Correction to : “Linear fractional transformations of continued fractions with bounded partial quotients”

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

EFFREY

C. L

AGARIAS

J

EFFREY

O. S

HALLIT

Correction to : “Linear fractional transformations of

continued fractions with bounded partial quotients”

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 741-743.

<http://www.numdam.org/item?id=JTNB_2003__15_3_741_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

741-Correction

to:

Linear fractional transformations

of continued fractions with bounded

_partial

quotients

par JEFFREY C. LAGARIAS et JEFFREY O. SHALLIT

RÉSUMÉ. Nous comblons un trou dans la démonstration d’un théorème de notre article cité dans le titre.

ABSTRACT. We fill a _{gap in} the

_proof

of a theorem of our _paper cited in the title.

The

_proof

of Theorem 1.1 of our _paper

[1]

has a _gap, which we fill below.

The _gap is that in the subcase

_{c ~}

0 and qa - 0 the

inequality

(4.11)

given

in

_[1]

does not follow from the

_{inequality immediately preceding}

_it,

and _may not be valid. We

_give

here a corrected

_argument,

which also

clarifies the

_{logic treating}

this subcase

_together

with the subcase

_{c ~}

0 and

qa-pc=0.

Theorem 1.1 asserts the

_inequality

_(4.1),

which states that

It can be derived as follows. Define

The case c = 0 is treated as in the _paper.

_{Suppose c ~}

0. We either have

an infinite _sequence of distinct

_{approximations}

x =

qllqol

I

with

with E -

_0,

or else we have a

_{single approximation}

with

L(1b) = §;

we

reduce the second case to the first

_{by viewing}

it as an infinite _sequence

with the same value

of q repeated

over and over.

_{By taking}

a suitable

sub-sequence we _may reduce to the case where either all of the

approximations

have qa - pc = 0 _or all of them have

qa -

pc #

0,

with E -+ 0.

742

In the first

_subcase,

where all qa - pc =

_0,

the

inequality

(4.7)

in

[1]

yields

since

_{c ~}

0.

_Letting

e - 0

_yields

_(4.1).

In the second

_subcase,

where all qa -

pc #

0,

we obtain as in

_[1]

the

inequality

(4.10),

which states that

We then continue

_{by using}

the

_{triangle inequality}

Here the first term on the

right

side is

equal

to

q[

I while the second term

_{is 1}

where x :=

Qllq1/JlI,

provided

we

pick

the

integer

_p

properly,

and we then have

Multiplying

this

_{by q}

and

_substituting

it in

_(4.10)

_gives

the

_inequality

which

_replaces

_(4.11).

Now

which when used in the

_inequality

_{(4.11’ )}

yields

Now there are two cases.

First,

if

L(8) &#x3E; L(~),

then the

inequality

(4.1)

holds

_trivially,

since 1.

_Second,

_L()

_{then, letting}

E 0 in the

preceding inequality,

_{P g q Y} the becomes 1 in the

L()-E

743

References

[1]

J.C. LAGARIAS, J.O. _SHALLIT, Linear Fractional _{Transformations of} Continued Fractions

with Bounded Partial Quotients. J. Théor. Nombres Bordeaux 9 _(1997), 267-279.

Jeffrey C. LAGARIAS

AT&#x26;T Labs-Research

Florham Park, NJ 07932-0971, USA

E-mail: jclGresearch.att.com Je$rey O. SHALLIT

Department of _Computer Science

University of Waterloo

Waterloo, Ontario N2L _3Gl, Canada