Remarkable sets of algebraic numbers in complete fields

RX}/IARKABLE SETS OT AIGNBRAIC NINtsERS

IN COMP]-,ETE FIE]-,DS

by

C.

J.

^{STIYTH B.A. (Hons.} ) (a.N.U. ₎

Departnent

of

Mathematics,

University

of

^Adelaide.

Ihesis

submitted

for

a l\[aster

of Arts

^Degree

at

the University

of

Adel-aiclet

November 1969.

c01{îriNTs

Chapter 1 : INÎRODUCTI0N

1A 1B 1C 1D

Definitions and Sim.ple Properties of ^S ancl f Historical tsackground

Generalisati-ons of S and f ; ^Aim of this ^Thesis Contents of this ^Thesis

Chapt

er 2:

^{NOIATION Al'{D AUXIIIARY IIIXORH'ÏS}

1

1

3 + 6

2A 2B 2C 2D

Notation

Prelininary ^Lemmas

Theorems from Algebraic Number Iheory Artinf

s

Decompositlon Theorem

B B

11

1g 20

2+

24 27 31 36

+2 42 +3

chapter

3:

^{THE SETS Sq(I{,4) AND Tq(X,n)}

3¡, Definitions

^{and nxamPles}

3B

^Hierarchy

of

the Sq(K'A) and the Tq(K,A)

3C

Non-enptiness

of

the Sq(I{rA)

3D

^Contents

of

the ^Tq(K,A); the sets ^Tq(K)

Chapter

4:

DISTRIBITIIOI{AI PROPERTIES 0F THn

sq(K,A) ^{AlrD THE} rn(r,A) Introduction

+A

4B The sets ^Sq(K)

4C 4D

6A,

6B 6C 6D

References:

I/IORE PROPERTINS OF S

IOR K q-FRIENDIY

Generalisecl Distributlon ^Theorems A Seconcl lormulation

+9 60

Chairter 5: ^{CIOSURI RISUITS FOR S}

(r,4)

q 5A

5B

tr^

5D 5E qTt

5G

Chapter 6:

Introducti-on

Definitions

and Prel-iminary lemmas

Compactness Results

Closure

of

the sets SfCr,¿) Friendly

lields

Closure

of

the ^Sn(f ,A)

for

c¿-friendly Fiel-ds A Possible nxtension

(r,n)

AND

r

^(rc,A)

6+

6+

6+

69 76 77 B2 B5

BB BB B9 103 108

110

cI q.

I ntroduction

Preliminary ^Results

Derived Sets

ot

^S,r(f _,A)o

for

I{ q-friendly

The Derived Set

of

Tq(K,A)o

for

K q-fz'iendly

ÁDE

Ø of F

t)

ù

eo

AI ]L F

SUII'IARY

lYe denote b¡r $ the set of algebraic ^numbers I, of

mo'ihrlus greater than one, for which the differences ^Ut-rr.

betweun gt _and the nearest ra bionaf in'teger a' tend to

zero. This set, for ^which there aTe several ^equivalent definitions, has been studied e;rtensively by Pisot ^(see for

example tAJ), ^Salem [B']rVi jayaraghavan ando-bhers. It ^has

many interesting properties: for ^instance it is cl-oseil, ^and has non-empt¡r derived sets of all finite orders. ^Closely

associ-a'bed wi-bÌr S is another set of algebraic ^numbers It

whose el-ercents have somewhat sj-mil-ar proilerties to those of S (for a definition of ^f , see fu-l). ^{For" exampfe,} for ^any

ÐøT ancl ^€7 O, there are ^{numbers À such} that the ^{numbers À9n}

(n=Or1 ,... ) are ^al-l- wittrin t of an integer.

lhe sets S and T are defined rel-ative to the ^rational- field, and, Iie Ín the real com¡rletion of that fiel-cì. ^the aim of this thesis is to ^s'budy sets Sn(fC,A) tend Tn(f'/t)

r¡¡hich are defined relative to ^an arbitr:ary algebraic ^number fieldZyand 1ie in a finite ^extension A of the compl-etion

of K at a spot er ^these sets not only inclr"rile as particul- ar ^cases ttre sets ^{S and} T, but al-so incorporate several particular generalisations of ^{them which have been stuclied'} by Chabauty, Grandet-Hr"rgot and others. ^{trTe nse} the ídeas of

these and other authors (notably Bertrandias)

to

^extend

results for

S and T as

far

as possible

to

^{general S^(KrA)}_q

and

Tn(f,n).

This

at

the ^same time ^provides a unified

treatment

of

those resul-ts which have already been extended

to particular

generalisations

of

^S and T.

In

the

first

two chapters ^we give the introduction, notation and preliminary

resul-ts.

ï/e define the sets

: S^(KrA) _qq and 1^(f rA)

in

Chapter

3,

^{and. prove soine results}

about the hier:a,rchy

of

incl-usions betureen these

sets,

^and

consider the qr-testion

of their

non-emptiness

(or

oth.erwise);

rÃre also give

a

structure theorem

for

the el-ements of

f^(f

_{rl ,A)} which actualJ-y

lie in K. In

^{Cha;cter 4}

r

^we

general:lse some

of

the

distributionel results

which char-

acterise

^ef.ements

of

S and T,

to

^{Sq(K,A) and.} tn(t<,A)

.

^For

some

of

these resufts ^{lve introduce}

a set

S^(i{) _q which ^plays the ^same rol-e

for

^{S^(x,A)} _qq ^a,nd, T^(KrA) as the rational

intege.rs played

for

$ and ^T.

In

Chapter 5r ^{we prove closure}

of

the sets ^{S^(frA),}_rI under eertain assumptlons on

the

nature

of

the base fiel-d ^I(.

Under these ^same assumpti-ons, we give

in

Chapt

er

6 various

results

concerning the cleriveil sets

of Sn(rrl)

ana Tn(K,Â):

we prove

that

the derivecl

set of

^{T^} _q^(lcr

¡)

^{contains rrmostrl}

elements of S^(f q". ,A), ^and that S^(K,A) ^q has ncn-empt¡r derived

sets of all finite ^orders.

References

A C. Pi-sot _, ilQuelques aspects de Ia théorie des entiers

a1.q:6briqLlestr (l\Tontráal _{1963 )} .

B a R. Salem., 'rAlgebraic l{unbers and Fourier Anal-.ysistl

( lleath ¹ 963 )

Stateaent.

I

hereby

certify that this thesis

contains ^no

material- v'rhich has been accepted

for

the award

of

any other degree

or

diploma

at

any

unlversity.

ffurthermorer

to

^the

best

of

my knowledge and belief

, it

^{contai-ns no traterial}

previously published or v'rritten by any other ^persont

except wl-rên d.ue reference

is

^made

in

the

text of

the thesis.

C.J. ^SI{YTH.

Acknowleclqement.

I

wish

to

express my sincere appreciation,to ^my

supervisor,

Ðr.

Jane Pitman

for

her hel p and advice ^vrhile working

for this de.lree. Partlcularly

while

Ï

^u/es

vrriti ng un

this thesis,

her

helpful

suggestions and

constructive

criticsism

have been

of

^{r:reat value.}