2) 0Sah Z &(' ξ i@j€k ]l 2 M 4 F+G 4 F M / ¬ ?aH@¶ ' δ α >

(Hajime Kaneko)

1

! "#%$

&('

x

^)+*,.-(/

x = u(x) + ε(x)

0.1325476

'

u(x)

^8:9<;

&7'

ε(x) ∈ [−1/2, 1/2)

=<>@?:A3)CBED5FCG

u(x)

⁰

x

^H

6 'JILKNM

95;5/

ε(x)

0

x

^H7O

'JILKNM

B:P3F+GNQRQTS(/

|ε(x)|

⁼

x

⁾

UWVYX:Z 6 '\[^]

x

D+SLH<_L`aSabJF+G

1

^9dcfeNg

Z

&R'

α

M

0

^S\h

Z

&L'

ξ

^i(jEk

].l

2 M 4

FYG^Q<H

M

gT/

'

m

(ε(ξα n )) ∞ n=0

^ian(o

[−1/2, 1/2)

^pJSL>@q

K(r 4 F

[EsJtT[

H7uJBavxwW=

>xyN)L=5zR{a|3S}baFYGa~

]

)/€Q<H

' m H

limit point

L‚WHxƒ:„\i\nxo

[−1/2, 1/2)

^)<8

Z

-…R†

[Ws(t.[ V

><y€)5=+‡

]ˆl

- Z h Z

GxQCH5O@‰LS:=+/

baFTŠR‹

0C1W274(Π' A '

α > 1

^)T*,-

|ε(ξα n )|

^H

U

e:H

limit point

0

Ž [^]ˆ( 4

F:Q

M 0J‘“’ M 4

FCG(”xH

2

P\)T/

Π' A '

α

0• Z 2 ':– 0

—J˜ 4

F+G

QYH@™aH5šc“S3=/€9}›œ‡

]l

- Z

FTžxŸW8a95; :¡W)+¢

4 Q M

i(Sdg@FY£

& 0¤R¥x4

FCG\D¦</

α

^i7>5y:H

&7'

H5§x„W)C¨J©

4

F+ž(Ÿ

0.ªL«

FCG

0

^SJh

Z &('

ξ

^0f¬ ?d)+­\®7S:¯5B°, 2 M

g7)T/L±

M ® s

4L«

-LH

&('

α > 1

^)+*

,f-

(ξα n ) ∞ n=0

^H@O

'LIxK

i7><q K<r 4

FJQ

M i

Koksma [7]

)R9@²T-@³:´a~

l 2

Gxµd)

1

^9WcˆeEg

Z ¬

?aH@¶

'

α

0

¯5B 4 F M

/J±

M ® s 4L«

-LH

&('

ξ

^{)5· Z} -\¸Yq}H7žRŸWi@¨a©

4

F:Q M V

¸7¹\)T¢d~

l 2 G M

Q(º(i7/5»3H(¼

½

i5¢

4 9 t )

Koksma

H7B(¾E)78:¿xFCÀJÁRƒL„}=5>@yE)(=5ÂRS3=xh Z G

ÃRÄ

1.1. 1)

&('

α > 1

^i@j€k

]l 2 M 4

F+G

4 F M /

&('

ξ 6= 0

^S(Å

Ž H

Æ7ÇaÈ 01\2@4 V

HLi7É3hE›

M VYÊRË(ÌLÍRÎ 4

F+G

lim sup

n→∞

ε(ξα n ) ≤ 1

2α − 2 . 2) 0

^Sah

Z &('

ξ

^i@j€k

]l 2 M 4

F+G

4 F M / ¬

?aH@¶

'

δ

^)+*,.-7¶

'

α > 1

^{S(Å Ž H Æ7ÇaÈ 01\2@4 V HLi7É3hE›}

M

VYÊRË(ÌLÍRÎ 4

F+G

lim sup

n→∞

ε(ξα n )

≤ 1 + δ 2α − 2 .

1

Å Ž

α

⁼

1

^{9WcˆeEg A SabJF} F+GaQRQTS(/ A SabJF

Pisot-Vijayaraghavan

'

(

^{Å Ž}

PV

'M 4 F

)

^839@;

Salem

' 0

B

4

F+G

Π' A 6 '

α

ⁱ

PV

'

SabJF

M

=/

α

ⁱ

1

^9WcˆeEg

Z &('

SN, [ V

α

0

›:HR*\i 4L«

-

1

^9WcœO€~

Z Q M 0 t

GE)

2

^ÅLpJH

' =

PV

'

SabJF+GdD

2 /

1

^9WcˆeEg

Z Π' A 6 '

α

ⁱ

Salem

'

SabJF

M

=/

K 0

›

α

^H:HR*\i

4L«

-

1

^{Å Ž Sab^c /}

[

·R*

i

1

^{H 0 É3hE›}

M V

1

^·a·WQ

M

0 t G

PV

'

α

^)+*,.-(/

α n

^{H Li}

6 '

)7hJF:Q M+[^]

lim n→∞ ε(α n ) = 0

M

hLFJQ M i

KJ[

FCG

PV

' H «

g"!d)·

Z

-@/

Hardy [6]

^{=YÅ Ž H5ž7Ÿ 0.ª}

«:2

G

#$

1.2.

Π' A '

α > 1

^i7jdk

]fl 2 M 4

FTG

4 F M /

0

^S3h

Z &x'

ξ

ⁱ

ÍRÎ

,.-(/

n→∞ lim ε(ξα n ) = 0

M

hLF&%'(

K

Š(‹€=T/

α

ⁱ

PV

'°M

hRFJQ

M

S:bRFCG3QTH

M g

ξ ∈ Q(α)

^S

bJF+G

Pisot [8]

^=aQYH@žxŸ

0 Å Ž

H}9

t

))+*°, 2 G

#$

1.3.

Π' A '

α > 1

^8a95;

0

^Sah

Z &('

ξ

^i@j€k

]l 2 M 4

F+G 4 F

M / ' m

(ε(ξα n )) ∞ n=0

^{i-,/. Ì H}

limit point

^,

[ 2 h Z 2

PRH%'(

K Š

‹d=/

α

ⁱ

PV

'

Sab^c

[ ·

ξ ∈ Q(α)

M

hJF:Q

M

SabJF+G

0

v(/

lim n→∞ ε(ξα n ) = 0

M

hJF21 3

'

α > 1

^H

ÍRÎ 4

= Æ

´(SabJF+G

Salem

' H «

g5!E)-6E,.-3=Å

Ž

H@žxŸ\i5‡

]l

- Z F+G

#$

1.4.

Π' A '

α > 1

^i@j€k

]l 2 M 4

F+GQYH M g ¬

?aH@¶

'

δ

^)+*

,.-

0

^Sah

Z &('

ξ = ξ(δ)

0

­+7EQ

M

)L9x²-

lim sup

n→∞

ε(ξα n )

< δ

M

S^gxF

2

P:H %'(

K

ŠL‹N=5/

α

ⁱ

PV

' D 2 =

Salem

'M

haF\Q M S

bJF+G

2

8:9<;

Π' A '

α > 1

^i5jWk

]ˆl 2 M 4

FG}QxQ+S

α

⁼

PV

' S V

Salem

' S V

h Z M 4

F+GJB(¾

1.4

^)L9Wc

lim inf

ξ∈R × lim sup

n→∞

ε(ξα n )

> 0

SabJF+G°QYH

È

)78:¿xF=+>aH-

0 Ž [^]ˆ(

,59

t G

α

^i-,x¾

'

H@§L„}=

Dubickas [4]

^{)L9x²-(Å Ž} H@žxŸ\i5³3´3~

l 2 G

#$

2.1.

^,R¾

'

α = p/q > 1

^i7jdk

]fl

2 M 4 F

(p, q ∈ N)

^{G 4 F}

M / ¬

?aH

0

^Sah

Z

ξ

^)+*,.-

1

q

,/ 6

E 1 (X )

^{=Å Ž} )L9x²-(j€k F+G

E 1 (X) = 1

2X 1 − (1 − X )

∞

Y

m=0

1 − X 2 m

! .

QHTž<Ÿ

0

>Yy@»

'

d

^H

Π' A '

α

^{) )*}

4

FfG“Q5QfS

È

ρ i (X 1 , · · · , X d )(i = 0, 1, . . .)

^{0 Å Ž H ÇaÈ )L9WcœB+ 4 F+G}





d

Y

j=1

(1 − X i Y )





−1

=

∞

X

i=0

ρ i (X 1 , · · · , X d )Y i .

α

^H-

0

α 1 , . . . , α d

M

8:gf/

α

^H

U

O È 0

a d X d + a d−1 X d−1 +· · · + a 0 ∈ Z[X ]

M 4

F+G

#$

2.2.

Π' A '

α > 1

^{i@Å Ž H5Šx‹}

01\2@4 M 4

F+G

1. |α i | > 1(i = 1, 2, . . . , d);

2. P d

i=1 |a i | ≤ |a 0 |;

3. 0 < ρ n+1 (α −1 1 , · · · , α −1 d ) ρ n (α −1 1 , · · · , α −1 d ) ≤ 1

2 (n = 0, 1, . . .).

4 F M /

0

^Sah

Z ¬

?aH

&('

ξ

^)+*,.-

lim sup

n→∞

ε(ξα n )

≥ 1

|a 0 | E d (α −1 1 , · · · , α −1 d ).

2

,/ 6 '

E d (X 1 , · · · , X d )

^=Å

Ž

H}9

t

)TB+W~

l

F+G

E d (X 1 , · · · , X d ) =

d

X

j=1





 Y

1≤i≤d i6 =j

1 X j − X i





 X j d−1 E 1 (X j ).

3

Π' A '

α

^)<·

Z

-(/R™aH7B(¾

2.2

^S

• Z 2 0

”xH€DRD

• Z

F+GdQ

H@™:S3=

α

^H:HR*€=

4L«

-

1

^9WcˆeEg

Z M

B 4

F+G@B(¾

2.2

0

³3´

4 F 2

P\)T/

α

^0•

Z 2 ':–

S

ξα n

^H7O

'JILK

0J4

G

D¦</

α

ⁱ

'

N (≥ 2)

^Sxb(Ff§@„

0!

k<FG

N

^"#$

M

=/ ƒ7„

{N i |i ∈ Z}

0&%('

)

V)

/

{0, 1, . . . , N − 1}

0

digit

M 4

F*#+$7S:bLFCG

ξ

0

N

^,

4 F M

Z t.- Ë =

ξ

^H

N

"(#/$3)78:¿xF

digit

^H

shift

M

*(0,.-

Z

F+G

4

h1 ) /

¶ '

ξ

^H

N

"(#/$xH32 M · 0

ξ =

m

X

i=−∞

s i N i

8d› GEQxQCS7/

s i

⁼

ξ

^H

N

"#+$a)@8J¿(F

i

^3H

digit

^FCG

ξN

H#/$:=

ξN =

m+1

X

i=−∞

s i−1 N i

)79²œ-jak ] l

FG.9²œ-</

ξN n

^HO

'7I@K 0 @4

F(Q M =

,

' m

(s i−n ) −1 i=−∞

0a«

F:Q

M ) 0 h ] h Z

G5>@yL»

'

d

^H

Π' A '

α

)-6E,.-(/\QYH#/$xH 0

¨ 4

F+GNDC¦</

%'

{ν i |i ∈ Z}

^{0 Å Ž H}9}

t

)TB:P3F+G

ν i = − 1 a 0

d

X

j=1

α i+1 j Y

1≤l≤d l6=j

1 α −1 j − α −1 l .

QYH

%'

=Å

Ž

H

½ 01\2@4

>LH

%' M

,.-\¿Sdg@F+G

Ä

3.1. 1)

¬

?aH 6 '

i

^)+*,.-

a d ν i+d + a d−1 ν i+d−1 + · · · + a 0 ν i = 0.

2) ν 0 = 1 a d

, ν −1 = ν −2 = · · · = ν −d+1 = 0.

·7)T/

ξ > 0

^)+*,.-€QYH

':–

H#/$3)78:¿xF

i

^\H

digit

0 Å Ž

H}9

t

)TB:P3F+G

s i = a d u(ξα −i ) + a d−1 u(ξα −i−1 ) + · · · + a 0 u(ξα −i−d ).

i

^\H

digit

^)<·

Z

-(/RÅ

Ž

H

½

i5¨:©

4

F+G

Ä

3.2. 1)

⁽

K

eEg<h 4L«

-LH '

i

^)+*,.-

s i = 0.

2) d ≥ 2

^h

]

|s i | < 1 2

d

X

l=0

|a l |.

3) α

ⁱ

'

S:h Z h

]

/ ¬

?:H 6 '

M

^)*°,f-

' m

(s i ) M i=−∞

^=:A

S3=xh Z G

9x²-R/RÅJp:H

½

839@;5»3H(¼

½

)J9€cˆƒJ„

{ν i |i ∈ Z}

0*%3'

) ) /

ƒL„

{j ∈ Z ||j| < 1 2

d

X

l=0

|a l |}

0

digit

M 4 F #/$Ri@B+LSdg@F+G

ÃRÄ

3.1.

¬

?aH 6 '

n

^)+*,.-

ξα n =

∞

X

i=−∞

s i−n ν i .

~ ] )

u(ξα n ) =

∞

X

i=0

s i−n ν i ,

ε(ξα n ) =

−1

X

i=−∞

s i−n ν i .

½ 0f• Z M

i digit s i

0 7 4 M

2.2

0 —

[1] Bertin M.-J. et al,Pisot and Salem numbers, Birkh¨ auser Verlag, Basel, 1992.

[2] Boyd David W., Transcendental numbers with badly distributed pow- ers, Proc. Amer. Soc. 23 (1969) 424-427.

[3] Dubickas Art¯ uras, Arithmetical properties of powers of algebraic num- bers, Bull. London Math. Soc. 38 (2006) no.1 70-80.

[4] Dubickas Art¯ uras, On the distance from a rational power to the nearest integer, J. Number Theory 117 (2006) no.1 222-239.

[5] Dubickas Art¯ uras, There are infinitely many limit points of the frac- tional parts of powers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no.4, 391-397.

[6] D. H. Hardy, A problem of diophantine approximation, Collected pa- pers of G. H. Hardy I, Clarendon Press, Oxford 124-129.

[7] J. F. Koksma, Ein mengen-theoretischer Satz ¨ uber Gleichverteilung modulo eins, Compositio Math. 2 (1935) 250-258.

[8] Pisot C, R´epartition (mod 1) des puissances des nombres r`eels, Com- ment. Math. Helv. 19(1946) 153-160.

[9] T. Vijayaraghavan, On the fractional parts of the powers of a number, J. London Math. Soc. 15 (1940) 159-160.

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606-8224

=

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4

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e-mail: kanekoha@math.kyoto-u.ac.jp