Golomb's self described sequence and functional differential equations

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Golomb's self described sequence and functional differential equations

PÉTERMANN, Yves-François, RÉMY, Jean-Luc

PÉTERMANN, Yves-François, RÉMY, Jean-Luc. Golomb's self described sequence and functional differential equations. Illinois Journal of Mathematics , 1998, vol. 42, no. 3, p.

420-440

Available at:

http://archive-ouverte.unige.ch/unige:12429

Disclaimer: layout of this document may differ from the published version.

1 / 1

ILLINOIS JOURNAL OF MATHEMATICS Volume42,Number 3, Fall 1998

GOLOMB’S SELF-DESCRIBED SEQUENCE AND FUNCTIONAL DIFFERENTIAL EQUATIONS

Y.-F.S.

PTERMANN

ANDJEAN-LUC

RMY

A

sequence(word) W ofpositive integers is

self-described

or self-generating if r(W) W,where r(W) isthe sequenceconsisting of the numbers ofconsecutive equal entries of W.

A

famous self-generating ^bounded sequence is Kolakoski’s 1, 2,2, 1, 1, 2, 1, 2, 2,-.. (see [Ch]). In this paper we consider Golomb’s

I, 2, 2, I, 1, 2,

sequence F, which is the only nondecreasing self-generating sequence taking all positiveintegral values, 1, 2, 2, 3, 3, 4, 4, 45, 5, 5,6, 6, 6, 6, Let

4

^{denote the}

I, 2, 2, 3, 3, 4,

goldennumber. We provethat F(n)

2-4n4- + _l-gn

^h

log

4 +

^{0 log logn}

log²n

where the realfunctionhiscontinuousand satisfiesh(x) -h(x4- 1) (x >_ 0). The methodofproofisintimatelyconnected with the moregeneral problemof character- ising thesolution

E

of an approximate functional integral equation of thetype

E

^(t)

^--ql-4t4-2 f2

^42-*t-’

^E

^(u)du

⁺

^{0 log}

^t*-

²

⁾

whichwediscuss inthe second part of thepaper.

1. Introduction

In

the problem section of the American Mathematical Monthly in 1966, S.W. Golomb [Go] considered the unique nondecreasing sequence

{F(n)}n>_

1,2, 2, 3, 3, 4,4, 4,5, 5, 5, 6,6, 6, 6,7...} "self-described"by thetwoconditions F (1) andF (n)

I{rn

F (m) n

}1

(n ^> 1). At^thetimeheonly requestedan asymptotic expression for^Received1991 Mathematics Subject Classification. Primary 11B83;^{May 1, 1997.} F(n)^F(n)as n

--

^cncxz.

-

^We

⁺

^E(n),haveSecondary11B37, IN37,34K15. ^(1.1)

@^{1998by theBoard}ofTrustees oftheUniversityof Illinois ManufacturedintheUnitedStatesof America

420

where c

t ^2-b, t (v/

--1--

1)/2

^{is the} golden number, and E(n)

o(n4-).

Thefirstpublishedcomplete proofof(1.1)isduetoN.J. Fine [Fi];D. Marcus [Ma]

proposed^aclever heuristical argument: see[P62]foraproofbased on

Marcus’

idea.

More

recently, I.Vardi [Va] asked fora more preciseestimatefor the errorterm E (n)of(1.1). Onthe one hand he could establish

E(n) O

logn] (1.2)

onthe other hand heconjecturedthat estimate(1.2)isoptimal.

CONJECTURE 1. Wehave

E(n)

S2+

\logn (1.3)

Vardi’sConjecture isbasedon a heuristic argument,whichledhim tobe more precise.

CONJECTURE2. Forn > 2wehave

where h(x)

satisfies

and

\ ^logb

--I--

^O (1.4)

E (n)

logn log²

h(x

+

^1)=-h(x)

for

^{x >0,}

Ih(x)l > 0

forx

^{(0, 1). (1.6)}

However, as Vardi himself pointed out, he was

"not

even able to show that lim

supn_

IE(n)l

cxz".

This wasproved by Y.-ES.P6termann[P61],whoshowed thatE(n)

2+(n 4’--)

^forevery > 0. Recently J.-L. R6my [R6]succeededin verifyingthe truth ofConjecture ^1.

Inthispaperweareconcerned withConjecture^2.

We

prove:

THEOREM 1. Wehave

n4,-1 h _{og n}

o-

_{log logn}

+

^{O (1.7)}

E (n)

logn log²n

whereh(x)

satisfies

and

h(x

+

^1)=-h(x)

forx

^>0,

hiscontinuous (1.8)

422 Y.-r.s.

PITERMANN

AND JEAN-LUC

RIMY

Remark 1. Conjecture1, whichweknow is true fromR6my’sresult[R6],implies thatthe function h in Theorem is notidenticallyzero. Itshould alsobe noted that assertion (1.6) in Conjecture 2 is false: we can indeed prove that h(0) 0 (see Remark2 inSection3),andthisshowswith(1.5)and(1.8)that thereisa numberu with0<ot < andh(ot)=0.

Henceweproposetomodifyassertion(1.6).

CONJECTURE3.

If

^{hisas inTheorem thereis anumberotwith0 <ot < and}

[h(x)[ > 0

for

^{x 6 (or,}

+

^or).

^(1.6’)

TherestofSection3 isdevotedtofour other remarks.

The proof of Theorem contains in fact an almost complete treatmentof the approximatefunctionalintegral equation

E(t)

-c-4’t

^4’-2 E(u)du

+

^{0 (1.9)}

,/2

,

^log2

and its summatory counterpart (2.1) in Section 2 just below: this is discussed in Remarks3 and 4.

Butthe errortermbound of Theorem isnotasgoodasthe boundconjecturedin Conjecture2: weextensivelydiscuss in Remark5 theexactand non trivial functional equation

d(t) -t 4’-

.]

^d(u)du

⁺

^{1, 1.1O)}

whichisofatypesimilar to(1.9),and for thesolutionofwhichthe better errorterm boundholds. (By

"non

trivial" wemeanthatthefunctionhassociated tothesolution is notidenticallyzero).

InTheorem 2 of Remark 3 we show that the solution of an equation of type(1.9) satisfies(1.7),with(1.5)and(1.8): ^{a sort}of restricted converseto this isobtained in Theorem 4 ofRemark 6,showingthat a functionE (t)

-

^h(loglog

^/

^log

^{) /}

^log

satisfies(1.9)formanychoicesof the function h.

Notation.

In

thesequel ^weshallfrequently appealwithoutcomment toclassical properties^{of the}goldennumber

4.

^{Althoughall theseproperties}can easily be inferred from the definitionof4,westatebelowafew of them for convenience.

q2= q +

^{1, q(q- 1)= 1, (q-}

¹⁾²⁼

^2-

^q,

^{(4- 1)}

³⁼

^{2q- 3.}

Acknowledgements. Thiswork was made possibleby^atemporarypositionas an invitedprofessor grantedtothefirstauthorbytheUniversit6Henri Poincar6-NancyI.

Both authors arevery gratefultothis institutionfor thisopportunitytoworktogether.

Theyalsowish tothank Ilan Vardi for histhoroughreading of themanuscript^{and his} usefulcommentsandsuggestions.

2. Proof of Theorem 1

Our startingpointisthe heuristicformula that ledVardi to hisConjecture 2.

LEMMA

1. Wehave

E(n)

-c-ePn

^-2 E(m)

+

^{0 (2 1)}

m<cn4- log²

Proof ^Every

^{n > canbe writtenuniquelyas n G(m) r,where0} < r <

F (m),and whereG (m)denotesthepositionof the last occurence of the integerrnin thesequence F.

We

have

F(n) F(G(m) r) m. (2.2)

Nowifwe putR(m)

Zk<m

^E(k),then, from Vardi’s result(1.2),weclearlyhave

R(m) 0

(lor4’m)

^(2.3)

Moreover,

E(n) E(G(m) r) R(m)

(,+o(IR(m)l))

cme - ^{m4’ +}

^0(1).

^(2.4)

(Equation(2.4)caneasily be derived from Vardi’spaper [Va]" see(10)of thatpaper;

or seeLemma3 in[P61].) Thus, with(2.3),(2.2)and(1.2),wemaywrite E(n)

Thisproves

Lemma

1. Nowwereplacethesuminthe approximate functional equation ofLemma byan integral, which is smoother(differentiable)andthus easiertohandle.

Forthis weusearesult ofSegal’s [Se]. ^12]

LEMMA A.

Let

f

^(n)bea

function of

^apositive integralvariable,and suppose

Z ^f(n)

^z(x)

⁺

^E(x),

n<x

424 Y.-F.S.

PITERMANN

AND JEAN-LUC

RIMY

where z(x) is twice continuouslyderentiable, and

z"(x)

is

of

^constantsign

for

x > 1. Then

Z

^E(n)

z(x) ⁺

^{(1 [x})E(x)}

⁺

^{E(t)dt /}

^O(Iz’(x)l)

^{/ O(1).}

LEMMA2.

/f

^the

definitions of

^the

functions

^FandE

of

^{1.1 areextendedtothe}

realarguments ^> byputting F(t) F([t]) ct 4’-

+

^{E(t), thenwehave}

E(t)

-c-4t

^ok-2 E(u)du

+

⁰

a2 log² (2.5)

Proof

^Ifweput

^f(m)

^{whenm orm G(k)+l} for somek, and

f

^(m) 0 otherwise in

Lemma A,

then F(x)

,,,<. f

^{(m) z(x)}

+

^E(x)where

z(x)^Nowcx^ifwe

-

^{letThehypotheses}Do(t) ^of

-c-4t

^Lemma4-A

I

^{arect4-satisfiedE(u)du,andLemma2isproved.(2.6)}

wehave of course

t- )

E(t) Do(t)

+

⁰

\log2t

so that from(2.6)and(2.7)wehave

Do(t)

-c-et

^e-2 Do(u)du

+

⁰

a2 log²

Similarlyif weput

ct

-

D(t)

--c-t

^-2

f

_d2 ^DO(U)du,

wemaywrite

and

D(t) Do(t)

+

^O

\log²

(2.7)

(2.8)

(2.9)

(2.10)

fete-’ (t.O)

D(t)

-c-4t

^4-z D(u)du

+

^{0 (2.11)}

d2

I,

log²

We shall work with the approximate functional-differential equation (2.11), rather than with(2.5)^or(2.8),whichare "functional" butnot "differential".

By

(2.7)and (2.10)it isindeedsufficient toprovethe theorem forDinsteadofE. Ifwewrite

t4,-

D(t) K(t), (2.12)

log

this defines a function K(t) O(1) which is differentiable for every >_ 2. If similarly^wewrite

Do(t) K0(t), (2.13)

log thenby (2.10)wehave

Ko(t) K(t)

+

^{O(l/logt). (2.14)}

LEMMA

3. Wehave

tK’(t) +

^{K(t) 4-K(ct}

^‘-)

^{O(l/logt). (2.15)}

Proof

^{First we}differentiate expression(2.12).

t- t-2 t-2

K’(t)

4- (qb- l)K(t) K(t). (2.16)

D’

(t)

log log²

Thenwedifferentiate(2.9).

ct4,-I

D’(t)

-c^{c -4 4-2}

-

⁽⁴

^(ct4,-1

^2)t0-3

^{)-I .]} Ko(ct-)c(q5

^Do(u)du l)t^-2

log(ct-)

(q

2)D(,_) ^t0-2

^logtKo(ct

4’-’) +

⁰

(

^\log,4,-22

)

^(2.17)

Equating

(2.16)

and(2.17) (with^{a use}ofdefinition(2.12)) yields log

/t) ^t*- ogtt*- ( ^t- )

tl’t) + + ^{---loto-) o}

log² andadivisionby

4-2/IOg

with anappealto(2.14)finishestheproof.

A

keystep in theproofofTheorem isto ensurenowthatK (t)issufficientlynear K

(ct4’-).

This willbe donebyan induction argumenton mfor in intervalsof the form [Nm,Nm+], where

No, N, N2

^{is a}sequenceof positive real numberswith

b--! ^C

Nm cN,+,

^thatis

Nm+ -2Nm , ^(2.18)

forrn >0. Weneed a lower bound on

Nm.

426 Y.-F.S. PiTERMANNANDJEAN-LUC

RMY LEMMA

4.

If No

^{ischosenlargeenoughwehave}

Nm

^{> 3}

’’’

^(2.19)

Proof.

^If

^Q

^:=

4

^-2

1/c

^wehave

N N]

c

^{--q-ONo (4-2N0)}

^(QNo)

,

and ingeneral,ifQ0

Qo2

(=

1/4),

wehave

Nm

^(QNm_)4

(Q(QNm_2)4)

^4,

Q4+42++4’"No Q4N Qo .... _No

> (QoNo)⁴

....

_>

₃₄

⁴

....

provided

No

^{ischosenlargerthan32}

!

^Qo

43 .

^{And wehave}

b2((m

^|)=m(2(|-

c-m)) m

ifm > 1. Thisconcludes theproofif we notethat since

No

^{> 3,(2.19) alsoholds}

form 0. ^[21

We

are now inpositiontoprove:

LEMMA

5. For ^> 3wehave

Proof.

^Let

^{No, N,} N2

^{be a} sequence of positive real numbers satisfying (2.18)form > 0, and with

No

large enoughtoensurethe validity of(2.19). Weshow that thereis anabsolute positiveconstantCandasequenceof realpositive^numbers M0,

M, M

^with

(

^C

)

^{(m>0) (2.21)}

Mm+l

^(C

+

^mm)

+

log

Nm

and

IK(t)

+ ^K(ct4)-)l

^<

^Mm

^(2.22)

log

for 2 < <

Nm.

^Theproofisbyinduction onm. Estimate

(2.22)

holds forrn 0 if we choose

Mo

large enough.

Assume

it is satisfiedfor some m > 0, and let

Nm

^{<_ S}

< Nm+l.

Then we haves

c4 -

^{0and cs4- for some with <}

^Nm.

Andby (2.11)and(2.12)wemaywrite

u

- ^s0-

^{B(s), (2.23)}

D(s) -c -4’ s 4’-2 K (u)

logu

^du

+ .log

s

where B(s)l

<

^{Bfor anabsoluteconstantB. First weshowthat thereisanabsolute}

constant

C

^{such that}

t

^{K (u)}

^u4-

^logudu

^c -

^{K (t)logss}

^{+ O}

^(s)4)c

-

^logs2s

^{(CI+Mm)(I+ C}

^(2.24)

for some function 0, with

10

(s)l ^{< 1.} Integrating byparts in(2.23)andreplacing log by

4

^-1logs

+

^O(1)weobtain

K (u)

b/b-

log^udu

^ckc

^-1

^K(t)logs sis ⁺⁰

^log2s

t

lftu -’( ^uK’(u) K(u))

^du,

4

^{logu logu}

and thusinordertoobtain(2.24)it isclearlysufficienttoshow that

( 1 t

u0-1 _IK(u)I

du <

2C-1

^{S (Co}

-+-

^Mm)

+

I "=

logu ^luK’(u)l

^q-

log--u--

log²s

(2.25) for some absoluteconstant

Co.

^Now there are absolute constants

A

and

A,

with

IK(u)l

^< A (u ^> 2)and, by

Lemma

3, with

luK’(u)l AI

log^u

+ ^IK(u) + ^K(cu4’-*)l

^(u

>

^2).

Thusonappealingtothe induction hypothesiswehave u 4-!

I ^< (A+A-k-Mm)

logZu

^{du < (AI+A+Mm)}

(2c-1

^{logs2s 0}

t

^logs

^S tt

whence(2.25)and

(2.24)

follow.

Now

weuse(2.12)and(2.24)in(2.23)andobtain

s

e-

_s

-

^s4’-

logsK(s)

-K(t)log

^s

⁺

^0(s)log²s(C2

+

^Mm)

)1 ⁺

^(2.26)

where

C2

^{is anabsoluteconstantand some}real functionwithIO(s)l 1. Thus IK(s)

-+- ^K(cs4’-)l <

C2

logs

(

(C2

--

^Mm)logs

--

^logN,,,

^mm+

^logs

provided C has been chosen withC >_ C2, whence theproofofthelemma willbe completeif we ensurethat asequenceof numberssatisfying (2.21 also satisfies

Mm

^{O(m) (m > 1). (2.27)}

428 Y.-ES.

PITERMANN

AND JEAN=LUC

R.MY

Indeed, by Lemma4 we have >

30

ifNm_ < <

Nm

^{(m > 1), so that}

rn < loglog t/log

4 +

^{O(1),and thuswith(2.27)we willhave}

’K(t)+K(ct4’-’)]<- logMm--t

^O

(lg "log

^lg

)

Infactweprovethat

mm

<(m+l)C3

H

^(1+C4

^-i)

^{(m>0), (2.28)}

O<i <m

where

C3

denotes max(C,M0). This is sufficientto ensurethat

(2.27)

holds" the productin(2.28)isboundedbyan infiniteproductthatconverges,since

4

^{> 1.}

^We

prove (2.28) byinduction on m.

It

holds forrn 0.

Assume

itis satisfied for some m ^> 0. Thenby Lemma4wehave

< (C

+

^Mm)

+

mm+l

^(C

+

^mm)

-+

_1o

+m+l)C3

H

⁽¹

^+C4)-i)(1

^+Cdp-m).

O<i<m

TheproofofLemma5 isnowcomplete. ^[21 Nowwedefinethefunctionkby

k(

^lg

^)log

^lg

^q

^{K (t)}

andweprove:

LEMMA6. Forx > wehave

k(x)

+

^{(x )}

^O(x4-).

^(2.29)

Proof.

^{Firstnotethat}

log log(ct^o-I

log

b _--lglgt-log4) l+O(l-g t)

Thennotethatby Lemmas^{3 and 5 we}have

K,(t)=O(lglgt)

log which with

K’(t)

k’

(log.log

^logb

)

log log

q

impliesthat

k’(

^{lg lg}

)log ₄

^{O (log logt), (2.30)}

whence, in particular,

k( ^lglgt-log4 l+O())--k( ^lglgt-logb l)+o(lglgt) "logt

Thus

+ _,_ _(,o,o),o + (’’-,I (’’’t,o ^{+ o} ,o

An appealtoLemma 5,andthechangeof variable x log log t/log

q,

conclude the proof.

Letnow_x0 > 0 be fixed, and consider thesequences

xi :=x0

+

^{2i and} Yi :=k(xi) (i ^>0). (2.31) WiththehelpofLemma6,by addingand subtractingtermsof the form(-

)J+

k(xo

+

j) (j ^> 1),it is notdifficulttoseethat {yi isaCauchy sequenceand thusconverges tosome real number, which we call h(xo). Similarly, forx0 >_ _{1, Yi} converges to h(xo 1),where

y

k(xi 1) (i >_ 0).Andagain by Lemma6weseethat

h(xo)

+

^{h(xo 1) lim(k(xi)}

^-I-

^{k(xi 1)) 0. (2.32)}

Weneedaprecise bound for the quantity Ih(x) k(x)l.

LEMMA7. Wehave

’’t,o-- _o-- (,o,o),o ^{+ o} (,o,o),o

Proof.

(2.33)

Put

x _x0 loglog

/

log

4.

Then with the notation

(2.31

wehave

Ih(x) k(x)l lim

lYi Yol.

NowwithLemma6wehave

lYi Y0l < lY Y0l-I-lY2

yl-I-’"-I-lYi

Yi-l

O(i>_o(XW2i)e-(X+2i)’g )

O(xe-xlog4),

and the lemma isproved. ^[21

430 Y.-F.S.

PTERMANN

AND JEAN-LUC

RIMY

In

ordertoconclude theproofof Theorem itthus remainstoprove:

LEMMA

8. The

function

^h is continuous.

Proof. ^Suppose

^{on thecontrarythath is not}continuous atsome x. Then there is asequence6i --> 0with

[h(x)- h(x

+

^{i)l > C > 0,}

whereCis somepositiveconstant.

By

(2.31),foreverypositiveintegerj wehave

Ih(x +

^{2j) h(x}

+

^2j

+ ⁾¹

^>_

^c.

Now

ifwewrite x

+

^{2j log log}

^{tj/log q,}

^{thenby Lemma 7 wehave}

h(x

+

^{2j) h(x}

+

^2j

+

^{i) k(x}

+

^{2j) k(x}

+

^2j

+

ⁱ⁾

+ ^A(tj,

^{i)log logtj}

logtj where

IA(tj,

i)l _< AforsomeabsoluteconstantA. Butwe canchoose j j0 such that for _tj,, wehave

whence

log log C

logt ²

Ik(x +

^{2j0) k(x}

+

^2j0

+

^{ei)l >} C

-

foreveryi, which contradictsthe continuity of k.

Theproofof Theorem isnowcomplete.

3. Remarks

Remark2.

As

wementioned it in the introduction(Remark 1),Vardi’s assertion in hisConjecture 2that

Ih

^{(x)l > 0 for x 6 (0, 1)is}not correct,and thisfollows from h(0)

-

^0.

^We

very brieflyindicatehow thislatter fact can beproved. If^wewrite

4’-I

(loglogt)

E(t) log ^k log

p

thenR6myprovedin[R6]that

Ikl

^{(u)l >_ a (u [m}

+

^{or, m+/];m >_ m0),}

wherea,ot

and/3

^areexplicitreal constantswithot

</3

^{anda > 0.}

By

refiningthe computationsneeded toachievethat, it ispossibletochoose theconstants a, u and

/3

^{in sucha}way that the additional conditionc < 0 <

/

be satisfied. (Weobtain c -0.0018

and/3

^0.2948,witha 0.0007486.) ThusIk(m)l ^> aform >_m0, whence

Ih(0)l- Ih(m)l- lim

Ih(m)l-

lim Ikl(m)l >_ a > 0.

m--+x nl--+

Remark3. Itshould be noted that inSection2 weprovemorethan just Theorem 1.

In

fact wegavean almostcomplete proofof thefollowing^result.

THEOREM2.

solution

of

^someapproximate

functional

^equation

of

^{the type}

E

^(t)

-c-4t

^4-2

E

^(u)du

+

⁰

,/2 log²

Then theconclusions

of

^{Theorem hold}

for E

^instead

of

^E.

Let

E

^beanintegrable real

function defined

^{on [2, )which is a}

(3.1)

Proof.

^{We haveto ensure that theproof of Theorem is} valid from equation (2.6) on,if wereplace E by

El.

^Fromthispointontheonly property ^ofE we use (otherthanequation (2.5)) isVardi’sresult(1.2) (whichisusedinthederivationsof equations

(2.17)

and(2.24)). Soweonlyneedtoshow the following.

LEMMA

9.

If E

^isasolution

of

^anequation

of

^{type(3.1),then}

El(t) 0

\logt (3.2)

Proof of

^{the lemma.}

^{Let No,} N

^{beasequenceof realnumbersasin(2.18),}

with

No

^{largeenoughtoensure}the validity ofLemma4. We prove byinduction on m thatif

e

E [2,

Nm

^],then

IE(e)I

^_<

Am

^(3.3)

log

e

for someincreasing sequence^ofpositivereal numbersexceeding 1, A0,

A

^with

Am+l

^Am(l

+ C ^-m)

for an absoluteconstantC.This will implythat thereis anabsoluteconstantAwith

Am

^{< Aforeverym,}whence the lemma. For

A0

large enough, (3.3)holds form 0 andm 1.

Suppose

(3.3)holds for somem > 1, and let L [Nm,

Nm+].

^Then L

c4-

^{and cL}

-

^{for some 6}

^INto-I,

^{Nm]. Notethat fromLemma4we}

can inferlog ^>

4 m-.

Forsomeabsoluteconstants B,

B, B2

^and

B3

^wehave

f2

^L

-

IEI(L)I

^< IE(u)lduWB

ceO-I

log²L

432 Y.-F.s.

PITERMANN

AND JEAN-LUC

RIMY

Am ( ^B )

^L

-

<

log²L

A,,,LO-’( B2) ^Z,nL

^c-’

<

+

^{< (1}

+ ^B3-")

^<

Am+

log L log L

providedChasbeen chosen withC^>

B3.

log

L’

Remark 4. Ifafunction

E

of the positiveintegerssatisfiesafunctional equation of thetype

n4-I )

E

⁽ⁿ⁾

-c-en

^4-2

Z ^E

^(k)

⁺

^{O (3.4)}

k_<cn- log²n

thenit is not difficult toshow,byan argument similar^tothat ofLemma9, that

E

(n) O

,

^logn

Itthen easily follows that the extended function

El

^(t)

E

^{(It])satisfiesafunctional}

equationoftype (2.5),^{and thus}that Theorem2appliesto

E.

It should be noted atthis point that if one replaces the error term of (3.4) by

O(ne-/log

^+’n) where 0< e < 1, then Theorem 2 can still beprovedfor

E,

the onlydifferenceinitsconclusionsbeingthat the errorterminthe expression of

E

ⁱⁿ

termsof h isnowO(n^-I log logn

/

log^I+’n).

In the remark on page 3 of [Va] a relation less precise ^than (3.4) (without er- ror term and with the symbol instead of =) is displayed, followed by an as- sertion, the most natural interpretation of which appears to be that the function n

O-h

(log logn

/

log )

/

logn=:

Ez

^{(n)isasolutionto(3.4)}(possibly with^alarger^er- rorterm)when h(x

+

^-h(x).

^But

^{wejustsawthatfor E2,withh(x}

+

^-h(x),

tobeanasymptotic solution of(3.4),itis atleastnecessarythath be continuous.

And infact there arecontinuous functionsh withh(x

+

^{-h(x)such that}

E2

^is

not asolutionof(3.4). Inordertoverify^thelatter, firstnotethatanecessarycondition for

^O(n-/log

afunction²E2(n)^{n). Thus,}O(n^if _Ez(n)

- ^/

^{log n)n4,-!tobe}_h

(lg

^{asolutionlognof}

]

^(3.4)isE2(n

⁺

^E2(n)

log

n \

^log

q ]

anecessaryconditionis

h

(

^{lg lg(n}

^{q +} )

^h

(

^{lg lglogbn}

) ^o (

^logn

)

Wecan find asequenceof positiveintegersni ---+ ^O (i --+ Cx) such that theintegral parts [loglogni/log q] are all of the same parity, and such that the fractional parts

6i :=

{loglogni/log} decrease ^to0as

--

^cxz.

^By

^{making the} sequence sparse enough^wemayalso assume that_{i+ <}

3i

^<i,wherei := {log log(n/+ 1)/log

4}.

Thus if weput h(6i)

l/ni

and h(i)

l/ni +

^{l//logni, we can completethe}

constructionofacontinuous functionhwithh(x

+

^{1) -h(x)} and such that

log log

Notethatitis even possibletoensurethat h be indefinitelydifferentiable(withthe restriction

hk)(0)

cxzfork 1,2 ).

Remark5. TheerrortermestimateO(t^4’- loglogt/logt)weobtaininourThe- orem is not asgoodas theerror termasserted boundO(t^4-

/

log t)of Conjecture 2.

Herewegive^anexampleofa(nontrivial) functional equation of type(3.1)forwhich thebettererror termbound holds. Inthis casewe canexhibit anexplicit expression forthefunctionh(here: g),which in additionpermits^arather easy(compared^tothe argument in [R6])derivationofanf2-estimate asstrongasthat of Conjecture 1.

THEOREM3. Thereisauniquesolution

for

^theexact

functional-differential

^equa-

tion

d(t) -t

e- f

_ao ^d(u)du

⁺

^{1. (3.5)}

This solution

satisfies

^anequation

of

^type(3.1). Moreover,

for

^{u > e, wehave}

g

+

^{O (3.6)}

d(u) logu log

q

log²u

where

and

g(x)---g(x

+

^{1) (x >0) (3.7)}

and wherein addition

g is continuous, (3.8)

g is notidentically zero. (3.9)

Proof

^{We first}establish theexistenceofasolution. Theargumentis similar to thatin [Yo, Chapitre 4,p. 151-153]. We arelooking fora solutionof(3.5)of the form

d(t)

Z

^{e,,(t), (3.10)}

n>0

434 Y.-F.s.

PITERMANN

AND JEAN-LUC

RIMY

where for the time being we assume that the sumconverges absolutelyanduniformly ineach bounded interval [0, t]. Thenbyusing(3.10)in(3.5)^weobtain

Z

^>0e,,(t)

Z

^>0

^t*-2 fo ^en

^{(u du.}

Weseekasolutionsatisfying and

eo(t)-- teP-

en(t) --t 4-

/o

^{e_(u)du (n > 1), (3.11)}

withe(t) ,,t

’’,

and where or,

and/,

^arereal numbers.

An

elementarycomputa- tionyields

0 1,

fl0=0,

n ^{(n-I-- l)(-}

l)--(1- 2)= (4- 1)/,_

+

^{(4- 1)}

O/n-I

o. /- +

Thesequence/30,/ satisfiesafirstdegreerecurrence relationwhose solution is

k ^--(

^1)k+l

+ ⁽¹

¹⁾

⁽¹

^{1) (3.12)}

whence

(-])"

izi

or,,=

4"

^(3.13)

Notethat as n --+ the productin (3.13) remains bounded. Also note that0 <

/n

^<

4

^{1. Nowthereremains toensure}that with these values of c,

and/,

^{the sum} in(3.10) does indeedconvergesabsolutely anduniformly in eachbounded interval [0, t]. Thisfollows from

c,,tt,, =o(l+t0-).q,,

^(3.14)

Nowweshow that thesolutionof(3.5)isunique. Let

d

^and

d2

^{betwosolutions}

of(3.5)and consider their differencee "=

d -d2. Let to

^{> 0andMbe the}sup[e(u)[

foru 6 [0,to]. Then if <

to

^wehave

t-I t4-I

le(t)l ^<

t0-2 f0

^{le(u)ldu <_ Mtok-2}

^f

^{dO du}

^Mt24-3

^Mt

Now let

(k

/k+l

^and

Mk MI/I

(k >_ 0) (3.15)

Thenit iseasytoseebyinduction onk,with(3.14)and(3.15),thatif <

to

^then le(t)l

< ^Mkt

^4k

--MIck+lt/’+’

^< C

k+

forevery positive integerk, where C is apositive constant depending only^on

to.

Hence

e(t) 0 if <

to

^and

d d2

^d.

Nowwe write

4’-’ 4’-’

(log

^log

t)

d

1-g

^J

l--g J

logq (3.16)

andweshow that

J(t)

+ ^J(t4’-)

⁰

(1o-)

Using (3.14)in(3.16)^wemaywrite

(3.17)

J (t)

otnt

^{logt, (3.18)}

tl>--

where, byconvention,^or_ 0.

Hence

wehave

J(t)

+

^J(t

^4-)

^06,

+

logt --logt

y(-l)"y,,

>_0

(3.19) where on appealingto(3.15)we seethat

and forn > 1,

l)t

When >

(2) 43,

thesequence {y,,} in(3.17) isunimodal’,i.e., thereis aninteger

no

such thaty,,+ > y,, forn <

no

^and?’,,+ _< y,, forn >_

no.

^Indeed Yo ^{< Y,and for} n>_l,

Yn+l x

y,,

)2

.1 U v(u)

436 Y.-F.S.

PITERMANN

AND JEAN-LUC

RIMY

whereu

-,,-2

andx

t-;

and forxfixed thefunctiono(u)is strictlyincreasing

from-2 ^to+oforu

^{[0, 1). It}follows that thereis auniqueu0 > 0witho(u0)- 1.

And thusn0 is the smallest positiveinteger satisfying

-""-e

^{u0. Now withthe}

unimodality of

{,}

^and(3.19)weseethat

IJ(t)

+ J(t-)l

V,,,logt. (3.21)

Thequantities n0 and _u0 depend on t. As increases to

+,

n0 alsoincreases to

+,

and _u0decreases to0. Thus from thedefinitionofu0, i.e. v(uo) 1, we see that

t""

remainsbounded as

,

whence, since_u0 > 0, _u0 O (1

/

logt). With

(3.20)

itfollows that

V,,,,

<< -2""t--""- ^<< u ^<< _log2t

andthis with(3.21)implies(3.17).

Nowwe show that dsatisfies anapproximate functional-differential equation of type (3.1). AsinLemma3 weobtain

tJ’(t)+J(t)+J(t-l)= O(),

^(3.22)

andwith(3.17)this implies

J’(t)-

0

)lgi

^{thatis, (u) 0(1). (3.23)}

With(3.5)^wehave

-’ d(cOt)

c-Or -

^{d2 d(u)du C}

⁺

^O(t

^-2)

^(3.24)

and,with(3.23)and(3.16),

d(ct)

^c

^logt -’

^j

(loglogt

^log

^+O ()) ^+O (t

^lg

^-’ 2 )

d(t)

+

^{0 (3.25)}

log

ThusTheorem 2 applies to e. Butwemust ensurethat thebetter estimate(3.6) holds, with(3.7)and(3.8). Tothatpuwosewereplace Lemma5by

J(t)+J(ct-’)= O(),

^(3.26)

whichfollows from (3.17)and(3.23). Thenwith(3.23)insteadof(2.30)^wereplace

Lemma

6by

j(x)

+

^{j(x 1) O(e}

^-go)

^(3.27)

and (with g instead ofh)

Lemma

7by

J

_logq ^g _log4

+

^{O (3.28)}

The continuity of g ^isobtainedexactly^asinLemma8.

Thereremains toprove (3.9). Infactweprove^that log log 4)

)

g 2

+

logq g(0.47998...) 0. (3.29)

We

deriveaclosed formula forg(2

+

^loglog 4)/log 4)).

As

intheproofof(3.17),we showthatthe series(3.18) forJ (t) is analternatingseries whosetermsinmodulus constitutea unimodalsequence. The maximal value of the moduli

Io,,t

log

tl

of thetermsin(3.18)isreachedwhen

no

^isthe smallest integer such that t,,,, ^{> t, where} t,,

Nowif wewriteu,, log log t,,/log

4

^{weclearlyhave}

log log 4)

u,, n

+

³

+ +

^{o(1), (3.30)}

log 4) andwemaywrite

j (u,,) J(t,,) (et,, logt,,

=0

Z

^{Ol +n _cb---,,}

e=-,, or,,

t,,

^{or,,logt,, =:}

e=-

(-

),,+eae.,,

(3.31) (where ae.,, ⁰ifg _< -n). Itis not difficult toseethat

and

1)"

ot,,logt,,=Fl4) log 4)v,1, where

FI’--H(I--*-’)-’

and whereke.,,,Ie.,, andv,,converge, uniformlyin

g,

to as n --+ o. Itfollows that thetermsae.,, of(3.31) convergeuniformlyingto

FIq

3-e-2-’ log 4),whence

,}im(-l)"j(u,,) ^1-14

^Iog4

^{(-l)e4 -e-o--’}

0.001289257.-..

438 Y.-F.S.

PITERMANN

AND JEAN-LUC

RIMY

Now

by (3.28)wehave j(u,,) g(u,,)

+

O(l/Iogt,,),andthus, with (3.30), 0.001289257

,,lim

(-l)"j(u,,)

,,lim

(-l)"g(u,,) -g

(2 ⁺

and

(3.29)

^isproved.

Remark 6. It is easy to see that ifX is any real number the exactfunctional- differential equation

dz(t) -t^4)-2

Jo

^dz(u)du

⁺

^(3.32)

hasexactlyonesolution, which isgivenby

dz(t)

d

(t), (3.33)

where

d

^{d is the solution of (3.5).} Returning to the error term E of (1.1), it is tempting ^to hope that E behaves similarly as

dz

for some ,k, and in particular that the functions h and g have the same zeros. But experimental data, although supportingtheconjecturethatthereis

some/4

^{with 0}

</4

< such that Ig(x)l > 0 for x 6 (/4, +/), alsostrongly indicatesthatprobably

1

^isdistinct fromtheot of Conjecture 3. Inthe accompanyingfigurethe dotted curverepresentsthe function

k

of Remark 2(approximatingthe function h of Theorem 1),and the continuousline representsthe function

d),,

(t)of(3.33),where

.0

1.054559132..- is chosen in such away that the amplitudes of bothfunctionsareequal.

Note

that itappears unlikely that the limitfunctionhwillbeatranslationof

dz,,.

^{Infact the}followingresult shows thatmany verydifferent functions

E

^{cansatisfy an}approximatefunctionalequation of type(3.1).

THEOREM4.

If f

^{is arealor}complex-valued

function defined

^{on the realnum-}

bers,satis.,ing

.t

^(x

+

⁾

-f

^(x) (x >_ o), and

[ f

^hasaFourier seriesrepresentation

f(x)

ak

errikx

k odd

withthepropertythat

y-k

_odak

lakl

^< cx, then the

function

-’ (log )log

^log

satisfies

^anapproximate

functional

^equation

of .

^{pe(3. l).}

Figure

Proof.

^Let

E

^beasinthe theorem. Then

ct-I

Z f2

^ct-I

I

-c-t

^-2

_E

(u)du

-c-t

^e-2 at

d2 k odd

U

-

^(logU)

^"*-

^du.

Nowthe lastintegral^is

(log u)^"-7;-2du

Thus

t- Z

^ak

er

^{k ,,e-,,}

⁺

⁰

_\logZt

log _o

440 Y.-F.S.

PITERMANN

AND JEAN-LUC

RIMY

t- ,,,,,,

(t - ⁾

+

^O

logt _odd log

2t

and the theorem isproved. ^I-I

REFERENCES

[Ch] V’sekChvfital,NotesontheKolakoskisequence, DIMACSTechnicalReport93-84, 1994.

[Fi] N.J.Fine,Solutiontoproblem5407,Amer.Math.Monthly 74(1967), 740-743.

[Go] S.W.Golomb, Problem5407,Amer.Math.Monthly73 (1966), 674.

[Ma] DanielMarcus,Solutiontoproblem 5407,Amer.Math.Monthly74 (1967), 740.

[P61] Y.-F.S.P6termann, On Golomb’sselfdescribing sequence,J.NumberTheory53 (1995), 13-24.

[P62] On Golomb’sselfdescribingsequenceII,Arch.Math. 67 (1996), 473-477.

[R6] Jean-Luc R6my,Sur lasuite autoconstruitede Golomb,J.NumberTheory66(1997), 1-28.

[Se] S.L.Segal, Anote ontheaverageorder[’numbertheoretic errorterms,DukeMath.J.32 (1965), 279-284;erratum sameissue,765.Erratum33 (1966), 821.

[Va] Iian Vardi,Theerrortermin Goiomb’ssequence,J.Number Theory 40 (1992), 1-1I.

[Yo] K6sakuYosida,Equationdidrentielles^etintdgrales, Dunod,Paris 1971.

Y.-F.

S. P6termann, Universit6 de Genbve, Section de Math6matique 2-4, rue du Li6vre,C.P. 240, 1211 Gen6ve24,Suisse

peterman^@ibm.unige.ch

Jean-Luc R6my, Centrede Rechercheen Informatiquede

Nancy

(CNRS)et

INRIA

Lorraine,B.P.239, 54506Vandceuvre-ls-NancyCedex,France