CC BY
9
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 1.
УДК 510 DOI 10.22405/2226-8383-2018-20-1-224-247
О множествах ограниченных остатков для (t, .^-последовательностей I
Мордехай Б. Левин
Левин Мордехай Борисович — кандидат физико-математических наук, Факультет математики, Университет Бар-Илан, Рамат-Ган, Израиль. e-mail: mlevin@math. Ыи.ас.И
Аннотация
Пусть (x„)„>o — s—мерная последовательность типа Холтона, полученная из глобального функционального поля, b > 2, 7 = (71, ...,7S), 7i € [0,1) с Ь-адическим разложением Ъ = + 7i,2&-2 + i = 1,..., s.
В этой статье мы докажем, что [0, 71) х ... х [0, 7S) — множество ограниченного остатка относительно последовательности (x„)„>0 тогда и только тогда, когда
max maxjj > 1 | =0} < то.
Мы также получим аналогичные результаты для обобщенных последовательностей Ни-деррайтера, последовательностей Хинга — Нидеррайтера и последовательностей Нидер-райтера — Хинга.
Ключевые слова: множества ограниченных остатков, (t, в)-последовательности, последовательности Холтона .
Библиография: 15 названий. Для цитирования:
Мордехай Б. Левин О множествах ограниченных остатков для (t, з)-последовательностей I // Чебышевский сборник, 2019, т. 20, вып. 1, с. 224-247.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 1.
UDC 510 DOI 10.22405/2226-8383-2018-20-1-224-247
On a bounded remainder set for (t, s) sequences I
Mordechav B. Levin
Levin Mordechay Borisovich — candidat of physical and mathematical Sciences, Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel. e-mail: mlevin@math. hiu.ac.il
Abstract
Let x0, xi,... be a sequence of points in [0,1)s. A subset S of [0,1)s is called a bounded remainder set if there exist two real numbers a Mid C such that, for every integer N,
|card{n < N | x„ € S} — aN| <C.
Let (x„)„>^ an s—dimensional Halton-type sequence obtained from a global function field, b > 2, 7 = (j1,...,js), 7j € [0,1), with 5-adic expansion 7i = 7i,1&-1 + + ...,
i = 1,..., s. In this paper, we prove that [0,71) x ... x [0,7S) is the bounded remainder set with respect to the sequence (x„)„>0 if and only if
max max{j > 1 | 7^ =0} < to.
We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.
Keywords: bounded remainder set, (t, s) sequence, Halton type sequences.
Bibliography: 15 titles.
For citation:
Mordechay B. Levin, 2019, "On a bounded remainder set for (t,s) sequences I", Chebyshevskii sbornik, vol. 20, no. 1, pp. 224-247.
Dedicated to the 100th anniversary of Professor N.M. Korobov
1. Introduction
1.1. Bounded remainder sets. Let xo, xi,... be a sequence of points in [0,1)s, S ç [0,1)s,
N-1
A(S, (xra)^=-o1)= £ (1S(x,) - A(5)),
ra=0
where (x) = 1, if x G 5, and (x) = 0, if x f Here X(S) denotes the s-dimensional Lebesgue-measure of S. We define the star discrepancy of an N-point set (x^)^—1 as
^((xn)^-1) = supo<yi,...,„,<! |A([0, y), (x^-D/N|,
where [0, y) = [0, y1) x ■ ■ ■ x [0, The sequence (xra)ra>0 is said to be uniformly distributed in [0,1)s
if Dn ^ 0. In 1954, Roth proved that limsupwN(lnN)-2D*((xra)^="01) > 0. According to the well-known conjecture (see, e.g., fl, p.283]), this estimate can be improved to
limsup^ N (ln N)-^*((xra)^=o1) > 0. (1)
See [2] and [7] for results on this conjecture.
A sequence (xis))ra>0 is of low discrepancy (abbreviated l.d.s.) if D^x^)^-,1) = 0(N-1(ln N)s) for N ^ œ. A sequence of point sets ((x^w)^="01)^'=1 is of low discrepancy (abbreviated l.d.p.s.)
if ^((x^y)^="01) = 0(N-1(lnN)s-1), for N ^ œ. For examples of such a sequences, see, e.g., fl], [3], and'fll].
Definition 1. Let x0, x1,... be a sequence of points in [0,1)s. A subs et S of [0,1)s is called a bounded remainder set for (xn)n>0 if the discrepancy fu,notion A(5, (x^)^—1) is bounded in N.
Let a be an irrational number, let I be an interval in [0,1) with length |/1, let jna} be the fractional part of na, n = 1, 2,... . Hecke, Ostrowski and Kesten proved that A(5, ({na})^=1) is bounded if and only if |/1 = {fco;} for some integer k (see references in [4]).
The sets of bounded remainder for the classical s-dimensional Kronecker sequence studied bv Lev and Grepstad [4]. The case of Halton's sequence was studied by Hellekalek [5]. Let & be a prime power, 7 = (71, ...,^s), li S (0,1) wit h 6-adic expansion
li = li,ib-1 + %2b-2 + ..., i = 1,..., s,
and let (x„)ra>o be a uniformly distributed digital Kronecker sequence. In [7], we proved the following theorem:
Theorem A. The set [0,71) x ... x [0,7^) is of bounded remainder with respect to (xn)n>0 if and only if
max maxjj > 1 | 7i,j = 0} < œ. (2)
In this paper, we prove similar results for digital sequences described in [3, Sec. 8]. Note that according to Larcher's conjecture [6, p.215], the assertion of Theorem A is true for all digital (t, s)-sequences in base b.
2. Definitions and auxiliary results.
2.1 (T, s) sequences. A subinterval E of [0,1)s of the form
E = nM-di, (Oi + 1)b-di),
oib-di 1
i=1
with ai, di G Z, di ^ 0, 0 ^ ai <bdi,ioi 1 ^ i ^ s is called an elementary interval in base b > 2.
Definition 2. Let 0 ^ t ^ m be integers. A (i, m, s)-net in ba se b is a point set x0,..., xbm-i in [0,1)s such that #{n G [0,bm — 1\lxn G E} = bl for every elementary interval E in base b with vol(^) = bt-m.
Definition 3. Let t > 0 be an integer. A sequence x0, xi,... of points in [0,1)s is a (t, s)-sequence in base b if for all integers k ^ 0 and m > t, the point set consisting ofxnmth kbm < n < (k + 1)bm is a (t,m, s)-net in ba se b.
Bv fNi, p. 56,60], (t, m, s) nets and (t, s) sequences are of low discrepancy. See reviews on (t, m, s) nets and (t,s) sequences in [3] and [11].
Definition 4. ([3, Definition 4.30]) For a given dimension s > 1, an integer base b > 2; and a function T : No ^ N0 with T(m) < m for all m G N0, a sequence (x0, x1,...) of points in [0,1)s is called a (T, s)-sequence in base b if for all integers m > 0 and k > 0; the point set consisting of the points Xkbm, ...,Xkbm+bm-1 forms a (T(m),m,s)-net in base b.
Definition 5. ([3, Definition 4.47]) Let m,s > 1 be integers. Let C(1'm),...,C(s'm) be m x m matrices over F&. Now we construct bm points in [0,1)s. For n = 0,1,...,bm — 1; let n = Y=o a3(n)b^ be the b-adic expansion of n. Choose a bijection : Z5 := {0,1,...., b — 1} ^ F5 with 0(0) = 0; the neutral element of addition in F&. We identify n with the row vector
n = (ao(n),..., am-1 (n)) G F™ with ar(n) = <fi(ar(n)), r G [0, m). (3)
We map the vectors
<x
= (»£,..., y^m) := nCyg = £ Tir(n)$ S ¥b, (4)
r=0
to the real numbers
to obtain the point
№ = £ , = ) (5)
i=i
Ma^,...,^) G [0,1Г. (6)
The point set (xo,...,x™^} is called a digital net (over ¥/,) (with generating matrices
(£(1>m),..., с(s>m))).
For m = we obtain a sequence x0, x1,... of points in [0,1)s which is called a digital sequence (over ¥b) (with generating m atrices (С (1>те),...,С
We abbreviate С(г'т) as С « for m G N and for m = oo.
2.2 Duality theory (see [3, Section 7]).
Let N be ад arbitrarv F^-linear subspace of F^m. Let H be a matrix over F& consisting of sm columns such that the row-space of H is equal to N Then we define the dual space С of N to be the null space of H (see [3, p. 244]). In other words, is the orthogonal complement of N relative to the standard inner product in F^m,
^ = (A G F£m | В ■ A = 0 for all В G^}.
Let С(1),...,C(s) G be generating matrices of a digital sequence (xn(C))n>0 over F&.
For any m G N we denote the m x m left-upper sub-matrix of С(i) bv [С(t)]m. The matrices [C(1)]m,..., [C(s)]m are then the generating matrices of a digital net. We define the overall generating matrix of this digital net by
[CV = (Р(1)Ш[С(2)]Т|...|[C(e)]^) G F™X*m (7)
for any m G N.
Let Cm denote the row фасе of the matrix [C]m i.e.,
m— 1
cm = {( T $ ar(n)) | 0 < n<6m).
I V ' J'' /1< j<m,1<i<s )
1<j<m,1<i<s
r=0
The dual space is then given by
Ci = (A G | B • = 0 for all B G Cm}.
Lemma A. ([3, Theorem 4.86]) Let b be a prime power. A striet digital (T, s)-sequence over F^ is uniformly distributed modulo one, if and only î/liminf— T(m)) =
2.3 Admissible sequences.
For x xjb--*, and y ^J2j>1 Vjwhere Xj, yj G Z^ := (0,1,...., b
— 1}, we define the (^adic) digital shifted point v by v = x © y := ^j>1 vjwhere vj = xj + yj (mod 6) and Vj G Z6. Let x = (^ 1),...,^s) ) G [0,1)s, y = (y(1),..., y(s)) G [0,1)s. We define the (^adic) digital shifted point vbvv = x © y = (^1) © y(1),...,#(© y(s)). For n1, G [0, bm), we défi ne n1 © n2 : = (<V&m © U2)bm)bm.
For x ^J2j>1 xib-\ where Xi G Z&, Xi = 0 (f = 1, Xk+1 = 0, we define the absolute
valuation ||.||6 of £ by \\x\\b = 6-fc-1. Let ||n||6 = bk for n G [6fc,6fc+1).
Definition 6. point set (xn)0<n<i)m in [0,1)s ¿s d—admissible in base b if
s
min ||xra e xfc ||6 >b-m-d where ||x||6 := ftL?
0<k<n<bm -"--Ml J
i=1
A sequence (xn)n>0 m [0,1)s ¿s d—admissible in base b if infn>k>0 \\n © k\\b x \\xn © xk||fe > b d.
By [8], generalized Niederreiter's sequences, Xing-Niederreiter's sequences and Halton-tvpe (t, s) sequences have d—admissible properties. In [8], we proved for all d—admissible digital (t, s) sequences
(x,n)n>0
max ND*((xn © w)o<n<N) > Kms l—N <bm —
with some w and K > 0. This result supports conjecture (1).
Definition 7. A sequence (xn)n>0 in [0,1)s is weakly admissible in base b if
s
Km := min \\xn © xk\\. > 0 Vm > 1 where \\x\L := TT .
0—k<n<bm 0 0 J-J- II b
i=l
Let m > 1, Tm = [logg(nm)] + m, w = (w(l), ...,w(s)), w(l) = (w(t), ...,Wt!),
9w = (A > 1 | 4™A>j = wf, j e [1,Tm], i e [1, s]} and 9w = 0 V wf e Zb. (8)
Theorem B. (see [9, Proposition]) Let (xn)n>0 be a uniformly distributed weakly admissible digital (T, s)-sequence in base b, satisfying (8) for all m > m®. Then the set, [0,71) x ... x [0,7s) is
(xn)n>0
2.4 Notation and terminology for algebraic function fields. For the theory of algebraic function fields, we follow the notation and terminology in the books [14] and [13].
Let b be an arbitrary prime power, ¥b a finite field with & elements, F^(^) the rational function field over Fb, and F^] the polynomial ring over F&. For a = f/g, f,g e F^], let
v^(a) = deg(g) — deg(/)
be the degree valuation of F^(^). We define the field of Laurent series as
ro
F&((^)) := { ^ aixi | m e Z, ai e F6}.
i=m
A finite extension field F of F^(^) is called an algebraic function field over F^. Let F^ be algebraically closed in F. We express this fact by simply saving that F/Fb is an algebraic function field. The genus of F/Fb is denoted by g.
A place V of F is, by definition, the maximal ideal of some valuation ring of F. We denote by Op the valuation ring corresponding to V and we denote by Pf the set of places of F. For a place V of F, we write ^-p for the normalized discrete valuation of F corresponding to V, and any element t e F with up (t) = 1 is called a local parameter (prime element) at V.
The field Fp := Op /V is called the residue field of F with respect to V. The degree of a place V is defined as deg(P) = [Fp : F&]. We denote by Div(F) the set of divisors of F/Fb.
The completion of F with respect to up will be denoted by F(v\ Le11 be a local parameter of V. Then F(v) is isomorphic to Fp((t)) (see [13, Theorem 2.5.20]), and an arbitrarv element a e F(p) can be uniquely expanded as (see [13, p. 293])
ro
a = ^ Siti where Si = Si(t, a) e Fp C F(p).
i=v-p (a)
The derivative ^j, or differentiation with respect to t, is defined by (see [13, Definition 9.3.1])
, ro
di = e 1. (9)
i=v-p (a)
For an algebraic function field F/Fb, we define its set of differentials (or Hasse differentials, H-differentials) as
Ap = (y dz | y G F, 2 is a separating element for F/Fj,}
(see [14, Definition 4.1.7]).
Lemma B. ([14, Proposition 4.1.8] or [13, Theorem 9.3.13]) Let z G F be separating. Then every differential 7 G A^ can be written uniquely as 7 = y dz for some y G F. We define the order of a d^ at V by
vv(a d£) := vv(a d£/di), (10)
where i is any local parameter for V (see [13, Definition 9.3.8]).
Let Qp be the set of all Weil differentials of F/F&. There exists an F—linear isomorphism of the differential module A^ onto Qf (see [14, Theorem 4.3.2] or [13, Theorem 9.3.15]).
For 0 = w G Qf, there exists a uniquely determined divisor div(w) G Div(F). Such a divisor div(w) is called a canonical divisor of F/F&. (see [14, Definition 1.5.11]). For a canonical divisor W, we have (see [14, Corollary 1.5.16])
deg(VF) = — 2 and l(T^) = (11)
Let a d^ be a nonzero H-differential in F and let w be the corresponding Weil differential. Then (see [13, Theorem 9.3.17], [14, ref. 4.35])
vv (div(w)) = vv (a d£), for all Pg Pf . (12)
Let a d^ be an H-differential, t a local parameter of V, and
a d£ = ^ Sitidt G F(p).
i=v-p (a)
Then the residue of a d^ (see [13, Definition 9.3.10) is defined by
Resp(a d£) := TrFv/Fb(5-i) G Fb.
Let
Res-p,t (a) := Res-p (adi). For a divisor V of F/F&, let denote the Riemann-Roch space
£(£>) = (y G F \ 0 | div(y) + V > 0} U (0}. (13)
Then £(P) is a finite-dimensional vector space over F, and we denote its dimension by l(^). By [14, Corollary 1.4.12],
l(£>) = (0} for deg(P) < 0. (14)
Theorem C (Riemann-Roch Theorem). [14, Theorem 1.5.15, and 14, Theorem 1.5.17 ] Let W
be a canonical divisor of F/F5. Then for each divisor A G div(F), 1(A) = deg(A) + 1 — g+— A); and
1(A) = deg(A) + 1 — g for deg(A) > — 1.
3. Statements of results.
3.1 Generalized Niederreiter sequence. In this subsection, we introduce a generalization of the Niederreiter sequence due to Tezuka (see [3, Section 8.1.2]). By [3, Section 8.1], the Sobol's sequence, the Faure's sequence and the original Niederreiter sequence are particular cases of a generalized Niederreiter sequence.
Let b be a prime power and let pl,...,ps e F^] be pairwise coprime polynomials over F^. Let ei = deg(^i) > 1 for 1 < i < s. For each j > ^d 1 < i < s, the set of polynomials (Vi,j,k(x) : 0 < k < ei} needs to be linearly independent (mod Pi(x)) over F^. For integers 1 < i < s, j > 1 and 0 < k < ei, consider the expansions
over the field of formal Laurent series
F6((c-1)). Then we define the matrix C^ = )j>l,r>0 by
cfr = a(i)(Q + 1,k,r) e Fb for 1 < i < s, j > 1, r > 0,
where j — 1 = Qei + k ^^^^^h mtegers Q = Q(i,j^d k = k(i,j) satisfying 0 <k<ei.
A digital sequence (xn)n>0 over F^ generated by the matrices C(l),..., C(s is called a generalized Niederreiter sequence (see [3, p.266]).
Theorem D. (see [3, p.266] and [7, Theorem 1]) The generalized Niederreiter sequence (xn)n>0 with generating matrices, defined as above, is a digital d—admissible (t, s)-sequ,ence over F^ with d = e0, t = e0 — s and e0 = el + ... + es.
In this paper, we will consider the case where (x,pi) = 1 for 1 < i < s. We will consider the general case in [10].
Theorem 1. With the notations as above, the set [0,7l) x ... x [0,7s) is of bounded remainder ( x n ) n> 0
3.2 Xing-Niederreiter sequence (see [3, Section 8.4 ]). Let F/Fb be an algebraic function field with full constant field F^ and genus g. Assume that F/Fb has at least one rational place Pro, and let G be a positive divisor of F/Fb with deg(G) = 2g and Pro e supp(G). Let Pl, ...,Ps be s distinct places of F/Fb with Pi = Pro for 1 <i<s. Put ei = deg(P^) for 1 <i<s.
By [3, p.279 ], we have that there exists a basis w0, wl, ...,wg of £(G) over F^ such that
vp^ (wu) =nu for 0 <u < g,
where 0 = n0 <nl < .... < ng < 2g. For each 1 < i < s, we consider the chain
£(G) C C(G + Pi) C C(G + 2Pi) c ...
of vector spaces over F^. ^v ^^^^^^^^rom the basis w0,wl, ...,wg of C(G) and successively adding basis vectors at each step of the chain, we obtain for each n e N a basis
(w0,WI, ...,Wg,kf , ...,k!n^i}
of C(G + nPi). We note that we then have
kf e C(G + ([(j — 1)/ei + 1)]Pi) for 1 <i<s and j> 1. (15)
Lemma C. ([3, Lemma 8.10]) The system (w0,wl,...,wg} U (kji"l}l——s,j>l of elements of F is linearly independent over F&.
Let z be an arbitrary local parameter at Pœ. For r G No = N U (0}, we put
(16)
zr if r G («0, «1, ..., %},
\wu if r = nu for some u G (0,1,...,$}.
Note that in this case up^ (zr) = r for all r G N0. For 1 < i < s and j G N, we have to -' G £(G+nPi)
for some n expansions
for some n G N and also P^ G supp(G + nPi), hence (fc(î)) > 0. Thus we have the local
= £ aSfor 1 < i < s and j G N, (17)
r=0
where all coefficients a^ G F^. Let ^ = N0 \ = (ft(0), ft(1),...},
= (no,ni, ...,%}.
For 1 < i < ^d j G N, we now define the sequences
= 41(r), = (cS>, , ...) := (flj>)neNa\{na,...,n9} = (41(r))r>0 (18)
= (n(i) n(i) n(i) n(i) n(i) n(i) ) <= FN
where the hat indicates that the corresponding term is deleted. We define the matrices C(1),..., C(s) G fNxN by
C« = (c(;),c^,4°,...)T for 1 < i < s, (19)
1.e., the vector c^- is the jth row vector of C(l) for 1 < i < s.
Theorem E (see [3, Theorem 8.11] and [7, Theorem 1]). With the above notations, we have that the matrices C(1), ..., C(s) given by (19) are generating matrices of the Xing-Niederreiter d—admissible digital (i, s)-sequence (xn)n>0 with d = e1 + ... + es, t = g + e1 + ... + es — s.
In order to obtain the bounded remainder set property, we will take a specific local parameter
2. Let Po G PF, Po £ (A,... , Ps, Px>}, Po G supp(G) and deg(Po) = eo. By the Riemann-Roch theorem, there exists a local parameter z at P^, with
* G £((25 + 1)Po — Px>) \ £((2$ + 1)Po — 2P^). (20)
Theorem 2. With the notations as above, the set [0,71) x ... x [0,7s) is of bounded remainder with respect to (xra)ra>o if and only if (2) is true.
3.3 Generalized Halton-type sequences from global function fields.
Let q > 2 be an integer
n = ^2 eq,j (n)^-1, eq,j (n) G (0,1,...,<? — 1}, and ipq (n) = ^ eqJ (n)g-J. i>1 i>1
Van der Corput proved that (<pq(n))n>o is a 1—dimensional l.d.s. Let
Hs (n) = (n),...,^s fa^ n = 0, 1, 2, ...,
where q1,...,qs > 2 are pairwise coprime integers. Halton proved that (iis(n))ra>o is an s—dimensional l.d.s. (see [11]).
Let Q = (qi, q2,....) and Qj = qiq2....qj, where qj ^ 2 (j = 1,2,...) is a sequence of integers. Every nonnegative integer n then has a unique Q-adic representation of the form
<x
n = J>jqi ■ ■ ■ qj-i = ni +n2q\ + n3qiq2 +----,
j=i
where Uj G {0,1,..., qj — 1}. We call this the Cantor expansion of n with respect to the base Q. Consider Cantor's expansion of x G [0,1) :
^ Xj/Qj, Xj G {0,1,..., qj — 1}, Xj = qj — 1 for infinitely many j. Q— x
Vq{ £nj h ••• 1j-i) = £ U
j=1 - ' j-i^ ■ V
Let pi,j > 2 be integers (s > i > 1,j > 1) g.c.d.(pi,k,pj,i) = 1 for f = j, Pi,0 = 1, ^¡J = Ui<k<3 Pi,k, i G [1, s], j > 1 V% = fri,i,Pi,2,...), P = (Vi,.., Vs).
In [5], Hellecaleq proposed the following generalisation of the Halton sequence:
Hv = (№ (n),...,VPs (n))~=o. (21)
In [Te], Tezuka introduced a polynomial arithmetic analogue of the Halton sequence : Let p(x) be an arbitrary nonconstant polvnomial over F&, e = deg(p),
n = a0(n) + ai(n)b + ■ ■ ■ + am(n) bm.
We fix a bijection 0 : Z ^ F^ with 0(0) = 0. Denote vn(x) = a0(n) + ai(n)x + ■ ■ ■ + am(n)xm, where ar(n) = 0(ar(n)), r = 0,1,...,m. Then fn(x) can be represented in terms of p(x) in the following way:
vn(x) = r0(x) + ri(x)p(x) + ••• + rk (p(x))k, with k = [m/e]. We define the radical inverse function (pp(x) : F^[x] ^ F&(x) as follows
<fP(x)(vn(x)) = ro(x)/p (x) + n(x)/p2(x) +-----+ rk/(p(x))k+i.
Let pi(x),..., ps(x) be pairwise coprime. Then Tezuka's sequence is defined as follows
xn = (^i(^pi(x)(n)),...,^s(^Ps(x)(n)))n=0,
where each ai is a mapping from F to the real field defined by <Ji(J2j>w ajx-j) = J2j>w 0-i(aj)b-j■ By [Te], (xn)n>0 is a (t, s) sequence in base b.
In 2010, Levin [7] and in 2013, Niederreiter and Yeo [12] generalized Tezuka's construction to
F
Let F/Ffc be an algebraic function field with full constant field F& and genus g. We assume that F/Fb has at least one rational place, that is, a place of degree 1. Given a dimension s > 1, we choose s + 1 distinct places P^ of F with deg(P^) = 1. The degrees of the places Pi,...,Ps
are arbitrary and we put ei, = deg(Pi) for 1 < i < s. Denote by Of the holomorphv ring given by Of = f]P=Pco Op, where the intersection is extended over all places P = P^ of F, and Op is the valuation ring of P. We arrange the elements of Of into a sequence by using the fact that 0F = Um>0 £(mP^). The terms of this sequence are denoted by f0, fi,... and they are obtained as
follows. Consider the chain £(0) C F(Fro) C F(2Fro) C ■ ■ ■ of vector spaces over F^. At each step of this chain, the dimension either remains the same or increases bv 1. From a certain point on, the dimension always increases by 1 according to the Riemann-Roch theorem. Thus we can construct a sequence t>o,^i,... of elements of Of such that ..., vl(TOps+1)-1} is a F^-basis of £(mPs+i).
We fix a bijection 0 : Zb ^ F^ with 0(0) = 0. Then we define
ro ro
fn = ^ 0r(n)vr G Of with 0r(n) = 0(ar(n)) for n = ^ ar(n)6r .
r=0 r=0
Note that the sum above is finite since for each n G N. We have ar(n) = 0 for all sufficiently large r. Bv the Riemann-Roch theorem, we have
{/ | / e£((rn + g — 1)Fs+i)} = |/n | n G [0,6m)} for m > g.
For each i = 1,..., s, let pi be the maximal ideal of Of corresponding to Pi. Then the residue class field Fpi := Of/pi has order bei (see [14, Proposition 3.2.9]). We fix a bijection aFi : Fpi ^ Z^. For each i = 1,..., s, we can obtain a local parameter ti G Of at pi, bv applying the Riemann-Roch theorem and choosing ti G £(fcFro — P%) \ £(fcFro — 2 Pi) to a suitably large integer fc. We have a local expansion of fn at pi of the form
fn = E ¿Swith all G Fp, n = 0,1,... .
j>0
We define the map £ : Of ^ [0,1]s bv
ro ro
e(/n) = ( E ^pi (/ni])(^ei )-J-1,..., E ^ (/Sx^ r^).
Now we define the sequence x0, xi,... of points in [0,1]s by xn = £(/n) for n = 0,1,... . From [12, Theorem 1], we get the following theorem :
Theorem F. With the notation as above, we have that (xn)n>0 is a (i, s)-sequence overFb with t = g + ei + ... + es — s.
The construction of Levin [7] is similar, but more complicated than in [12]. However in [7], we can use arbitrary pairwise coprime divisors ,..., Ds instead of pi aces Pi,...,Ps.
In this paper, we introduce the Hellecalek-like generalisation (21) of the above construction:
Let Pf := {F|F be a place of F/Fb}, P0, Fro G Pf, deg(P^) = 1 deg(fb) = e^ P0 = Pij G Pf for 1 < j, 1 < i < s, Ph ,jl = Pi2,j2 ^ ^i = ¿2) Pi,j = ^0, Pi,j = ^TO for all nitj = deg(Fi;j), m,j = deg(Fj,j), V0,j = Fj,
Vi,0 = 1, Vi,j = Pi,k, = deg(Fi,j) = nij-i + nitj, «¿,0 = 0. (22)
i<fc<j
Let i G [0, s]. We will construct a basis (wjl))j>0 of Of in the following way. Let
Litj = £((mtj + 2^ — 1)Fro) = C(Aij), = (mtj + 20 — 1)Fro, (23)
Lij = £((«»,,■ + 20 — 1)Fro — Vij) = £(Bi,j), Fi,j = Kj + 20 — — Vid, Li,j = £(Kj +20 — 1)Fro — Vij-i), Bi,j = (mj +20 — 1)Fro — Vijj-i.
Using the Riemann-Roch theorem, we obtain
deg(Ai,j) = ni,j + 2g — 1, dim(Lij) = ni,j + g, deg(Bi,j) = 2g — 1, (24)
dim(Li ,j) = g, deg(Bij) = fii,j + 2g — 1, dim(L j) = fii,j + g.
Let (ujJ)J=;l be a F& linear basis of Lij. By (23) and (24), we get that the basis (uj^)9=i can be
extended to a basis (v ji, ••• , vj,] ., uj\, ••• , uj^) of L ,j
Bearing in j G [1, uUi,j]. So
/) ... v(i) ui) ••• ui)'
Bearing in mind that (ujJ)J=i is a F& linear basis of L,j, we obtain that vG L,j for
vf?9 G L,j := Lj \ Lfor j G [1,Ui,j]. (25)
Let
Vhj := {^£9 | 1 < j < nhk, 1 < k < j} U {u« | j = 1,..., 0}. (26)
We claim that vectors from Vij are F^ linear independent. Suppose the opposite. Assume that there (i)
exists byk J G F5 such that
j n i,k g
u + u = ^ where u = ^ w, w = ^ bfy^j, u = ^ ^ji. (27)
k=i 9=i 9=i
Let Wi = 0 for some I G [1, j] and let wk = 0 for all k G [1, Z). Using (23) - (25), we get
wi G Li,j = C((Ui,i + 2g — 1)P^ — Vi,i-i) \ C((Ui,i + 2g — 1)P^ — Viti). Applying definition (13) of the Riemann-Roch space, we obtain
wi G £((ui,j + 2g — 1)PX — Viti-i) \ £((m,j + 2g — 1)PX — Viti). But from (27), (22) and (25), we have
i j
—wi = u + u — ^2,Wk = ^ Wk + u G C((ni,j +2g — 1)P^ — P^). k=i k=Z+i
We have a contradiction. Hence vectors from Vij are F^ linear independent. By (23) - (26), we have Vj,,j C Lij and
card( Vi,j ) = i,k + g = + 5 = dim(Lj,j ). k=l
Hence vectors from Vi,j are the F^ linear basis of Li,j.
Now we will find a basis of Li opposite. By (23) and (24), we get
Now we will find a basis of Lij-2g. We claim that ujJ G Lij-2g for j G [1, g]. Suppose the
^ G Lij-2g n Lij = C((nitj-2g + 2g - 1)Pœ) n C((mtj + 2g - 1)PTO - Vitj) = C((mj-2g + 2g - 1)P^ - Pi,,-) = C(T).
By (22), deg(T) = Ui,j-2g + 2g -1 -Uij < 0. Hence £(T) = {0}. We have a contradiction. Bearing in mind that Vy is F& linear basis of Lij, we obtain that a basis of Lij-2g can be chosen from the
set (^i,...,^n.i,...,vj'j,...,vyli]) = Vitj \ {uj^ | p = 1, ...,#}. From (23) - (25), we get
v^ G Li,k C Lj,j-2fl for p G [1, ni,k] and 1 < fc < j — 2$.
Hence vectors
(i) (i) (i) (i) _(i) _(i) _(i) (i)
1 < P < for some p G [1, nj,k] and fc G (j — 2g, j] are an F& linear basis of Li,j-2g (0 < i < s).
Therefore (^k^)^^ fc k>1 is the F^ linear basis of Of = Uj>iLj,j. We put in order the basis (4^)i<M<ni , fc,k>i as follows
w.
(i\3-1+^-i = , with m,0 = 0, 1 < p < nitj, 0 < i < s. (28)
So we proved the following lemma :
Lemma 1. For all i G [0, s] there exists a sequence (wj^ )j>0 such that (wjl))j>0 is a F^ linear basis of Of and for all j > 1a F^ linear basis of Li,j can be chosen from, the set {w0i), ...,^^¿+29^}'
Bearing in mind that (wjl))j>0 is the F5 linear basis of Of, we obtain for all i G [1, s^d r > 0
(j) (j)
that there exists cj 'r G F5 and integers lr such that
¡(i)
wr] = E $ ^j-)i, cj0j)-i = 1, and cj0) = 0 for j — 1 = r. (29)
j=i
Let n = Y^r>0 af (^)&r- We fa a bijection 0 : Z5 ^ F5 with 0(0) = 0. Then we define
ro
/n = E 0r(n)^r0) G Of with 0r(n) = 0(ar(n)) for n = 0,1,... . (30)
r=0
By (29), we have for i G [0, s]
00 00
fn = E °r(n) E cS ^j-i = E wj-i E °r (n)4> = E ^n?j wj--i (31)
r=0 j=i j=i r=0 j=i
where ¿j = Er>0 0rMcjJ G F&, ^j = dj-iM-We map the vectors
ytt = (yin),i,y%,--) (32)
to the real numbers
= E ^(¿S w j>i
to obtain the point
xn :=(^ni),...,^ns)) G [0,1)s. (33)
Theorem 3. With the notations as above, the set [0,7i) x ... x [0,7s) is of bounded remainder ( x n ) n> 0
Remark. It is easy to verify that Hellekalek's sequence and our generalized Halton-tvpe sequence (xn)n>0 are l.d.s if
s m s m
lim sup y^log(pij) < ^ and lim sup y^deg(Pjj) < to.
m m
%=i j=1 %=i j=1
3.4 Niederreiter-Xing sequence (see [3, Section 8.3 ]). Let F/Fb be an algebraic function field with full constant field F and genus g. Assume that F/Fb has at least s + 1 rational places. Let P1,..., Ps+1he s + 1 distinct rational places of F. Let Gm = m(P1 + ... + Ps) — (m — g + 1)FS+1, and let ti be a local parameter at Pi, 1 < i < s +1. For any f e C(Gm) we have uPi(/) > — m, and so the local expansion of f at Pi has the form
f = £ fi,jt3i, with fi,j e F6, j > — m, 1 < i < s.
j=-m
For 1 < i < s, we define the Fj,-linear map ^m,i : £(Gm
) ^ Fm by
^m,i (/) = ( fi,-1,..., fi,-m) e Fm, for feC(Gm).
Let
Mm = Mm(Pi ,...,Ps;Gm) := {Ww(/) (/)) e FT | /e£(Gm)}.
Let
e f(°X( the generating matrices of a digital sequence xn(C)n>0, and let (Cm)m>:i be the associated sequence of row spaces of overall generating matrices [ C]m, m = 1,2,... (see (7)).
Theorem G. (see [3, Theorem 7.26 and Theorem 8.9]) There exist matrices C(1), ...,C^ such that (xn(C))n>0 is a digital (t, s)-sequence with t = g and C^ = Mm(P1,..., Ps; Gm) for m > g + 1, s> 2.
In [8, p.24], we proposed the following way to get xn(C)n>0 :
We consider the ^-differential dtLet w be the corresponding Weil differential, div(w) the divisor of w, and W := div(dts+1) = div(w). By (9)-(ll), we have deg(W) = 2g — 2. We consider a sequence v0, v1,... of elements of F such that {v0, v1,..., Vi((m-g+1)ps+1+w)-i} is an Fb linear basis of Lm := C((m — g + 1)Ps+i + W) and '
vr e Lr+1 \ Lr, uPs+1 (vr) = —f + 9 — 2, r > g, and vr,r+2-g = 1, vr,j = 0 (34)
2 < < + 2 —
vr := ^ vr, jt-+_1 for vrj e F;, and r > g.
j<r-g+2
i e F (1 < < )
d ts+1 = Tid^, for 1 <i<s.
Bearing in mind (10), (12) and (34), we get
vPi (vjn) = vPi (vjTidti) = up. (vjdts+1) > up. (div(dis+1) — W) = 0, j > 0.
We consider the following local expansions
(
VrTi := ^ c^T1, where all eg e Fb, 1 < i < s, j > 1. (35)
3 = 1
Now let C(i) = (
¿/r)j-i,r>0) 1 < i < s, let (C^)m>i be the associated sequence of row spaces of overall generating matrices [C]m, m = 1, 2,... (see (7)).
Theorem H (see [8, Theorem 5]). With the above notations, (xn(C))n>0 is a digital d—admissib le (i, s) sequence w ith d = g + si = g, an d C^ = ^TO(Pi,..., Ps; Gm) for a 11 m > g + 1.
We note that condition (34) is required in the proof of Theorem H only in order to get the discrepancy lower bound. WThile the equality = ..., Ps; Gm) is true for arbitrary sequence
■¿0, ?)i,... of elements of F& such that for all m > 1
{¿0,^i, ...,v£((m-g+i)pa+1+W)-i} is a Fb linear basis of Lm. (36)
In order to obtain the bounded remainder property, in this paper, we will construct from (t>n)n>0 a special basis (iin)n>0 as follows:
Let P0 G Pf, F0 = Pi (i = 1,..., s + 1), and 1 et i0 be a local parameter of P0. For simplicity, we suppose that deg(P0) = 1. Let
Lm = £((m — g + 1)Ps+i + W), = £((m + 2)Ps+i + ^ — mf0),
= £((m + 2)Ps+i + W — (m + 1)fb). (37)
It is easy to verify that
deg(Lm) = 2^, dirn(Lm) = g + 1, deg(£m) = 2^ — 1, dim(£m) =
for m > 0, deg(LTO) = m + 0 — 1, dim(LTO) = m, for m > g . (38)
Using the Riemann-Roch theorem, we have that there exists
and wm G LTO+g+i, m = 0,1,... . (39)
According to Lemma B, we have that there exists r0 G F, such that dis+i = r0di0.
Let u G Lm = £((m — g + 1)Ps+i + W) with m > 0. Bearing in mind (10), (12), (37)-(39) and the Riemann-Roch theorem, we get
^p0(w-0) = vp0(ur0di0) = vp0(«dis+i) = ^p0(div(-u) + W) > 0 (40)
and
^p0(wmT0) = ^p0(div(wTO) + ^) = m for m = 0,1,... . (41)
We consider the sequence (vj)j>0 (34). By (36), (-¿j)™=-)i is an Fb linear basis of Lm. Let
j-i
vj = {i)j + E bkvk I bk G Fb, k G [0, j)}, a(j) = max ^(^0). (42)
k=0 J
It is easy to verify that a(j) = ) for f = j. We construct a sequence (j)j>0 as follows :
j = 1)0, Vj G {v G Fj | ^0(ur0) = a(j)}, j = 1,2,... . (43)
It is easy to see that (j)j>0 satisfy the condition (36). Bearing in mind (40)-(42) and that j G Lm for j < m, we get
^0 (jj ^0) = (jk ^0) for j = fc, and (j T0) = a(j) > 0, j > 0. (44)
Hence, for all / G Lm, we have
^0(/7-0) G {a(0),a(1),...} =: H.
Taking into account (41) and (44), we obtain
R = {n In > 0} = N0. (45)
Suppose that a(j) > j+g. Bv (36) - (38), fy e Lj+1 = C((j —g + 2)Ps+1 + W). Hence vj e C(X), with X = (j — g + 2) Ps+1 + W — (j + g + 1) P0.
Bearing in mind that deg(P0) = deg(Ps+1) = 1 and deg( W) = 2g — 2, we get deg(X) = —1. Therefore C(X) = {0} and we have a contradiction. Hence
a(j) <j+ g. (46)
By (45), we have that for every integer k > 0 there exists r > 0 with a(r) = k. Therefore the map a : N0 ^ N0 is an isomorphism. Hence there exist integers P(k) > 0 such that
P(k) = a-1(k), a(fi(k)) = k and P(a(k)) =k for k = 0,1,... . (47)
From (46), we get for j = P(k)
k = a(P(k)) = a(j) <j+g = P(k) + g. (48)
Let
Bj = {r > 0 | a(r) < j}. (49)
Taking r = P(k), we get a(r) = k and
B3 = {P (0),P(1),...,P(j — 1)} for j> 1. (50)
Suppose j e Bj+g+1 for some j, then j = fi(j + g +1) for some I > 1. Using (48) with k = j + g + l, we obtain
3+1 = (j + 9 + l) —g<P(j+g + l)= j. We have a contradiction. Hence
j e Bj+g+1 for all j > 0.
We consider the local expansion (35), applied to i = 0 :
(
VrT0 := ^ ¿^t0-1, where cf} e F6, j > 1, C(0) = (c^r ),-1,r>0. (51)
=1
Let (x^0 (C(0)))n>0 ^e the digital sequence generated by the matrix C(0).
Now we consider the matrix C(i) = (eg)j-1,r>0, obtained from equation (35) and (51), where
we take vr instead of vr ( i = 0,1,..., s). ^^^^g Theorem H, we obtain that (xn0>\C(0)), xn((7))n>0 is ( , + 1) =
Lemma 2. There exists a sequence (i>j)j>0 such that (x^(C(0)), xn((7))n>0 is the digital (t, s + 1)-sequence with t = g and {0,1,..., m — 1} C Bm+g.
In §4.4, we will prove
Theorem 4. With the notations as above, the set [0,71) x ... x [0,7S) is of bounded remainder with respect to (xn(C))n>0 if and only if (2) is true.
4. Proof
Consider the following condition
liminf(m — T(m)) = to. (52)
We will prove (52) for the generalized Halton sequence in §4.3. For other considered sequences, assertion (52) follows from Theorem D, Theorem E and Theorem H.
The sufficient part of all considered theorems follows from Definition 2 and (52). Therefore we need only consider the case of necessity.
4.1 Generalized Niederreiter sequence. Proof of Theorem 1.
From Theorem D, we have that (xn)n>0 is the uniformly distributed digital weakly admissible (i, s)-sequence in base b. By Theorem B, in order to prove Theorem 1, we need only to check
condition (8). By ( 8, p.26, ref 4.6 ), we get
^ = ^ (^ g * =^ <53>
m—1
with I = ) + 1, n(^) = ^ a,j(n)^J+2 and dj(n) = Res 1).
n Pc-O
3=0
We take yi,j,k(®) = xmyijik(®) instead of yi,j,k(#)• Now using Theorem D, we obtain from (53), (4) - (6) that (Xn)0<n<b™ is a (i, m, s) net for m = STm +1 with x^j = 1 (2)^-) = nJ- Bearing in mind that xcn = xbmn, we obtain (8). Hence Theorem 1 is proved. ■
4.2 Xing-Niederreiter sequence. Proof of Theorem 2.
Bv Theorem B and Theorem E, in order to prove Theorem 2, we need onlv to check condition (8). *
From (3) - (6), we get that in order to obtain (8), it suffices to prove that
#{n G [0,6M) | yg = «f, 3 G [1,7-m] for i G [1,s], and a,—i(n) = uf j G [1,m]} > 0, with M = STm + (m + 2^)(2^ + 1)e0 + m0, (54)
m0 = + 2 + e1 +-----+ es, for all uf G F6.
Let
¿(1) = i1 if T iStrUe, 0, otherwise.
Let fc^ = zh(i) = zh(^) for j G #1 with ffi = N0 \ ff = {to(0),to(1),."}> ^2 = {n0,m,...,ns}• From (17), we have
40 = — 1 = r G i > 1.
Let 40 := ag)(r). By (4), (5) and (20), we get
= ^ — 1 = ¿3 = E dr (n)c5> = dMi—1)(n) (55)
r0
40) = Eah0—1)(™W and fcf = zh(>—1) G — 1)(2g + 1)^), j > 1.
3>1
DiUiiCli 5Ct|UCHtC (X^ )
Let n = Er=01 ar (n) br and let
So, we obtain a digital s + 1-dimensional sequence (xn), xn)n>0.
n = E ar(n)br, n= ar(n)br, U = {n|n e [0, bM)}, U = {n|n e [0, bM)}.
reHi reH2
By (4), (18) and (55), we get
¿j = J2ar(n)= E ar(n)c2 + E ar(n)c2 = V$j + V$j,
r>0 reHi r€H2
ie [1,^ ¿3 = = aмi-l)(n), = 0 j>1.
We fix n e U. Let
Au,n = {n e [0,6M) I yV = uf, j e [1, g, i e [1, s],
u . _ ^0) n [1 m
It is easy to verify that statement (54) follows from the next assertion
V(t°)a = uf0, j e [1, m], n = n}. (56)
# Au,n > 0 V uf e Fb, n e U. (57)
Taking into account that y^- = + Viij > we Set
Au,n = {n e U I y£)3 = Uf, j e [1, Tm], i e [1, s], y$ = uf, j e [1,m]},
where ug = ug — y^y
According to (4), (18) and (55), in order to prove (57) , it suffices to show that the vectors
KM(cg) = (eg,..., cjM-1) e FbM, with 1 < j < di, 0 < i < s, (58)
di, = rm, 1 < i < s and d0 = m, are linearly independent over F&.
To prove this statement, we closely follow [3, p.282]. Suppose that we have
m s rm
T, fi0)KM(cf) + £ Y.ST-M(c?) = 0 e FbM
3 = 1 i=1 3 = 1
for some /« e Fft with £,7=1 I^-1(/f)I + £>1 I^H/f)I > 0. We put fj0 = 0 for r > m. Hence
m s Tm
Ef{30)¿0! + ££ 1f4 = 0 for re[0,m).
3=1 i=1 3=1
Bv (18) and (55), we obtain eg = ag^) for 1 < i < s and c^l = 5(j — 1 = r)). Therefore
nm S Tm s Tm
0 = E f^t U — 1 = + EI] f^r) = fi0l)+1 + EI] f^^ir) (59)
3 = 1 i=1 3 = 1 i=1 3 = 1
for r G [0, M).
Now consider the element a G ¥b given bya = a1 + a2, where
m—1 s rm s rm g
«1 = E/h0r)+1-h(r), «2 = EI]/ii)^Si) — EI]/fE41(6°)
r=0 i=1 j=1 i=1 j=1 u=0
Using (17), we get
s Tm x g s Tm
«2 = EE/f(E4?* — E4!= E (EE^4?)*.
i=1 j = 1 r=0 «=0 rEHi i=1 j=1
From (18), (59) and (60), we obtain
a = E (/$0+1 + EE 4i)a(ih(r)) ^h(r) = E №+1 + EE 4i)aS(r)) ^M.
r>0 i=1 j = 1 r>M i=1 j=1
Hence
^ (a) > M. (61)
Furthermore, (15), (16), (20), (55) and (60) yield
s
«1 G£((m + 2^)(25 + 1)f0), «2 G + E([^m/ei] + 1)p). (62)
i=1
Combining (61) and (62), we obtain
s
a G £(G + E([^m/ei] + 1)Pi + (rn + 2^)(25 + 1)fb — .
i=1
But from (54), we have
s
deg (G + E([^m/ei] + 1)Pi + (m + 2^)(25 + 1)^0 — MP^)
i=1
s
= 2^ + E([^m/ei] + 1)ei + (m + 2^)(25 + 1)e0 — M
i=1
< 2g + srm + e1 + ••• + es + (m + 2^)(2^ + 1)e0 — M < 0.
Hence
s
£(G + E([rm/ei] + 1)pi + (m + 2^)(2^ + 1)^0 — = {0}
i=1
by (14) and therefore we have a = 0.
By (15), we have vPo(fcj^) > ^d vPo(wu) > 0 to all i,j,-u. According to (60), we get ^p0 (a2) > 0. Suppose that a1 = 0. Taking into account that z0 = zno = w0 = %(r) for r > 0, we obtain from (60) that uPo (a1) < 0. We have a contradiction. Hence a1 = 0 and a2 = 0. From Lemma C, we conclude that /J(i) = 0 to all f, j. Hence the system (58) is linearly independent over
Thus (54) is true and (xn)n>0 satisfies the condition (8). By Theorem E, (xn)n>0 is the d—admissible uniformly distributed digital (i, s)-sequence in base b. Applying Theorem B, we get the assertion of Theorem 2. ■
4.3 Generalized Halton-type sequence. Proof of Theorem 3. Lemma 3. The sequence (xn)ri>0 is uniformly distributed in [0,1)s .
Proof. By Lemma A, in order to prove Lemma 3, it suffices to show that m — T(m) ^ to for m ^ to. Let Rk = maxi<i<sni,k, k = 1, 2,... .We define ji,k from the following condition nijiik >Rk > ni,jik-i. Let Rk = TJs=lnl,3^.
( , m, )
s di
E = b-di, (ai + 1)b-di), with ai = ^aitjbj-1, aid e Zb, di > 0, i=i j=i
1 ^ i < s, d1 + ■ ■ ■ + ds = Rk, we have
#{n e [0,n e E} = #{n e [0,bm) | = Uf, j e [1,dt],i e [1,s]} = bm-Rk, where m > Rk + (3g + 3)eo, ug = ^(a^) e F6, (63)
j e [1,di], i e [1,s].
By Definition 2, we get that (x^n>0 is a (T, s)-sequence in base b with m — R(k) > T(m) for m > Rk + (3<jf + 3)e0. Bearing in mind that R(k) ^ to for k ^ to, we obtain the assertion of Lemma 4.
Taking into account that di < Rk < ni,ji k for 1 <i<s, we get that in order to prove (63), it suffices to verify that
#{n e [0,n | y« = ug, j e [1,nitjiik], i e [1,*]} = (64)
for all ug e Ffc, with j e [1,nijifc], i e [1, s].
Let ^ = ( m0e0 + 2g — with m0 = [m/e0] — 2g — 1.
Bv Lemma 1, we obtain that there exist sets H1 and H2 such that H1 U H2 = {0,1, ...,m — 1},
(0)
H1 n H2 = 0, (f«} ))reH1 is the F^ linear basis of and #H2 = m — m0e0 — g =: with
r=0 ar (
<jf1 — e0(2g + 1) — g e [0, e0). Let n = £^=01 ar(n)6r and let
n = E ar(n)br, n = E ar(n)6r, U = {nIn e [0, bm)}, U = {n|n e [0, bm).
reHi reH2
So
m— 1
fn = E a-(n)f(0) e £(.M) ^ n = n, for n e [0,6m).
r=0
We fix n e U. Let
Au,n = {n e [0,bm) I y® = uf, j e [1,m,jiik], i e [1,s], n = n}. It is easy to see that statement (64) follows from the next assertion
#Au,h = bm-AkV uf e F6, n e U. (65)
Taking into account that yn^- = y^i + Viij> we Set
Au,n = {n eU I = Uf, j e [1,n„.fc ], i e [1, s]},
where -u^ = -u(i) — y^^- • Let
to := (^nn ,-..J^mn1 ^ ,fc ,..,y{:?1,...,y(ni ,s , fc) g Ff.
We consider the map 0 : £(M) ^ F^fc defined by
■0(/) := 0(/n) where £(M) 9 / = /n with some n G
Note that 0 is a linear transformation between vector spaces over F&. It is clear that in order to prove (65), it suffices to verify that 0 is surjective. To prove this, it is enough to show that
dim (£(M)/ker($)) = Rk. (66)
Using (23), (25) and (28), we get that w(i) = 0 (mod Vitjik) for
I > ni,ji k. By (23), (25), (28), and (31), we derive that y^- = 0 fa all j e [1,ni,ji k] if and only if / = /nEE 0 (mod Pi^Jfor f G [1,4 ' ,
From the definition of 0 it is clear that
ker(0) = LL(ff ), with H = .M - Ê Vl>ni>H k.
¿=1
Using Riemann-Roch's theorem, we obtain that dim(M) = m0e0 + g = m — where deg(M) = m0e0 + 2g — 1 and
deg(#) = moeo + - 1 - ^ = m + g - 1 - ^i - Rk.
i=1
Hence dim(ker(0)) = m-Rk-> (3g+3)e0-> (3g+3)e0 - (2^+2)e0 - g > 1, dim(^) = m-and (66) follows. So is indeed surjective. Therefore (65) and Lemma 3 are proved. ■
Lemma 4. The sequence (x„)ra>0 satisfies condition (8). Proof. Let
s
M = ([M1/e0] + 35 + 1)e0, M1 = ^ where nji<m > rm > n^-1 (67)
¿=0
Bearing in mind that y^] = dj—1 (n), (j = 1, 2,...), we get from (32) - (33), that in order to obtain
for i e [1,s], n0jo,m = ([We0] + 1)e0 j0,m = [m/e0] + 1. Bearing in mind that y^] = (8), it suffices to prove that
#{n G [0, bM) | = uf, j G [1, ] for i G [0, s]} > 0 (68)
for all G F6. Let M = (([M^] + 1)e0 + — 1)P^. By (22), deg(P^) = 1. Hence deg(M) = ([M1/e0] + 1)e0 + 2$ — 1. Using Riemann-Roch's theorem, we obtain that
dim(.M) = ([M1/e0] + 1)e0 + g = M1 + 51 + g with 51 := ([M1/e0] + 1)e0 - M1. (69)
>v Lemma 1, we get that an Fb linear basis of L with M = ([M1/e0] + + 1)e0 = nQ,[Ml/eo]+3g+1.
Bv Lemma 1, we get that an F5 linear basi s of L(^) can be chosen from t he set {w0°\ ..., ^MM-1}
Let n = Z^V(n)br and let f.n = E^l-1^(n)w^0). We get that for all / e £(.) there
n [0, bM) such that f = /n. From (31), we have
oo
fn = £ C,W-1, 0 <f< s. (70)
3=1
Let
fn) := (y$,..., yinl OJOtm,..., y2,..., ) e Ff1. (71)
Consider the map $ : £(.) ^ Ff1 defined by
$(/) := fn) where f = /n with some n e [0, 6M).
We see that in order to obtain (68), it suffices to verify that $ is surjective. To prove this, it suffices to show that
dim (£(.M)/ker(V0) = M1. (72)
Using (23), (25) and (28), we get that wg = 0 (mod Vitjim) for k > ni,m. From (70), (23), (25) and (28), we derive that y^- = 0 to all j e [1,niyjim] if and only if fn = 0 (mod Vi,jim) for ie [0, s]. , , ,
From the definition of $ it is clear that
ker(V>) = £( H ), with H — M- E .
i=0
Using (67), (69), (22) and Riemann-Roch's theorem, we obtain that
deg(tf) = Mi + 9l + 2g - 1 nh3iim = 9l + 2g - 1
i=0
and dim(ker('0)) = gl + g. By (69), dim(^) = Ml + gl + g. Hence dim (£(^)/ker(^)) = Ml. Therefore (72) is true. So ^ is indeed surjective and (68) follows. Therefore Lemma 4 is proved. ■
Lemma 5. The sequence (xn)n>0 is weakly admissible.
Proof. Suppose that x^n = xg for some i, n, k. From (71) and (32)-(33), we get that y^ = yg for j > 1.
Using (70), we have
fn = Y1 vSj (n)wT-v
3>l
(o)
r
that n = k. By Definition 7, Lemma 5 is proved. ■
Applying Theorem B, we get the assertion of Theorem 3
m — / (n)w{~f>
Hence fn — fk- Taking into account that (w}Uj)r>^ an F^ linear basis of Of, we obtain from (30)
4.4 Niederreiter-Xing sequence. Proof of Theorem 4.
Similarly to the proof of Lemma 5, we get that (x„)ra>0 is weakly admissible. By Lemma 2, (:xn)n>o is the digital uniformly distributed sequence.
According to (4), (5), (8) and Theorem B, in order to prove Theorem 4, it is enough to verify that
#{n G [0, bM )| y^) = éf, j G [1, rm],i G [1, s], a, (n) = 0 for j G [0, m)} > 0 (73)
(i)
for all Uj G F&, where M = srm + m + 2g + 2.
Bearing in mind that by Lemma 2 (®n ), xn)„>^ a (g, s + 1) sequence, we obtain
#{n e [0,6Myy® = uf,j G [1,7m], i G [1,s], = 0,i G [1, m + g + 2]} > 0
for all e Fb.
Therefore, in order to prove (73), it suffices to verify that
if = 0 for j e [1,m + g + 2] then aj(n) = 0 for j e [0, m). (74)
Now we will prove (74) :
From (35) and (43), we have vrf0 = ^J>1 c^0?i0— 1 with ^Po(Vr0) = a(r). Hence c^0r) = 0 for j < a(r) and cf] = 0 for j = a(r) + 1.
Using (4), (47) and (49) we obtain cj°g(.,_i) = 0 and
fi = ZX (n)4> = Z ar (n)4> = Z (n)4°r> - L
r>° a(r)<j
We apply induction and consider the case j = 1. By (50), we see that ар(°)(п) = 0 if уП°1 = 0.
Suppose that ар(°)(п) = ■ ■ ■ = ару_1)(и) = 0 if y^l = ■ ■ ■ = = 0 for some I — 1 .Now let
(°) (°) (°)
Vni = = УЩ = <i+i = 0 see
0 = Д+i = Z arW+i.r = Z ®r(«)ci+)i,r = aP(i )(«)ci+)i)p(o-
r€Bm reBl+1\Bt
Bearing in mind that c^ р^ = 0 we get ap(i)(n) = 0.
Therefore if уП°] = 0 fa all 1 < j < m + g + 1, then ap(j_i)(n) = 0 fa all 1 < j < m + g + 1. Using Lemma 2, we get ar(n) = 0 for all 0 < r < m — 1. Hence (74) is true and Theorem 4 follows. ■
Aknowledgment. Parts of this work were started at the Workshop "Discrepancy Theory and Quasi-Monte Carlo methods"held at the Erwin Schrodinger Institute, September 25 - 29, 2017.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Beck, J., Chen, WT. WT. L. Irregularities of Distribution, Cambridge Univ. Press, Cambridge, 1987.
2. Bilvk, D. On Roth's orthogonal function method in discrepancy theory, Unif. Distrib. Theory 6 (2011), no. 1, 143-184.
3. Dick, J. and Pillichshammer, F. Digital Nets and Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.
4. Grepstad, S.; Lev, N. Sets of bounded discrepancy for multi-dimensional irrational rotation. Geom. Funct. Anal. 25 (2015), no. 1, К7 133.
5. Hellekalek, P. Regularities in the distribution of special sequences, J. Number Theory, 18 (1984), no. 1, 41-55.
6. Larcher, G. Digital Point Sets: Analvis and Applications. Springer Lecture Notes in Statistics (138), pp. 167-222, 1998.
7. Levin, M. B. Adelic constructions of low discrepancy sequences, Online J. Anal. Comb. No. 5 (2010), 27 pp.
8. Levin, M. B. On the lower bound of the discrepancy of (t, s) sequences: II, Online J. Anal. Comb. No. 5 (2017), 74 pp.
9. Levin, M.I?.. On a bounded remainder set for a digital Kronecker sequence, arXiv: 1901.00042.
10. Levin, M. B. On a bounded remainder set for (t, s) sequences II, in preparation.
11. Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, in: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, SIAM, 1992.
12. Niederreiter, H. and Yeo, A. S., Halton-tvpe sequences from global function fields, Sci. China Math. 56 (2013), 1467-1476.
13. Salvador, G. D. V. Topics in the Theory of Algebraic Function Fields. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 2006.
14. Stichtenoth, H. Algebraic Function Fields and Codes, 2nd ed. Berlin: Springer, 2009.
15. Tezuka, S. Polynomial arithmetic analogue of Halton sequences. ACM Trans Modeling Computer Simulation, 3 (1993), 99-107
REFERENCES
1. Beck, J., Chen, W. W. L. Irregularities of Distribution, Cambridge Univ. Press, Cambridge, 1987.
2. Bilvk, D. On Roth's orthogonal function method in discrepancy theory, Unif. Distrib. Theory 6 (2011), no. 1, 143-184.
3. Dick, J. and Pillichshammer, F. Digital Nets and Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.
4. Grepstad, S.; Lev, N. Sets of bounded discrepancy for multi-dimensional irrational rotation. Geom. Funct. Anal. 25 (2015), no. 1, 87-133.
5. Hellekalek, P. Regularities in the distribution of special sequences, J. Number Theory, 18 (1984), no. 1, 41-55.
6. Larcher, G. Digital Point Sets: Analvis and Applications. Springer Lecture Notes in Statistics (138), pp. 167-222, 1998.
7. Levin, M. B. Adelic constructions of low discrepancy sequences, Online J. Anal. Comb. No. 5 (2010), 27 pp.
8. Levin, M. B. On the lower bound of the discrepancy of (t, s) sequences: II, Online J. Anal. Comb. No. 5 (2017), 74 pp.
9. Levin, M. B. On a bounded remainder set for a digital Kronecker sequence, arXiv: 1901.00042.
10. Levin, M. B. On a bounded remainder set for (t, s) sequences II, in preparation.
11. Niederreiter, H. Random Number Generation and Quasi-Monte Carlo Methods, in: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, SIAM, 1992.
12. Niederreiter, H. and Yeo, A. S. Halton-tvpe sequences from global function fields, Sci. China Math. 56 (2013), 1467-1476.
13. Salvador, G. D. V. Topics in the Theory of Algebraic Function Fields. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 2006.
14. Stichtenoth, H. Algebraic Function Fields and Codes, 2nd ed. Berlin: Springer, 2009.
15. Tezuka, S. Polynomial arithmetic analogue of Halton sequences. ACM Trans Modeling Computer Simulation, 3 (1993), 99-107
Получено 09.01.2019 г. Принято в печать 10.04.2019 г.
