A. DIJKSMA
B.BUOTHEEK TU Dem y ^954 11,0
naeo 651432
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECKNISChE WETEKSCHAPPEN AAN DE TECnNISCKE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR
MAGNIFICUS IR. H.R. VAN NAUTA LWÏKJL, hOOGLERAAR IN DE AFDELILG DER ELEKTROïECiiNIEK, VOOR EEN COmilSSIE UIT DE SENAAT TE VERDEDIGEN OP
WOENSDAG 20 JANUARI 1971 Ï'E H UUR
DOOR
AALT DIJKSMA wiskiojfidig i n g e n i e u r g e b o r e n t e N i j k e r k
PROF. DR. B, MEULENBSLD.
EN IS TOT STAND GEKOMEN ONDER DIRECT TOEZICHT VAN
DR. H.G. MEIJER,
LECTOR AAR DE TECHNISCHE HOGESCHOOL DELFT.
Ter nagedachtenis aan mijn vader
Aan mijn moeder
I Uniform distribution in topological groups 1
II Uniform distribution in GFjq,x| and GF[q,x] 4
III Consequences of the Weyl criteria 8
IV The mapping T s GFJq,x| - R 9
V Polynomial generated sequences 12
VI Some metrical theorems 16
VII The measure theoretic approach to uniform
distribution in GF[q,x] 18
References 19
Samenvatting 21
Biography 22
I. UNIFORM DISTRIBUTION IN TOPOLOGICAL GROUPS
In 1916 Weyl [21] introduced and studied the concept of uniform distribution modulo 1 of a sequence in R, the field of real numbers,
Definition (1 . 1 ) . Let A = (x.)._. be a sequence in R. Let a,p £ R and let n be a natural number. Put A(a,p,n) = card |i | 1 s i s n, a ^- jx.| < p|, where |x| denotes fractional part of x £ R.
The sequence A is called uniformly distributed modulo 1 if lim n A ( a , p , n ) = p - a
n-»co
for all [a,p) c [0,l).
Theorem (1.2). Let A be as in definition (I.1). Then the following statements are equivalent.
(a) A is uniformly distributed modulo 1. (b) (Weyl criterion) For all integers h ?^ 0
-1 "
lim n E exp (27tihx^) = 0.
n-«= j = 1 'J
(c) For every continuous complex-valued function f on [0,l] 1
1 ^ r lim n E f({x^|) = f(x)dx.
n-»oc i = 1 4-.
Eckmann [8] in 1943 extended Weyl's definition to compact groups,
Definition (1,3)« Let G be a compact group. Let v be the left Haar-measure on G such that V ( G ) = 1. Let A = (x.)-_-i ^e a sequence in G. Put A(M,n) = card ji | 1 g i s n, x. £ M} where M c G and n is a natu-ral numiber. If for every measurable subset M of G whose boundary has measure zero
lim n""" A(M,n) = V ( M ) n—00
then A is called uniformly distributed in G.
Theorem^ (1,4)« Let A be as in definition (I.3). Then the following statements are equivalent.
(a) A is uniformly distributed in G.
(b) (Weyl criterion) For each non-trivial irreducible representation D of G -1 "
lim n S D ( X . ) = 0, n-» 00 1 = 1
(c) Por each continuous complex-valued function f on G
-1 "" r
lim n E f(x^) = f dv.
n-oo 1 = 1 ^
More recently, 1965» Rubel [19] introduced and studied the
notion of uniform distribution of sequences in locally comipact groups.
Definition (1.5)• Let G be a locally compact group. Let H be a closed normal subgroup of G of compact index, i.e., ^/i[ is compact, and let cp % G -* G/g denote the natural homomorphism. Let A = (x.)-_-| ^e a
sequence in G, Then
(a) A is called uniformly distributed modulo H in G provided that the sequence (cp (x.))._ is uniformly distributed in G/g in the sense of definition (1.3).
(b) A is called uniformly distributed in G if it is uniformly distributed modulo H in G for every closed normal subgroup H of G of compact index.
The above definition is a generalization of Niven's notion of
uniform distribution of sequences in I, the ring of integers. See [l7j» Some of the results Niven obtained have been extended by Uchiyama [20]. We will discuss the notion of uniform distribution in I by starting from Rubel's definition.
Let I have the usual, that is the discrete topology. Then the additive group of integers is locally compact. Every (closed, normal) subgroup of I of compact index is of the form ml = jO, ±ri, ±2m,..,} for some
m£{l,2,,..}. That is, the class of subgroups of I of compact index equals the class of ideals of the principal ideal ring I, The correspon-ding compact quotient groups are the discrete, finite, cyclic groups I with m elements, n\£ {1,2,...}. The unique normalized Haarnieaaure v on
-1
I assigns measure m to each of the elements of I . It follows from
m "^ m these observations and definitions (1.5) and (1.3) that the uniform
distributivity of a sequence of integers is defined as follows.
Definition (1.6). Let c|i = (x.)._ be a sequence in I, Let n and m be natural numbers, m s 2. Put c|j(n,c,m) = card {i j 1 s i g n, x. E c (mod m)| where c £ I,
(a) If for every integer c with 0 ^ c ^ m-1
lim n c|j (n, c,m) = m
n—oo
then (Ij is said to be uniformly distributed modulo m in I.
(b) 4. is called uniformly distributed in I if it is uniformly distributed modulo m in I for all m £ J2,3,...|.
In case G is an Abelian group the non-trivial irreducible representations of G in theorem (1,4) become the non-trivial
continuous characters of G. The group of all continuous characters of I consists exactly of elements of the form exp (2nijm x) with j£{0,1,.,,,m-l}, m e j l , 2 , . . . ] . A character of 1 is non-trivial
if j / 0. Hence theorem (1-4) now implies the following Weyl criteria due to Niven and Uchiyama.
Theorem (1 .7)« The sequence <j; in definition (1.6) is uniformly distributed modulo m if and only if for every j £ |1,2,...,m-1]
n
lim n E exp (2Ti;ijm x, ) = 0. n->oo k=1
Corollary (l«8). The sequence i\> in definition (l.6) is uniformly distributed in I if and only if for every rational number r v;ith 0 < r < 1
-1 "
lim n E exp (27iirx, ) = 0. n-*oo k=1
II. UNIFORM DISTRIBUTION IN GF(q,x} AND GF[q,x]
In this dissertation we deal with uniform distribution modulo 1 of sequences in $' = GF{q,x|, the field of formal Laurent series in the indeterminate x over the finite field GP(q), which are of the form
n
(2.1) S a.x^
i=-oo
where n may assume any integer value and uniform distribution of sequences in the ring $ = GF[q,x] consisting of polynomials in x over GF(q). We consider $' as an extension of $. The elements of $' and f are denoted by lower Greek and capital letters respectively. The degree of 0 ^ a £ <ï>' , deg a, is defined as usual. We put deg 0 = - oo , We '
_1
say that a £ $' is rational if a = AB for some A and B in $. Let a £ <ï>' have the form (2.1 ). Then we define the integral part [a] and the fractional part [a] of a by
n
[a] = E a.x anda. J 1 [a] = a - [a] . i=0
We call a and P in $' equivalent modulo 1 if [a] = jpj.
Let the topology on $' be defined by the basis {§' + a | n £ I,a£ $'} where for each integer n
$' = { a | a £ $ ' , d e g a < n | .
One can show that with this topology $' is a locally compact Hausdorff field in which the elements of the basis, called discs, are open, closed and compact.
Let \i be the Haarmeasure on the additive group $' normalized so that |i($') = 1 . This normalization of \i implies that each disc $' + a has measure q , n £ I, a £ $'.
Let GF(q) = GF(p ) , for some prime number p and natural number r, be defined by a zero p of an irreducible polynomial over GP(p) of
degree r. Then a . £ GF(q) in (2.1) can be uniquely expressed as
a = b p"^~ +b „P"^" + • • • +^ ' ^- £ '^F(p), i = 0,1 , . . . ,r-1 % if in (2.1 ) n g -2 then we suppose that a . = 0,
Let a £ $ ' be given by (2.1 ). Then we define e(a) by e(a) = exp(27iib _ p )-For a £ $ ' let the complex-valued function e on $' be defined by
a e^(p) = e(ap).
We now are able to define and discuss uniform distribution of
sequences in <ï>' and $, We first consider Carlitz's definition concerning uniform distribution modulo 1 in $'. See [l].
Definition (2,1 ). Let i\, = {a.)._. be a sequence in $'. Let p £ $' be arbitrary and let k and n be positive integers. Put cj^. (n,P) =
card |i I 1 ^ i ^ n, deg {a.- p|< -k|. If for every positive integer k and every p £ $'
lim n" cl;j^(n,3) = q"
n—oo
then 4; is called uniformly distributed modulo 1 in $' ,
Furthermore Carlitz proved in a direct way the following Weyl criterion,
Theorem (2.2). The sequence i\> of definition (2.1) is uniformly distributed modulo 1 in $' if and only if for all A £ $, A / 0,
-1 "
lim n E e(Aa.) = 0,
n— oo i = 1
Since ie.] A £ $} is the group of all continuous characters of the compact additive subgroup $' of $', it follows from theorems (1,4) and (2.2) that definition (2,l) is a particular case of Eckmann's definition (l.3). Furthermore, if we consider the discs of <|)' as analogues of the intervals of R, then uniform distribution modulo 1 in $' is an analogue of Weyl's definition (l.l).
We now consider Hodges's definition concerning uniform distribution
in $. See [ll].
Definition (2.3)» Let Q = ( A . ) . _ be a sequence in $. Let i'l,C £ 4>,
deg M s 1 and let n be a natural number. Put e(n,C,M) = card ji | 1 ^ i ^ n, A. E C (mod M ) } .
(a) The sequence 6 is called uniformly distributed modulo M in $ if for every C £ $, deg C < deg M
lim n~ 0(n,C,M) = q~ n—oo
where m = deg M.
(b) 9 is said to be uniformly distributed in $ if it is uniformly distributed modulo M in $ for all M £ $ with deg M a 1.
In [5] we proved in a direct way the following Weyl criteria
Theorem (2.4). The sequence 9 of definition (2.3) is uniformly distrib-uted modulo M if and only if for each C £ $, C / 0, deg C < deg M
-1 " 1 lim n E e(CM A.) = 0 .
Corollary (2.5), The sequence 9 of definition (2.3) is uniformly distributed in $ if and only if for each rational a £ *1>' , a ^ 0,
o -1 "
lim n E e(aA.) = 0.
n—oo 1 = 1
The analogy of definition (2.3)> theorem (2.4) and corollary (2.5) with definition (1.6), theorem (1,7) and corollary (l.8) respectively is clear. In both cases the ideals of the principal ideal rings I and <|) and the characters of their quotient rings are considered. Every ideal in $ is of the form M$ and the characters of the additive group $/M(i) are of the form e -1 , M,C £ *, deg C < deg M,
As has been remarked before the class of subgroups of compact index of I equals the class of ideals of I. This equality does no longer hold if I is replaced by $, which we also give the discrete topology. The ideals of $ are of compact index, but these are not the only subgroups of $ that share this property. See examples below.
Therefore, it follows from theorems (2.4) and (I.4) that definition (2.3) is equivalent to definition (I.5) with G = $, provided that the term normal subgroup in Rubel's notion is replaced by ideal. If we Eiaintain Rubel's definition (I.5) as it stands, then, in case G = $, every „Rubel"-uniformly distributed sequence in $ is also „Hodges"-uniformly distributed, but the converse need not hold. In the next chapters when we say that a sequence is uniformly distributed in
GF[q,x] then we are referring to a sequence which is uniform.ly distrib-uted in the sense of definition (2.3).
Now we give two examples of subgroups of $ of compact index which are unequal to M$ for any M £ $. For the first one, let a £ $' be irrational with deg a = a s 1. Put cj;. = {[aZ] | Z £ $|. Then t\)^ is a subgroup and card ^/i\,^ = q as is easily verified. It follows from theorem (5.3) of [4] that ii. ^ M$ for all M £ ^. For the second example, let A and B be elements of $ such that 1 ^ deg A < deg B-1 and ( A , B ) = 1 . Then the subgroup (|) „ = ( BL + kA | L £ $, k=0,1 , . . . ,p-1 | is of compact index. Suppose that C|J„ = M<ï^ for some M £ $. Ttien A and B are elements of M$ and therefore divisible by M, Hence M divides ( A , B ) = 1, which implies tnat M £ GF(q) \ |0} and M$ = $. Clearly i>^ ^ ^. So i\> ^ ^ m
for any M £ $.
As is shown in [5] and [15] there exists a sequence ([aZ.])._. of elements of 4-' which is uniformly distributed in <1> in the sense of Hodges's definition. But evidently this sequence is not uniformly
distributed modulo i\, in the sense of definition (I.5). This illustrates the fact that in ^ Hodges's definition does not coincide with Rubel's definition,
If a sequence is uniformly distributed modulo M in $ and c|; is a subgroup of $ of compact index such that M$ c (|j (viz, 4jp where B$ c (jjp) "then this sequence is uniformly distributed modulo (jj in the sense of Rubel's definition. This follows from the following group isomorphism
*A =*/''^/v>
'M*and the fact that if a sequence of elements of a compact group (read $ / M $ ) is uniformly distributed in the sense of Eckmann's definition
then it is uniformly distributed modulo each of it's closed normal subgroups (read 'VM^') as Rubel has shown in [19].
Finally, we remark that uniform distribution of sequences in $ modulo the particular polynomial M = x has been studied by L.
III. CONSEQUENCES OF THE WEYL CRITERIA
From the Weyl criteria, theorems (2.2) and (2.4) and corollary (2.5), many interesting relations between uniform distribution in $' and <& follow. See [5] and [7]« These relations are analogues of theorems of Van den
Eynden [9] and Kuipers and Uchiyama [13] concerning uniform distribution in R and I. Some of the theorems we mention here have also been proved by Hodges [12] in a direct way.
Theorem (3.1 ) • Let (a.)._ be a sequence in $'. Let M £ «ï" have degree ^ 1. If ( M a.)- ^ is uniformly distributed modulo 1 in $', then ([a.1). . is
V l''l = 1 "^ ' ^L i-''i = 1
uniformly distributed modulo M in $.
Theorem. (3«2). If the sequence (a.)._ is uniformly distributed modulo 1 in $', then ([Ma.])._ is uniformly distributed modulo M in $ for all M £ $, deg M ^ 1. Conversely, if ([Ma.])._. is uniformly distributed
modulo M in $ for infinitely many M £ $, then {a.)._. is uniformly distrib-uted modulo 1 in $'.
Theorem (3.2) is the analogue of the following theorem of Van den Eynden.
Theorem (3.3). If the sequence (x.)._ in R is uniformly distributed modulo 1 then ([nx.])._. is uniformly distributed modulo n in I for all n £ I, n s 2. Conversely, if ([nx.])._. is uniformly distributed
/ \00
modulo n in I for infinitely many n m 1, then (x.j._. is uniformly distributed m^odulo 1 in R.
Theorem (%4)« The sequence (pa.)._. is uniformly distributed modulo 1 in $' for almost all (for every irrational, for all) p £ *' if and only if the sequence ([pa.])._. is uniformly distributed in $ for almost all (for every irrational, for all) p £ $',
In the last theorem the quantifier „alm.ost all" is to be taken with respect to the Haarm.easure \i defined in the previous chapter,
IV. THE MAPPING T S GF{q,x| -' R,
Let T be a one-to-one correspondence between GF(q) and the set of integers |0,1,,.,,q-1} such that T ( 0 ) = 0, We extend the domain and range of T to $' and R respectively by defining
n . n
T ( E a . x ) = E T(a.)q,
i=—oo i=—oo
We observe that T , restricted to $, establishes a one-to-one correspondence between $ and the set of non-negative integers. The mapping T is an
analogue of a mapping from the p-adic numbers into R defined and studied by Monna [16]. It is used in the papers [4], [5], [6], [7] and [l5].
In [15] Meijer and the author proved the following theorem.
Theorem (4.1). The sequence (A.)._, in <ï> is uniformly distributed modulo M in $ if and only if for h=1 , 2, , . . , q'^-l
n
lim n~ E exp [ 2TiihT(A . (M) )q~°'] = 0. n-oo j = 1 J
Here m = deg M 5 1, and A . ( M ) is the uniquely determined element of $ such that A. = A . ( M ) (mod M ) and deg A . ( M ) < m, j=1,2,... .
«J J O
This theorem establishes an interesting relation between uniform distribu-tion in $ and I. In particular, it implies the following result.
Corollary (4-2). Let k be a positive integer. Then tlie sequence (A.)._
k 1 1 - 1
in $ is uniformly distributed modulo x if and only if the sequence
( T ( A . ) ) . _ is uniformly distributed modulo q in I.
The following theorem resembles a theorem of Cugiani [3] concerning uniform distribution in R and in the field of p-adic numbers. The proof of our theorem is simpler,
Th eorem (4.3)» The sequence t\> = {a.)._^ in $' is uniform.ly distributed modulo 1 in $' if and only if the sequence <\> = {ria- )) . _^ in R is uniformly distributed modulo 1 in R.
Proof. It follows from theorem (3.2), corollary (4«2) and theorem (3.3) that the following four statem.ents are equivalent.
(a) 4^ is uniformly distributed modulo 1 in $'.
(b) ([x ot. ])._>, is uniformly distributed modulo x in $, k=1,2,,.. . (c) ([q T(a.)])-_^ is uniforffily distributed modulo q in I, k=1,2,... . (d) 4^ is uniformly distributed miodulo 1 in R.
The above theorem and corollary (4-2) lead to the following problem. / NOO
Is the sequence (A.;._ in $ uniformly distributed in $ if and only if the sequence (T(A.))°°_ is uniformly distributed in I ? There are many
/ ^oo / / \ \Oo
uniformly distributed sequences {k. ) . _^ in <ï> for which {i:{k.))._ is uniform^ly distributed in I. See tiie remarks written at the end of [5] and before theorem 8.4 in [4]«
The answer to the problem however is in the negative. In order to show this we need the following rather technical lemma,
Lemma (4,4). Let T be restricted to $ = GF[q,x] = GF[p , x ] . Let m > 1 be an integer such that (m,p) = 1 and (m., T(a)+T(-a)) = 1 for some a £ G F ( q ) \ j 0] . Then for every A £ $ and every k £ I there exists an F £ $ such that T ( F A ) E k (mod m ) .
Proof, Let A £ $ and k £ I be given. Let 9(m) be the number of positive integers g ra which are relatively prime to m. Let k ,k ,c..,k be
positive integers such that k^=1 and (k. ^-k. ) (p(m.) > deg A for m k. cp (m)
i=1,2,..,,m-1. Put G = E X . Then G £ $ and using the fact that i = 1
q9^ •^ E 1 (mod m) one easily verifies that T ( W G A ) = T ( G ) T ( W A ) E 0 (mod m) for all w £ GF(q).
Now, let GA = (a-,x +a-,_ x +...+a )x v/here a.. £ GF(q), i = 0,1,...,l, a, 7^ 0, a ^ 0 and j s 0. Then there exist u and v in GF(q) such that
1 1
ua-. = a and vaQ = -a. Since T ( V G A X ) = T(vGA)q E 0 (mod m ) , T ( U G A ) E 0 (mod m) and ax -ax = 0 in GP(q) we have
T ( ( V X " ^ + U ) G A ) = T(VX"^GA) - T(-a)q-^'^^ + T ( U G A ) - T(a)q'^"^^ E -(T(a) + T(-a))q ^ (mod m)
E s (mod m) with (s,m) = 1 .
Since (s,m) = 1 there exist an element n £ I such that ns E k (mod m). Similar to the construction of G we may construct H £ $ such that T ( H ) E n
(mod m) and T ( H B ) = T ( H ) T ( B ) where B = (vx + U ) G A . If we put F = (vx + U ) H G
then F has the desired property of the lemma. This completes the proof.
In order to substantiate the statement preceding lemma (4.4)? let
( B . ) . _ - be any uniformly distributed sequence in $ such that B. ^ 0 for i=1,2,... (e.g., B(i) = T (i), i=1,2,..., see chapter V) and let (P-)-_^ be a sequence of arbitrary elements of $. Put
1+deg B.
A. = B. + X ^ P. B. B B,,
, lOO
i = 1,2,... . Then the sequence 9 = (A.)._.. is uniformly distributed in $. For, if M £ $ and deg M i^ 1, then there exists a natural number k such that B, E 0 (mod M ) and this implies that for i ^ k A. E B. (mod M ) .
k ^ ' i- 1 1 ^ '
1+deg B.
We observe that the factor x guarantees that
1+deg B.
T ( A . ) = T ( B . ) + T ( F . B . B . , ... B, X ^)
^ i ' ^ 1 ' ' ^ 1 1 1 - 1 1 '
for i=1,2,... . Let 0 ^ a £ GF(q) and let m be a positive integer satisfying the conditions of lemma (4.4). It then follows from this lemma that for i=1,2,... there exists F. £ $ such that
1+deg Bj_
T ( F . B. B . ^...B,X ) = - T ( B . ) (mod m ) ,
^ 1 1 1 - 1 1 / \ -^z \ /
/ \00
Now, let the sequence (F.j. . be such that for each i F. satisfies this
1 1 = 1 1
relation. Then T ( A . ) E 0 (mod m) for all natural nuDibers i. Hence the sequence ( T ( A . ) ) . _ . is not uniformly distributed modulo m and thus not uniformly distributed in I.
V. POLYNOMIAL GENERATED SEQUENCES
In this section we state the m^ain results of [5]> [6] and [15]' We put Z. = T~ (i-l) for 1=1,2,... .
Theorem (5«1 )» Let f(Y) be a polynomial over $' of degree k with 0 < k < p (p is the characteristic of GF(q)). Then the sequence (f(Z.))._ is uniformly distributed modulo 1 in $' if and only if f(Y)-f(0) has at least one irrational coefficient.
For f(Y) = aY+p, a,p £ $, theorem (5''') has been proved in tv/o differ-ent ways, (a) In [15] Meijer and the author have proved the uniform
/ NOO
distributivity of (c£Z.+pj._ in a direct way by using Kronecker's criterion for irrationality of a. (b) In [5] the author proved this particular case by means of the following inequality
n , I E e(aZ. ) I < q
i = 1 ^
where deg {a} = -t and n is a natural number. Furthermore in [5] a generalization of this particular case has been proved.
The proof of the above theorem is based on an analogue of Van der Corput's difference theorem. See [2].
Theorem (5.2). Let g s $ $' be a function. Put gg(Z) = g(Z+B) -g(Z), B,Z £ i>. If (g-oi'Zi.)) . _. is uniformly distributed modulo 1 in <ï>' for all B £ $ \ | 0 | , then (g(Z. ))•_-, has the same property.
Part (a) of the following theorem appears in [5] and [15]; part (b) in [6] only.
Theorem (5» 3)» Let f(Y) be a polynomial over $' of degree k.
(a) Suppose k=1. Then ([f(Z.)])._ is uniformly distributed in $ if and only if either f(Y)-f(0) has an irrational coefficient or a rational one of degree ^ 0.
(b) Suppose 2 ^ k < p. Then (ff(Z.)])._ is uniformly distributed in $ if and only if f(Y)-f(0) has an irrational coefficient.
We have not been able to characterize an arbitrary polynomial f(Y) over $' for which (f(Z.))._ is uniformly distributed modulo 1 in $'. However, some extensions of theorem ( 5 . 0 can be given. First of all we observe that if f(Y) = E a-, Y and among a^,...,a . is at least one
1=0 ^ ^ P~'
in $'. This follows from theorem (5.2).. The fact that a is irrational P
is no guarantee for the uniform distributivity in $' of this sequence, as has been shown at the end of [6].
Secondly we have the following result,
Theorem (5,4)» Let a,p be elements of $'. The sequence (aZ. + pz. )°°_ is uniformly distributed modulo 1 in $' if and only if for each non-zero A £ $ tiiere exists an a £ GF(q) and an integer k, k ^ 0, such tnat
e((ba +ca)x ) / 1 where b is the coefficient of x of Aa and c is -k-1
the coefficient of x of Ap.
In order to prove this theorem we need the following lemmas.
Lemma (5.5). Let a,p £ $' be arbitrary and put for k=0,1,2,... k
"1 n S^.(a,p) = E e(o:Z^ + pZ.) .
1 = 1
Then (a) S (a,p) = 1 and S, (a,p) = 0 or q , k=1,2,..., (b) S^_^^(a,p) = S^(bx"\cx-M Sj^(a,p), k=0,1,2,...,
where b is the coefficient of x of a and c the coefficient of x of p.
Proof. Since Z = 0, S (a,p) = 1 . Since A^-B^ = ( A - B ) ^ for all A,B £ $, it is easily verified that |s, (a,p)| = q S, (a,p). From this equality
- 1 - 1
the second part of (a) followsv Evidently S.(a,p) = S (bx ,ex ).1
-1 -1 where b is the coefficient of x of a and c is the coefficient of x
of p. Hence we now only have to prove (b) for k=1,2,... . Using the Laurent representation of a and p,
n . m . a = E b. x and p = E c. x
1 '^ . 1
l = - o o 1 = — o o
and observing that if
1 . 1 .
Z = £ a.x^ then Z^ = E {a.)^x^^
1 i
i=0 "• i=0 we see that
e(azP+pz) = e(( E b a^ + c a )x ^) i = 0 ~^
= ^^(^-pl-1 ^1 -^ "-I-1 ^ 1 ^ ^ " ) ^^^i?1 % i - 1 ^ ? + "-i-1^i^^ Putting l=k we obtain (b) for k=1,2,... . For,
k q
^k+1 '^"'P) = ^ e'^(^_pk-1^^+^-k-1^)''~ ^ ^ e(aZ^+pZ.)
a £ GP (q) ^ i = 1
= S^(bx"\cx"^) Sj^(a,p). This completes the proof.
Lemma (5.6). Let a,p £ $' be arbitrary. Suppose there exist an integer
T — 1
k ^ 0 and an element a £ G F ( q ) such that e((ba +ca)x ) 7^ 1 , where b and c are as in lemma (5»5). Then for all positive integers n
I E e(azP + pZ )| < q^^\
' . . 1 1 ' 1 = 1
Proof. Let K £ $ be an arbitrary polynomial of degree k+1. Then in the / \oo k+1
sequence (Z.j._ the first q elements constitute a, complete residu
•^ •""" k+1
system modulo K as does each succeeding block of q elements. The ele-k+1
ments of such a block are given by BK+Z., i=1,2,,..q for some B £ $.
r -k-1-1 "^
Let n be given and put 1 = [nq J. Let B. £ $ be such that
i Z . | ( j - l ) q ^ " ' U l ^ i g jq^^^^l = I B ^ + Z . h ^ i ^ q^+^ | , j = i , 2 1. Then
n
k+1i S e ( a z P + p z . ) | = I S E e(a(B K+Z. )P+p ( B K+Z ))+ E e(azP+pZ ]
i = 1 j = 1 1 = 1 '^ ^ . ^ k+1 , "^ i = lq +1 -j_ k+1 < I E e(aBPKP+pB.K)| ! E e(azP+pZ.)l + q^""^ j = 1 ^ - ^ 1 = 1 ^ ^^ 1
\^^{<^,?)
-. q^""^
k+1 = q -1 -INThe last equality follows from lemma (5.5). For S, , ( a, p) = S, (a,P ) S.(bx ,cx ), and the hypothesis of lemma, (5*6) and lemma (5-5) (a) i^ply that
S^(bx"^ ,cx"'') = 0.
This completes the proof.
Proof of theorem (5.4). The ,4f"-part of the theorem follows immediately from lemma (5.6) and the Weyl criterion, theorem (2.2). To prove the „only if"-part we suppose there exists a non-zero A £ $ such that for each
k £ 10,1,2,...} and all a £ GF(q) e((ba +ca)x ) = 1, where for each k b and c are as in the theorem. Then it follows from lemi.ia (5.5) that S (Aa , Ap) = q for all m=0,1,2,... . Hence for each natural number n _ I A a . ALi ) = ( m
m n
S e(AazP + ApZ.) = n.
, . 1 1 ' 0y_ ü u 7 i = 1The Weyl criterion (2.2) now im.plies that (aZ.+pZ.)._. is not uniformly distributed modulo 1 in $'.
Remarks, (a) If in theorem (5.4) oc=0 then we have again proved that (pZ.)._ is uniforFily distributed modulo 1 in $' if and only if p £ $' is irrational. For, the statement that for each non-zero A £ $ there
— 1
exist a positive integer k and an element a £ GP(q) such that e(cax ) 7^ 1 » -k-1
where c is the coefficient of x of Ap is equivalent to the statement that Ap ^ $ for all A £ $, A ^ 0, i.e., p is irrational.
(b) If in theorem (5.4) P=0 then this theorem gives a necessary and sufficient condition on a for the uniform distributivity of the sequence
, p»00 ,
(aZf). ^. The set of a £ $ for which this condition holds has measure 1,
1 1 = 1
This follows from a theorem of Carlitz [1J, which states that if (A.j._, is a sequence of distinct elements of $, then for almost all a £ $'
the sequence (aA.)- •, is uniformly distributed modulo 1 in $'.
^ 1^1 = 1 "^
(c) Theorem (5.4) can easily be extended to polynomials f(Y) of the form 1 k
f(Y) = 2 ex Y^ + a .
VI. SOME METRICAL THEOREMS
In this section we summarize and improve some of the m a m results of [7]. The measure on $' which we use here is the Haarmeasure |i defined in section II.
In order to state the following theorems without taking up too much k
space we suppose that W is a disc having measure q for some integer k,
(
\00 c(p.). . is a sequence of continuous lunctions from T into $ , 4- • = ^ i ' i = 1 • ' i j j
+ + +
(p.-cp., I is the set of positive integers, A ^ I x I containing the diagonal, A(n) = card {(i5J)|(i,j) £ A, max (i,j) g n| and a. . = deg(4j. .(a)-4;. .(p)) - d.eg(a-R), where a for (i,j) ^ A will be
i>J i j j i j j supposed to be independent of cc and p in ^.
oo _ ,
Theorem (6.1 ). If E n ''^ A(n) < oo and a. . a -k for (i,j) ^ A, then oo ^='' "^' "^
((p.(a))._. is uniformly distributed modulo 1 in $' for almost all a £ "i'.
From this theorem due to De Mathan [14] and theorem (3-2) we obtain the following result.
oo ^ X
Theorem (6.2). If S n A(n) < oo and a. . — co as max (i,j) -* oo for (ii)j) t Aj then ([cp. (a) ]) • _^ is uniformily distributed in $ for almost all a £ I-'.
In [7] we introduced a notion of almost uniformly distributed se-quences in $', We now give a new and stronger definition of almost uni-form distribution in $' which is the analogue of Pjateckii-Sapiro's notion of almost uniform distribution modulo 1 in R. See [I8].
A sequence (a.)._^ in $ is called almost uniformly distributed modulo 1 in $' if there exists an increasing sequence (n, ), _, in I such that for every non-zero A £ $
-1 "^
lim n, E e(Aa. ) = 0. k-'oo 1 = 1
Put S (A,a) = £ e(Acp.(a)) where 0 ^ A £ $ and a £ T. Then analo-guous to the first ~ part of the proof of theorem (2.6) in [7] one can show that for all A £ $, A ?^ 0
f |s^(A,a)|2 dii ^ ci^ A(n).
If A(n) = o(n ) for n-oo, then it follows that lim | S 2 |n S (Z.,a)ld)i=0 J . /- n 1 ' n-»oo ^ 1 = 2
and by Patou's lemma we obtain the following ^ improvement of theorem (2.6) of [7]»
2
Theorem (6.3). If A(n) = o(n ) and a. . a -k, (i,j) £ A, then for 6o' ^
almost all a £ ^' the sequence (9.(a))._ is almost uniformly distributed modulo 1 in $'
/ \00
Let (A.j._ be uniformly distributed miOdulo M in $ for infinitely many
M £ $. Put ^ = $', 9.(a) = A.a for ex £ $' and A(n) = { (i, j ) | A. =A . , i,j£l"*'|. Then one can show that all conditions of theorem (6.3) are satisfied.
We thus have the follov;ing improvement of theorem (3.1 ) of [ 7 ] . Corollary (6.4). If (^•)-_^ is uniformly distributed modulo M in $
for infinitely many M £ $, then (A.a)._ is almost uniformly distributed modulo 1 in $' for almost all a £ $'.
The follov;ing theorem im,plies that the conclusion of corollary (6.4) and hence the conclusion of theorem (6.3) cannot be imiproved, i.e.
the term „almost" cannot be omitted.
Theorem (6.5). There exist a uniformly distributed sequence (A ) . ^ in $ and a set 0 c $' ha.ving positive measure such tnat for each a £ 0
-1 "
lim sup |n S e(A.a)| > 0.
VII. THE MEASURE THEORETIC APPROACH TO UNlP0Ri4 DISTRIBUTION IN GP[q,x].
In this section we will discuss the contents of [4]. We do this by describing each section of this paper separately.
1. Introduction. We warn the reader that the notation of this paper deviates strongly from the notations of the previous chapters. In particu-lar, the measure [x here is not the Haarmeasure in $' used in the fore-going sections.
2. The measure of Banach-Buck. Let R be the Boolean algebra generated by all finite subsets of $ and arithmetic progressions in $, i.e., subsets of $ of the form { A N + B | N £ * } , A,B £ $. Using an analogue of Buck's
density on R and the Caratheodory extension of this density on a class D of subsets of $ v/e obtain a finitely additive measure |i defined on D .
3. Construction of special sets. It is shown that R c D . For •^ o ^ o
instance the infinite set of irreducible polynomials in $ is an element of p
D \ R , and so is the set |A |A £ $| .
4. Limit sets. Here we prove the fact that the measure ^ on D assumes all values of the closed unit interval [0,l],
5. Non-measurable sets. The results of this section imply that D is properly contained in the powerset of $. If a £ $' is irrational and
deg a > 0, then the set {[C(N]|N £ $} is not measurable. This set is in fact
extremal, i.e., it has outermeaaure 1 and innermeasure 0. Other examples are also given.
6. Integration. Using the measure |i and the notion of a (finite) partition of $ into measurable sets, we define an integral on $ by means of upper and lower sums.
7. Measurable functions. A real-valued function cp on $ is said to be measurable if JA|(p(A) < c} £ D for all real c. A complex-valued function is measurable if its real and imaginary parts are. Examples are given,
8. Uniform distribution. Here the main theorems ares
Th eorem (7.1). A sequence (A.)._ is uniformly distributed in $ if and only
^^ lim n"^ ? X (A.) = n(ï)
n—oo i = 1
for each ¥ in D . Here X,„ denotes the characteristic function of ï c $ . o 'F
Theorem (7.2). The sequence (A.)°°_. is uniformly distributed in $ if and only if for each complex-valued integrable function cp defined on $
-1 ^ r
l i m n E cp (A. ) = cp dji .
The analogy of t h i s theorem with theorems ( 1 . 2 ) and ( I . 4 ) i s immediate.
Certain consequences of these -tineorems constitute the l a s t part of t h i s section of [ 4 ] .
REFERENCES
Buck, R.C., The measure theoretic approach to density, Amer. Journ, of Math,, 68, 560-580 (1946),
Carlitz, L., Diophantine approximation in fields of characteristic p, Trans. Amer. Math. S o c , 72, 187-206 (1952).
Corput, J.G. van der, Diophantische Ungleichungen I, Acta Mathematica, 56, 373-456 (l93l).
Cugiani, M., Successioni uniformemente distribuite nei domini p-adici, 1st. Lombardo Accad. Sci. Lett. Rend. A, 96, 351-372 (1962).
Dijksma, A., The measure theoretic approach to uniform distribution of sequences in GF[q,x], Mathematica, vol. 11 (34), 221-240 (1969).
, Uniform distribution of polynomials over GF[q,x|in GF[q,x], Part I, Nederl. Akad. Wetensch. Proc. Ser. A, 72, 4, 376-583 (1969).
, , Part II, Nederl. Akad. Wetensch, Proc. Ser, A, 73, 3, 187-195 (1970).
, Metrical theorems concerning uniformi distribution in GP[q,x] and GFJq,x}, Nieuw Archief voor Wisk. (3), 18, 279-293 (l970).
and Mei.ier, H.G. , Note on uniformly distributed sequences of integers. Nieuw Archief v. Wisk. (3), 17 , 210-213 (1969).
Eckmann, B,, Uber monothetische Gruppen, Comment. Math, Helv,, I6, 249-263 (1943/44).
Eynden, C.L. van den, The uniform distribution of sequences, Dissertation, Univ. of Oregon (1962).
Gotusso. , L, , Successioni uniform.emente distribuite in corpi finiti, Atti del Seminarie Math, e Fis., vol. 12, 215-232 (1962/63).
nd
Hasse, H,, Zahlentheorie, Akademie Verlag, Berlin, 2 ed.
Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, Oxford Univ. Press, 4"*^^^ ed. (196O).
Hodges, J.H., Uniform distribution in GF[q,x], Acta Arithmetica, 12 , 55-75 (1966).
, On uniform distribution of sequences in GF|q,x} and GF[q,x], Annali di Matematica, Ser 4, 85, 287-294 (l970).
Kemperman, J.H.B., On the distribution of a sequence in a com.pact group, Comp. Math., I6, 138-157 (1964).
Khintchine, A.Ya., Continued fractions, P. Noordhoff, Groningen (1963). Kuipers, L., A remark on Hodges' paper on uniform distribution in
Galois Fields (Abstract presented by title), Notices Amer, Math. S o c , 15, 120 (1968).
and Uchiyama, S,, Notes on the uniform distribution of sequences of integers, P r o c Japa.n Acad., 44, 608-619 (1968), Mathan, B, de. Sur un théorème miétrique d' équirépartition mod 1 dans un corps de series formelles sur un corps fini, C.R. Acad. So. Paris, Ser. A, 289-291 (l967).
Mei,] er, H. G. , On uniform distribution of integers and uniform distribu-tion mod 1, Nieuw Archief v. Wisk. (3), 18, 270-278 (1970).
and Dijksma, A., On uniform distribution of sequences in GF[q,x] and GF|q,x|, Duke Math. Journ,, 37, 3 (l970).
Monna, A.F., Sur une transformation simple des norabres p-adiques en nombres reels, Nederl. Akad. Wetensch. P r o c Ser. A, 55, 1-9, (l952). Nagell, T., Introduction to number theory, New York, John Wiley (1951). Niven, I,, Uniform distribution of sequences of integers, Trans. Amer. Math, S o c , 98, 52-61 (196I).
Pjateckii-Sapiro, I.I., On a generalization of the notion of uniform distribution of fractional parts, Matem. Sbornik, JO (72), 669-676 (1952) (in Russian).
Rubel, L.A., Uniform distribution in locally compact groups. Comment Math. Helv., 39, 253-258 (1965).
uchiyama, M. and Uchiyama, S. , A characterization of uniform.ly distributed sequences of integers, Journ. of the Fac. of Sci., Hokkaido Univ., Ser. 1, XIV, 238-248 (1962).
Uchiyama, S., On the uniform distribution of sequences of integers, P r o c Japan Acad., 37, 6O5-6O9 (1961).
Weyl, H. , IJber die Gleichverteilung von Zahlen mod Eins, Math. Ann., 77, 313-352 (1916).
SAMENVATTING
In dit proefschrift bestuderen we de gelijkverdeling van rijen van elementen uit twee speciale groepen. De eerste is de additieve groep van het locaal compacte lichaam GF{q,x| dat bestaat uit formele Laurent reeksen in x over het eindige lichaam GF(q). De tweede is de additieve groep van de discrete ring GF[q,x] van polynomen in x over GF(q).
In hoofdstuk II bestuderen we gelijkverdeling modulo 1 in GF{q,x| en gelijkverdeling in GF[q,x] als speciale gevallen van de theorieën van gelijkverdeling in compacte en locaal compacte groepen. Door echter ge-bruik te maken van de kenmerkende structuren van beide groepen kunnen we tot resultaten komen, die geen analogon in de algemene theorie hebben. Dit blijkt bijvoorbeeld uit de resultaten van hoofdstuk V. In dat hoofd-stuk onderzoeken we de gelijkverdeling van rijen voortgebracht door polynoffien over GF{q,x}.
Soms kunnen stellingen uit de klassieke theorie van de gelijkverde-ling modulo 1 in R, de reële getallen, en uit de theorie van de gelijk-verdeling in de gehele getallen, vertaald worden in stellingen over gelijkverdeling modulo 1 in GF{q,x| en gelijkverdeling in GF[q,x]. Zie de hoofdstukken II, III en IV waar de verbanden, die er bestaan tussen de vier begrippen, nagegaan worden. Sjjeciaal verwijzen we naar hoofdstuk IV waar we aantonen dat er een afbeelding x • GFJq,x| -• R bestaat die modulo 1 gelijkverdeelde rijen in GF|q,x| overvoert in modulo 1 gelijk-verdeelde rijen in R.
In hoofdstuk VI worden maattheoretische uitspraken over gelijkverde-ling in GF{q,x| en GF[q,x] bewezen. De maat die we daar beschouv;en is de Haarmaat op de additieve groep van GP{q,x|. In hoofdstuk VII definiëren we een eindig additieve maat op een ring van deelverzam.elingen van
GF[q,x] en gebruiken we deze maat om tot een integraal criterium te komen voor gelijkverdeling in GF[q,x].
BIOGRAPHY
The author of this thesis was born in Nijkerk, Gld., in 1944» In 1961 he finished the HBS in Amersfoort. During I96I-I962 he attended the highschool in Elkhorn, Wisconsin, as an American Field Service exchange student. He entered the University of Technology of Delft in I962. He graduated in miathematics in 1967. During the course of his study he attended the Southern Illinois University, Carbondale, Illinois, where he obtained an M.A.-degree in mathe-matics. He was appointed as a member of staff in the department of mathematics of the University of Technology, Delft, in I968.
• ^
3.
7. De Féjer transformatie g=Tf op L (-00,00) gedefinieerd door °° . 2,
M - ^ f
^
i
^
^ f(t)dt
.00 (x-t)'
heeft als rechtscontinue spectraalontbinding
I -I 0 2-2p, 0 ^ li g 2 ^ 5 2 li < 0 'V- =
waarbij D de Dirichlet transform.atie is gegeven door
oc
(D f)(^) = 1 r ^ i ^ « ( ^ - ^ ) f ( t ) d t
a ^ J X—ü— 0 0
Vgl. Dunford, N., Spectral theory in abstract spaces and Banach Algebras, Proc, Symp. Spectral Th, and Differential problems, Oklahoma Agricultural and Mechanical College, 1951.
Chako ) beschouwt de integraal U(k) = ƒƒ g(x,y)e ' dxdy, waarbij D
D een compact gebied is in E„ en de reële functies g en Ji^ op D aan
zekere voorwaarden voldoen. Zijn eerste stelling, dat de „belangrijkste" bijdrage tot U(k) komt van kleine omgevingen van de kritieke punten van 0 in D, is onjuist.
'') Chako, N. , Asymptotic expansions of double and multiple integrals occurring in diffraction theory, J. Inst. Math, Applies, ±, 372-422 (1965).
De onderafdeling der wiskunde van de TH Delft dient het aantal boeken en dictaten, dat zij momenteel voor het propaedeutisch onderwijs nodig meent te hebben, te beperken.
De q -dimensionale Euclidische ruimte E is ontbonden in twee orthogonale
o q "^ o
deelruimtes E en E met dimensies q^ > 2 en q„ ^ 2. Q zijn de
een-q-, ^2 ^ ~ ^ °i
heidsbollen in E , 1=0,1,2, en dw is het oppervlakte element van Q
^i q-i
Voor ^ op Q geldt o
« -g-q -1 ^ 0 , m oneven,
Q const- P (l-2t^), m = 2n.
Hierbij is ^ = tE,^ + l'l-t ^2» ° = ^ = ^ ' ^i °P ^ ' i = 1,2, en zijn
( \ i
P^°"P^(t) en C'^(t) de Jacobi en Gegenbauer polynoom.
Vgl. Braaksma, B.L.J. en Meulenbeld, B., Jacobi polynomials as spherical harmonies, Rederl. Akad. Wetensch. P r o c Ser. A, 72,384-389 (l968).
Zij f een continue functie op [-1,1]. Dan geldt voor Re a > en Re p >
--g--1 --g--1 n waarbij = H(f) P.^°''P^(l-2t^),
(f) _
ITTIr(a+^)r(p+j.)r(2a+2p+2) (2n)!
{ ^^^) c't^^^\t){^-t^f^^^
2 r ( a + p + l") r(2n+2a+2R+2) ^.
2n dtUit deze formule kan men een integraalvoorstelling afleiden voor het =t p("'P)(l-2t2) p(«'P)(l-2s2).
proauci
n n
Vgl. Dijksma, A. en Koornwinder, T.H.,Spherical harmonies and the product of two Jacobi polynomials, zal worden gepubliceerd.
Zij 6(n) = sup { |q n-4L(n,p)| ; p £ $', k = 1,2,...j de discrepantie van de rij (^ = (oi^T?^^ in $' en zij d(n) = sup { \ (p-a)n- 4 (a,p,n)| t [a,p) c [0,l)| de discrepantie van de corresponderende rij 4^ = (T(a.))°°_. in R. Dan geldt de volgende betrekking tussen 6(n) en d(n).
6(n) s d(n) < 6(n) $2 + f ^ ^ : ^ log rf-y ] .
l log q ^ 6(n) )
Stelling (4.3) van het proefschrift kan met behulp van deze relatie op een kwantitatieve manier bewezen worden.
Vgl. Meijer, H.G., The discrepancy of a q-adic sequence, Nederl. Akad. Wetensch. P r o c Ser. A, 21, 54-66 (1968).
Zij f een stijgende, differentieerbare, niet negatieve functie op [0,oo) die voldoet aan (i) f' is monotoon, (ii) lim f'(x) = 0 en ( i ü ) lim x f' (x) = 00.
X->oo X-»oo
Dan is de rij (T ([ f(i)]))._ gelijkverdeeld in $.
Zie Dijksma, A., Uniform distribution modulo 1 in R and GF|q,x} and uniform distribution in I and GF[q,x], zal worden gepubliceerd.
De stelling van Uchiyama ) dat, als A = (a.)._ een rij is van positieve gehele getallen en als B = |a. | 1 = 1,2,... | 3anacli-Buck maat 1 neeft, dan is A gelijkverdeeld in I, is niet juist. Wel geldt dat de elementen van B zo geordend kunnen worden dat er een gelijkverdeelde rij ontstaat.
) Zie [20] in de referenties van het proefschrift Vgl, Dijksma, A. , en Meijer, H.G., Note on unifo.rmly
distributed sequences of integers. Nieuw Arch. V. Wisk. (3), 11, 210-215 (1969).
Het bewijs van stelling 6 in het artikel van Kuipers en Uchiyama ) is onvolledig.
Bij de bepaling van de salarisanciënniteit van een werknemer bij een overheidsinstelling dienen de militaire dienstjaren van deze werknemer volledig te worden ir;eegerekend.
Het is verheugend, dat de lezers van de kranten Algemeen Handels-blad-NRC en NRC-Algemeen Handelsblad hun voorkeur hebben uitge-sproken voor een voortzetten van het publiceren van het
