇ α ℓ ´ a ℓ q ℓ ˇˇˇˇ ď q ℓ and gcd p a ℓ , q ℓ q “ , for some real α ℓ and integers a ℓ , q ℓ , ℓ “ , . . . , n . Then for the sum S “ M ÿ x,y “ e p α xy ` . . . ` α n x n y n q we have | S | k ď ` k log p Q q ˘ n { M k ´ k N k,n p M q n ź ℓ “ min " M ℓ , ? q ℓ ` M ℓ ? q ℓ * , where Q “ max t q ℓ : 1 ď ℓ ď n u . We also need the following simple result which allows us to reducesingle sums to double sums.
L. M´ERAI AND I. E. SHPARLINSKI
Lemma 2.4.
Let f : R Ñ R be an arbitrary function. Then for anyintegers M, N ě and a ě , we have ˇˇˇˇˇ N ´ ÿ x “ e p f p x qq ˇˇˇˇˇ ď M N ´ ÿ x “ ˇˇˇˇˇ M ÿ y,z “ e p f p x ` ayz qq ˇˇˇˇˇ ` aM . Proof.
Examining the non-overlapping parts of the sums below, we seethat for any positive integers y and z ˇˇˇˇˇ N ´ ÿ x “ e p f p x qq ´ N ´ ÿ x “ e p f p x ` ayz qq ˇˇˇˇˇ ď ayz. Hence ˇˇˇˇˇ M N ´ ÿ x “ e p f p x qq ´ M ÿ y,z “ N ´ ÿ x “ e p f p x ` ayz qq ˇˇˇˇˇ ď a M ÿ y,z “ yz ď aM . Changing the order of summation and using the triangle inequality, theresult follows. (cid:3)
Sums of binomial coefficients.
We need results of certain sumsof binomial coefficients. The first ones are immediate and we leave theproof for the reader.
Lemma 2.5.
Let n be a positive integer. Then(1) for any integer k ď n we have n ÿ i “ k ˆ ik ˙ “ ˆ n ` k ` ˙ ; (2) for any polynomial P p X q P Z r X s of degree deg P ă n we have n ÿ j “ p´ q j ˆ nj ˙ P p j q “ . Lemma 2.6.
For any n, k with k ď n we have ÿ ℓ ` ... ` ℓ k “ nℓ ,...,ℓ k ě n ! ℓ ! . . . ℓ k ! “ k ÿ i “ p´ q k ´ i ˆ ki ˙ i n . Proof. As ÿ ℓ ` ... ` ℓ k “ n n ! ℓ ! . . . ℓ k ! “ k n the result follows directly from the inclusion–exclusion principle. (cid:3) ISTRIBUTION OF INVERSIVE GENERATOR 7 Proofs of the main results
Proof of Theorem 1.1. As u n ` L “ ag n ` L ´ b ` c “ ag ´ L g n ´ bg ´ L ` c, we can assume, that L “ S h p , N q “ S h p N q . We can also assume, that a “ c “
0. Finally we assume, that N ě t { since otherwise the result is trivial, see (1.7).Define r “ t log 2log N “ ρ ´ log 2 , where ρ is given by (1.8). First assume, that r ě s “ Z t r ^ and κ “ R ts V ´ . Then s ą β, s ď N { , r ď κ ă s, if N is large enough. Indeed, s ě t r ´ “ log N ´ ě β ´ ą β and 2 s ď t r “ N { . Moreover, κ ě ts ´ ě r ´ ě r and κ ď ts ď p log N q p log 2 q s “ t r s ď s. Let τ s be the order of g modulo 2 s . As s ą β , g τ s “ ` w ¨ s with gcd p w, q “ x , we have11 ´ x ” ` x ` . . . ` x t ´ mod 2 t , L. M´ERAI AND I. E. SHPARLINSKI thus u n ¨ τ s ” ´ b ´ g n ¨ τ s ” ´ ´ p ´ b ` g nτ s q ” ´ t ´ ÿ ℓ “ p ´ b ` g n ¨ τ s q ℓ ” ´ t ´ ÿ ℓ “ p ´ b ` p ` w ¨ s q n q ℓ ” ´ t ´ ÿ ℓ “ ˜ ´ b ` n ÿ i “ ˆ ni ˙ p w ¨ s q i ¸ ℓ mod 2 t . Define F κ p n q “ κ ÿ ℓ “ p w ¨ s q ℓ t ´ ÿ j “ j ÿ ν “ ˆ jν ˙ p ´ b q j ´ ν ÿ i ` ... ` i ν “ ℓi ,...,i ν ě ˆ ni ˙ . . . ˆ ni ν ˙ . Then u n ¨ τ s ” ´ F κ p n q mod 2 t . The expression κ ! F κ p n q is a polynomial of 2 s n of degree at most κ .Thus we can define the integers a , . . . , a κ by κ ! F κ p n q “ κ ÿ ℓ “ a ℓ ℓs n ℓ . Then the coefficients satisfy a ℓ ” κ ! ℓ ! w ℓ t ´ ÿ j “ j ÿ ν “ ˆ jν ˙ p ´ b q j ´ ν ÿ i ` ... ` i ν “ ℓi ,...,i ν ě ℓ ! i ! . . . i ν ! mod 2 s . We have v p a ℓ q “ v p κ ! { ℓ ! q . Indeed, as w is odd and b is even, byLemmas 2.6 and 2.5 we get t ´ ÿ j “ j ÿ ν “ ˆ jν ˙ p ´ b q j ´ ν ÿ i ` ... ` i ν “ ℓi ,...,i ν ě ℓ ! i ! . . . i ν ! ” ℓ ÿ j “ ÿ i ` ... ` i j “ ℓi ,...,i j ě ℓ ! i ! . . . i j ! ” ℓ ÿ j “ j ÿ i “ p´ q j ´ i ˆ ji ˙ i ℓ ” ℓ ÿ i “ p´ q i i ℓ ℓ ÿ j “ i ˆ ji ˙ ” ℓ ÿ i “ p´ q i i ℓ ˆ ℓ ` i ` ˙ ” ´ ℓ ` ÿ i “ p´ q i ˆ ℓ ` i ˙ p i ´ q ℓ ” ˆ ℓ ` ˙ p´ q ℓ ” ISTRIBUTION OF INVERSIVE GENERATOR 9 (we note that the last several congruences are actually equations).Write ω ℓ “ v p a ℓ q . Then ω ℓ ď v p κ ! q ď Y κ ] ` Y κ ] ` . . . ă κ for ℓ ă κ and ω κ “ | S h p N q| ď s N ´ ÿ n “ ˇˇˇˇˇ s ÿ x,y “ e ˆ h t u n ` τ s xy ˙ˇˇˇˇˇ ` τ s s ď s N ´ ÿ n “ ˇˇˇˇˇ s ÿ x,y “ e ˆ h t ¨ g ´ n g τ s xy ´ bg ´ n ˙ˇˇˇˇˇ ` s ď s N ´ ÿ n “ ˇˇˇˇˇ s ÿ x,y “ e ˆ hg ´ n p a s xy ` . . . ` a κ κs p xy q κ q κ !2 t ˙ˇˇˇˇˇ ` N { , where the coefficients a ℓ “ a ℓ p bg ´ n q for ℓ “ , . . . , κ , are determined asabove with bg ´ n instead of b .Write hg ´ n a ℓ ℓs κ !2 t “ r ℓ q ℓ , gcd p r ℓ , q ℓ q “ , ℓ “ , . . . , κ, with(3.1) 2 t ´ ℓs ´ ω ℓ ď q ℓ ď κ !2 t ´ ℓs ´ ω ℓ ℓ “ , . . . , κ. Then(3.2) | S h p N q| ď s N ´ ÿ n “ ˇˇˇˇˇ s ÿ x,y “ e p f n p x, y qq ˇˇˇˇˇ ` N { , where f n p x, y q “ r q xy ` . . . ` r κ q κ p xy q κ . Put σ n “ s ÿ x,y “ e p f n p x, y qq . For κ , there exists a k P r κ , κ s such that for N k,κ we have thebound of Lemma 2.2 (with κ instead of n ). Then by Lemma 2.3 we have | σ n | k ď ` k log p Q q ˘ κ { p k ´ k q s N k,κ p s q κ ź ℓ “ min " ℓs , ? q ℓ ` ℓs ? q ℓ * , (3.3)where by (3.1) we have Q ď κ !2 t and thus(3.4) log p Q q ď log p κ !2 t q ď tκ log p κ q . By the choice of κ we have sκ ă t ď s p κ ` q . As ω ℓ ď κ ď s , under κ ` ď ℓ ă κ we have by (3.1) q ℓ ď κ !2 s p κ ` ´ ℓ q ď κ !2 ℓs and q ℓ ą s p κ ´ ´ ℓ q thus 1 ? q ℓ ` ? q ℓ ℓs ď ` κ ! ? q ℓ ď κ κ ´ s p κ ´ ´ ℓ q . Whence κ ź ℓ “ min " ℓs , ? q ℓ ` ℓs ? q ℓ * “ sκ p κ ` q{ κ ź ℓ “ min " , ? q ℓ ` ? q ℓ ℓs * ď sκ p κ ` q{ ź κ ă ℓ ă κ κ κ ´ s p κ ´ ´ ℓ q{ ď κ κ sκ p κ ` q{ ´ s p κ ´ qp κ ´ q{ . (3.5)By Lemma 2.2 we have(3.6) N k,κ p s q ď κ κ ks ´ . κ s . Combining (3.3), (3.4), (3.5) and (3.6), we have | σ n | k ď ` tk log p κ q ˘ κ { κ κ k s ` sκ p κ ` q{ ´ s p κ ´ qp κ ´ q{ ´ . κ s and therefore | σ n | ! t {p κ q s ´ s {p κ q . Since tκ ă p ts q s ă p r q s , then2 s { κ “ N rs {p tκ q ą N {p r q . Moreover t { κ ď N log t {p r log N q ď N log log N {p r log N q , whence | σ n | ! s N ´ ηρ , ISTRIBUTION OF INVERSIVE GENERATOR 11 for some η ą N is large enough. Thus by (3.2) we have | S h p N q| ď s N ´ ÿ n “ | σ n | ` N { ! N ´ ηρ ` N { ! N ´ η { r which gives the result for r ě r ă N “ X t { \ ρ “ log N t “ log 2129 ` O p { t q . As N ď τ ă t , we have(3.7) log N log N ą . Then | S h p N q| ď ÿ ď k ă N { N ˇˇˇˇˇ p k ` q N ´ ÿ n “ kN e p hu n { t q ˇˇˇˇˇ . Applying the previous argument to the inner sums, we get | S h p N q| ! NN N ´ ηρ ! N ´ ´ ηρ by (3.7). Thus replacing η to η { , we conclude the proof.3.2. Proof of Theorem 1.2.
By the Erd˝os-Tur´an inequality, see [5]for any integer H ě D p L, N q ! H ` N H ÿ h “ h | S h p L, N q| . Define H “ Z τ t ? N ^ , where τ t is as in Lemma 2.1.For a given 1 ď h ď H , write h “ d j with odd j and d ď log H .Then consider the sequence p u n q modulo 2 t ´ d . Then clearly S h p L, N q “ S d,j p L, N q . where S d,j p L, N q is defined as S j p L, N q , however with respect to themodulus 2 t ´ d .By the above choice of parameters, we have(3.9) t ´ d ě t ´ log H ě
12 log N ` β ą β by Lemma 2.1, thus(3.10) τ t ´ d “ t ´ d ´ β ` . Using (3.8), we have D p L, N q ! H ` N H ÿ h “ h | S h p L, N q|! H ` N ÿ ď d ď log H d ÿ ď j ď H { d j odd j | S d,j p L, N q| . (3.11)For fixed d and j put N d “ R Nτ t ´ d V and K d “ N ´ N d τ t ´ d . Then | S d,j p L, N q| ď N d ´ ÿ i “ | S d,j p L ` iτ t ´ d , τ t ´ d q|` | S d,j p L ` p N d ´ q τ t ´ d , K d q| . (3.12)If K d ă β , we use the trivial estimate | S d,j p L ` p N d ´ q τ t ´ d , K d q| ď K d ă β . As 8 β ă p t ´ d ´ β q by (3.9), we get(3.13) | S d,j p L ` p N d ´ q τ t ´ d , K d q| ď τ ´ η p t ´ d q ´ p log τ t ´ d q t ´ d . If K d ě β , then as K d ď τ t ´ d we also have (3.13) by Theorem 1.1.Thus by (3.12) we have | S d,j p L, N q| ! N d ¨ τ ´ η p t ´ d q ´ p log τ t ´ d q t ´ d ! N ´ η p t ´ d q ´ p log τ t ´ d q { log N . By (3.9) and (3.10) we have p log τ t ´ d q log N p t ´ d q “ p t ´ d ´ β q log N p t ´ d q ě p t ´ d ´ β q log N t ě ˆ log Nt ˙ “ ρ { , whence | S d,j p L, N q| ! N ´ ηρ { . Then by (3.11), D p L, N q ! H ` ÿ ď d ď log H d ÿ ď j ď H { d j odd j N ´ ηρ { ! ´p t ´ β q{ ` N ´ ηρ { log H ! t ` N ´ ηρ { log H ! N ´ ηρ { ISTRIBUTION OF INVERSIVE GENERATOR 13 if N is large enough. 4. Comments
We note that an extension of our results to the case of sequences (1.5)modulo prime powers p t with a prime p ě s -dimensional vectors p u n , . . . , u n ` s ´ q , n “ , . . . , N. Our method is capable of addressing this problem, however investigat-ing the 2-divisibility of the corresponding polynomial coefficients whichis an important part of our argument in Section 3.1 is more difficultand may require new arguments.We also use this as an opportunity to pose a question about study-ing short segments of the inversive generator modulo a large prime p .While results of Bourgain [1, 2] give a non-trivial bound on exponentialsums for very short segments of sequence ag n mod p , n “ , . . . , N , seealso [7, Corollary 4.2], their analogues for even the simplest rationalexpressions like 1 {p g n ´ b q mod p are not known. Obtaining such re-sults beyond the standard range N ě p { ` ε (with any fixed ε ą
0) isapparently a difficult question requiring new ideas.
Acknowledgement
During the preparation of this wok L. M. was partially supportedby the Austrian Science Fund FWF Projects P30405 and I. S. by theAustralian Research Council Grants DP170100786 and DP180100201.
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E-mail address : [email protected] I.E.S.: School of Mathematics and Statistics, University of NewSouth Wales. Sydney, NSW 2052, Australia
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