Palindromic Prefixes and Diophantine Approximation

DOI 10.1007/s00605-006-0425-5 Printed in The Netherlands

Palindromic Prefixes and Diophantine Approximation

By

Steephane Fischler

Universitee Paris-Sud, Orsay, France Communicated by J. Schoißengeier

Received December 8, 2005; accepted in final form May 16, 2006 Published online October 16, 2006#Springer-Verlag 2006

Abstract.This article is devoted to simultaneous approximation to and² by rational numbers with the same denominator, whereis an irrational non-quadratic real number. We focus on an exponent 0ðÞthat measures the regularity of the sequence of all exceptionally precise such approximants. We prove that₀ðÞtakes the same set of values as a combinatorial quantity that measures the abundance of palindromic prefixes in an infinite wordw. This allows us to give a precise exposition of Roy’s palindromic prefix method. The main tools we use are Davenport-Schmidt’s sequence of minimal points and Roy’s bracket operation.

2000 Mathematics Subject Classification: 11J13, 11J70, 11J06, 11J82

Key words: Diophantine approximation, simultaneous rational approximation, palindromic prefix

1. Introduction

Throughout this text, we denote by a real number, assumed to be irrational and non-quadratic (that is,½QðÞ:Q53). We study the quality of simultaneous rational approximants to and ², that is the possibility to find triples x¼ ðx0;x1;x2Þ 2Z³ with

LðxÞ ¼maxðjx0x₁j;jx0²x₂jÞ

very small in comparison withjx0j. In more precise terms, for any 0< "41 we consider the exponent_"ðÞdefined as follows:_"ðÞis the infimum of the set of allsuch that for any sufficiently large B>0 there existsx2Z³ such that

14jx0j4B and LðxÞ4minðB¹⁼;jx0j^"1Þ: ð1Þ If this set of is empty, we let"ðÞ ¼ þ1. For"¼0, we let

0ðÞ ¼ sup

0< "41

"ðÞ ¼lim

"!0"ðÞ

since for any fixed , the map "7!"ðÞ is non-increasing. We do not consider _"ðÞ for" <0, sinceLðxÞ4jx0j^"1 with " <0 implies^x_x²

0¼_x1

x0

2

ifjx0jis suffi- ciently large (see [6]); therefore these exponents merely concern rational approxi- mants to.

The exponents_"ðÞ, introduced in [8], generalize the classical exponent₁ðÞ (denoted by ww^⁰₂ðÞ or by 1=^₂ðÞ in [3], [2] and [10]) for which the following results are known:

₁ðÞ42 for any, by applying Dirichlet’s pigeon-hole principle;

1ðÞ5¼^1þ ffiffi

p5

2 ¼1:618. . .for any, proved by Davenport and Schmidt [6];

There existssuch that1ðÞ ¼, proved by Roy [11];

The set of values taken by1ðÞis dense in ½;2, proved by Roy [10].

The aim of this paper is to prove analogous results on₀ðÞ. One can prove easily (see [3]) that₀ðÞ ¼ þ1for almost allwith respect to Lebesgue mea- sure, and Davenport-Schmidt’s lower bound on₁ðÞyields₀ðÞ5 for any. Moreover Roy’spalindromic prefix method([11]; see also [8]) allows one to obtain asuch that₀ðÞ<2, starting from any wordwwith ‘‘sufficiently many’’ palin- dromic prefixes. In more precise terms, letw¼w₁w₂. . .be an infinite word on a finite alphabet. We denote by ðniÞ_i51 the increasing sequence of all lengths of palindromic prefixes ofw, in such a way thatnis among then_iif, and only if, the prefixw₁w₂. . .w_n ofw is a palindrome (i.e.,w₁¼w_n,w₂¼w_n1;. . .). We let

ðwÞ ¼lim sup

i!þ1

n_iþ1 ni

if the sequenceðniÞis infinite, andðwÞ ¼ þ1otherwise. The wordswsuch that ðwÞ<2 are studied in [7]. Roy’s palindromic prefix method enables one to con- struct, from any non ultimately periodic wordwsuch thatðwÞ<2, a real number such that ₀ðÞ<2. Moreover, the equality ₀ðÞ ¼ðwÞ holds if w is the Fibonacci word (see [11]), and more generally for any characteristic Sturmian wordw(see [3]). In this paper, we prove this equality for any non ultimately peri- odic wordw such thatðwÞ<2. We also show a reciprocal statement:

Theorem 1.1.We have

f0ðÞ:2Rirrational; non-quadraticg \ ð1;2Þ

¼ fðwÞ: w an infinite wordg \ ð1;2Þ:

The setS¼ fðwÞg \ ð1;2Þthat appears in Theorem 1.1 is studied in [7]. The least element of S is the golden ratio (as implied by the previously mentioned results). Apart from , the least element is 2 ¼1þ ffiffi

p2

2 ¼1:707. . ., then (apart from2) it is3 ¼^2þ ffiffiffiffi

p10

3 , and so on: there is an increasing sequenceðnÞof isolated points in S, that converges to the least accumulation point ₁¼1:721. . . of S.

This follows from a result of Cassaigne [4] and the property ([7], Theorem 1.3) thatS\ ð1; ffiffiffi

p3

coincides with the set of values ofðwÞcoming from characteristic Sturmian wordsw. Combining this property with Theorem 1.1 proves Theeoreeme 2.1 announced in [8], namely: all values of ₀ðÞless than ffiffiffi

p3

can be obtained from characteristic Sturmian words as in [3]. In particular, the following corollary shows that the situation is completely different from the case of the exponent₁ðÞ, which assumes [10] a set of values dense in½;2.

Corollary 1.2.There is no real numbersuch that < ₀ðÞ< ₂¼1:707. . ..

We shall deduce Theorem 1.1 from a ‘‘structure theorem’’ on numberssuch that₀ðÞ<2, namely that the sequence of all ‘‘exceptionally precise’’ approx- imants toand² satisfies the ‘‘same’’ recurrence relation as the sequence of all palindromic prefixes of some wordwwithðwÞ<2. This enables us to associate with each such that₀ðÞ<2 a wordw such thatðwÞ<2, in such a way that ₀ðÞ ¼ðwÞ. This gives a kind of converse to Roy’s palindromic prefix method.

As byproduct of our approach, we also obtain the following result:

Theorem 1.3.Letbe such that₀ðÞ<2.Then there exists"₁>0 (depending only on0ðÞ) such that"ðÞ ¼0ðÞfor any" < "1.

Throughout the text, we shall use the following notation. We denote byN¼ f0;1;2;. . .g the set of non-negative integers, and let N ¼Nnf0g. We write utv_t (andv_t ut) ifv_tis positive fort sufficiently large, andut=v_t is bounded from above ast tends to infinity. Whenutv_t andv_t ut, we writeut v_t. At last, when a point ofZ³ is referred to as an underlined letter (e.g.x), we denote by the corresponding uppercase letter its norm, that is the largest absolute value of its coordinates (e.g.X¼maxðjx0j;jx1j;jx2jÞifx¼ ðx0;x₁;x₂Þ).

The structure of this text is as follows. In Section 2 we recall the notation and results of [7] about words with many palindromic prefixes, classical estimates about approximants, and also the definition and properties of Roy’s bracket opera- tion. Our main tool is the sequence of minimal points introduced by Davenport and Schmidt. Section 3 is devoted to this sequence: we recall classical results and prove new ones (which may be of independent interest). We state and prove in Section 4 our main diophantine result, which is the crucial step in the proof (x5.3) of the results stated in this introduction. This enables us in Section 5 to give a precise account on Roy’s palindromic prefix method – which was the original motivation of this paper. At last, Section 6 is devoted to some open questions.

2. Palindromes, Classical Estimates, and Roy’s Bracket

In this section, we recall the properties [7] of words with many palindromic prefixes, and also the bracket operation introduced by Roy [12].

2.1. Words with Many Palindromic Prefixes.Let us recall the definitions and results of [7] that will be useful here (with minor changes intended to fit into the diophantine setting).

Let :N !N be a function such that

ðnÞ4n1 for any n51: ð2Þ

Denote byð#kÞ_k50 the sequence of all indices n51 (in increasing order) such that ðnÞ4n2. This sequence may be either finite or infinite; it is infinite for the functions of interest here (see (3) below). We shall use the following variant of Definition 4.9 of [7]:

Definition 2.1.A function is said to be asymptotically reduced if (2) holds and the associated sequenceð#kÞis such that

ð#kÞ< #_k1 and ð#kÞ 6¼ ð#k1Þ

for any sufficiently largek. When the sequenceð#kÞ is finite, we agree that is asymptotically reduced.

Let us denote by Fthe set of all functions :N !Nsuch that is asymptotically reduced:

There existscsuch thatic4 ðiÞ4i1 for any i51:

There are infinitely manyi such that ðiÞ4i2:

9=

; ð3Þ For 2F, the sequenceð#kÞis infinite, and we have#_kþ14#_kþc1 for anyk sufficiently large (since#_kþ1c4 ð#kþ1Þ< #_k).

Given an infinite wordw¼w₁w₂. . .on an arbitrary (finite or infinite) alphabet A, we denote byðiÞ_i51 the (finite or infinite) sequence of all palindromic pre- fixes of w, and by n_i the length of _i (in such a way that _i¼w₁w₂. . .w_ni is a palindrome for any i51). Fori⁰4i, _i⁰ is a prefix of _i, so there is a word b¼w_ni0þ1. . .w_ni such that_i¼_i⁰b. We denote by¹_i0 _i this wordb.

We let W be the set of all non ultimately periodic words w such that the increasing sequenceðniÞis infinite and satisfies lim supn_iþ1=ni<2. Then the fol- lowing result is proved in [7]:

Theorem 2.2.(i)For any w2Wthere exists 2Fsuch that

_iþ1 ¼_i¹ _ðiÞi for any i sufficiently large: ð4Þ (ii) For any 2Fthere exists w2Wsuch that(4) holds.

Theorem 2.2 implies that anyw2Wcan be written on a finite alphabet. So we may assume, throughout this text, that the alphabetAis finite.

The following relation follows from (4) and will be used repeatedly:

niþ1¼2nin _ðiÞ for anyi sufficiently large:

The three conditions that appear in (3) are of different nature. The last one corre- sponds, in the definition ofW, to the assumption that wis not ultimately periodic (which is equivalent townot being periodic: see Lemma 5.6 of [7]). The second one enables us to get rid of the case ð Þ ¼2; this is very useful since Theorem 2.2 generalizes only to a specific class of words w such that ðwÞ ¼2. At last, the assumption that is asymptotically reduced ensures that distinct functions (modulo the equivalence relationRdefined below) always correspond to distinct wordsw.

Example 2.3.The Fibonacci wordabaaba. . .on the two-letter alphabetfa;bg corresponds (in Theorem 2.2) to all functions such that ðnÞ ¼n2 for any sufficiently largen(see [7], Example 4.7). More generally, ifwis the characteristic Sturmian word with slope½0;s1;s2;. . .thenðwÞ ¼2 if, and only if, the sequence ðsnÞis unbounded. In the opposite case, wbelongs to Wand corresponds to any function such that ðnÞ ¼ns_k1 ifncan be writtens₁þ þs_kfor some k, and ðnÞ ¼n1 otherwise, fornsufficiently large (see [7], Example 4.6). We recover the Fibonacci word by considering the special cases₁ ¼s₂ ¼ ¼1.

Example 2.4.Letw2Wbe written on a finite alphabet, sayA¼ fa1;. . .;a_rg.

Let p₁;. . .;p_r be arbitrary palindromes, written on another alphabet A⁰. Then

replacing eacha_iwithp_iin the wordwyields a wordw⁰written onA⁰. Denote by ðiÞ_i51 the sequence of all palindromic prefixes ofw, and by⁰_i the finite word obtained from _i by replacing each a_i withp_i. Since p₁;. . .;p_r are palindromes, all⁰_i are palindromic prefixes of w⁰. However, in general, the sequenceð⁰_iÞ_i51 does not contain all palindromic prefixes ofw⁰. For instance, if allp_i have length greater than 2 then the first letter of w⁰ is a palindromic prefix of w⁰ but is not among the⁰_i.

Throughout this text, we are interested only in asymptotic properties, so the palindromic prefixes ofw⁰ that are missing in Example 2.4 are not a problem (as long as their number is finite). To take this observation into account, we define an equivalence relationRonFas follows:

Definition 2.5. For ; ⁰2F, we set R ⁰ if there exist and i1 such that ðiÞ i¼ ⁰ðiÞ ðiÞfor anyi5i1.

We now define an analogous equivalence relation on W:

Definition 2.6. Letw;w⁰2W, andðiÞ_i51 (resp.ð⁰_iÞ_i51) be the sequence of all palindromic prefixes ofw (resp.w⁰). We setwR⁰w⁰ if there exist 2Fand such that

_iþ1 ¼_i¹ _ðiÞi and ⁰_iþ1 ¼⁰_i⁰ _ðiÞþ^{1 0}_i

for anyi sufficiently large.

This definition means that wR⁰w⁰ if, and only if, there exist ; ⁰2F with R ⁰ such that_iþ1 ¼_i¹ _ðiÞiand⁰_iþ1 ¼⁰_i⁰0¹ðiÞ⁰_ifor anyisufficiently large.

Another way of stating this is the following:wR⁰w⁰if, and only if, after omitting a finite number of initial terms in the sequence ðiÞ_i51 or in ð⁰_iÞ_i51, we have _iþ1 ¼_i¹ _ðiÞiand⁰_iþ1¼⁰_i⁰ _ðiÞ^{1 0}_ifor some 2Fand anyisufficiently large.

Thanks to Definitions 2.5 and 2.6, Theorem 2.2 yields a bijective map W=R⁰!F=R: ð5Þ We now define a quantityð Þas follows:

Definition 2.7. For 2F, we let

ð Þ ¼lim supm_iþ1 mi

where ðmiÞ_i51 is any increasing sequence of non-negative integers such that m_iþ1¼2mim _ðiÞ for any isufficiently large.

The value ofð Þdoes not depend on the choice of a peculiar sequenceðmiÞ, thanks to Proposition 6.2 of [7]. In particular, if corresponds to a wordwas in Theorem 2.2 then one can choosemi¼ni, henceð Þ ¼ðwÞ. It follows from this remark and from Proposition 6.2 of [7] that:

The mapW! ð1;2Þ; w7!ðwÞfactors into a mapW=R⁰! ð1;2Þ.

The mapF! ð1;2Þ; 7!ð Þfactors into a mapF=R! ð1;2Þ.

These two maps, together with the bijection (5), make up a commutative diagram:

W=R⁰!F=R

# "

ð1;2Þ

ð6Þ

We have constructed in [7] (Remark 7.6) two words w and w⁰ such that ðwÞ ¼ðw⁰Þ ¼ ffiffiffi

p3

, but with w6¼w⁰ mod R⁰ (actually w is not episturmian whereas w⁰ is characteristic Sturmian). This proves that in the diagram (6), the mapsW=R⁰! ð1;2Þ andF=R! ð1;2Þinduced by are not injective.

2.2. Classical Estimates about Approximants.Throughout this text, we let for any" such that 0< " <1:

A"¼ fx2Z³nf0g; LðxÞ4X^ð1"Þg;

withX¼maxðjx0j;jx1j;jx2jÞ(as explained in the Introduction). By convention, for

"51, we let A"¼Z³nf0g. Moreover, we identify a point x¼ ðx0;x1;x2Þ 2Z³ with the symmetric matrix

hx₀ x₁ x₁ x₂ i

.

In this section, we state classical estimates which are due (mainly) to Davenport and Schmidt. For instance, the first inequality of the following lemma is proved in Lemmas 2 and 3 of [6], the second one in Lemma 4.

Lemma 2.8. We have

jdetðxÞj ¼ jx0x2x²₁j XLðxÞ:

Moreover,if x;y;z are such that X5maxðY;ZÞand LðxÞ4minðLðyÞ;LðzÞÞ, then we have

jdetðx;y;zÞj XLðyÞLðzÞ:

Let kxkdenote the norm of a vectorx2Z³ (that is, its greatest coordinate in absolute value, usually denoted byXin this text). LetVbe a sub-Z-module ofZ³, of rank 2. Thenkx^yk, computed for aZ-basisðx;yÞofV, does not depend on the chosen basis but only onV. This is theheight ofV, denoted byHðVÞ.

Letxandybe two linearly independent vectors inV. Thenkx^yk ¼N HðVÞ where N is the index, in V, of the subgroup generated by x and y. Moreover, Lemma 3 of [6] shows that

kx^yk XLðyÞ þYLðxÞ: ð7Þ

2.3. Roy’s Bracket Operation.The bracket operation½:; :; :introduced by Roy [12] will be used in a crucial manner in the statement, and proof, of our results. If x;y;zare three linearly dependent vectors inZ³, understood as symmetric matrices, then the matrixxJzJy

whereJ¼h0 1 1 0

i

is also symmetric, and it is denoted by ½x;y;z. The importance of this operation in our context is easily seen, for instance, by considering Lemma 5.5 below. The following relation holds:

detð½x;y;zÞ ¼detðxÞdetðyÞdetðzÞ:

Moreoverzcan be obtained back fromx,yand½x;y;zusing the following formula:

detðxÞdetðyÞz¼ ½x;y;½x;y;z ð8Þ which makes sense since x, y and ½x;y;z are linearly dependent (see [12], Lemma 2.1). At last, since ðJyÞ² ¼ detðyÞId we have:

½x;y;y ¼detðyÞx: ð9Þ

The main property of this bracket is the following slight generalization of [12], Lemma 3.1ðiiiÞ(with X¼maxðjx0j;jx1j;jx2jÞand so on).

Proposition 2.9. Let x;y;z be linearly dependent vectors inZ³.Let ¼ZLðxÞLðyÞ þLðzÞmaxðYLðxÞ;XLðyÞÞ:

Then the bracket u¼ ½x;y;zis such that

UXYLðzÞ þ and LðuÞ :

Proposition 2.9 will also be used through the following Corollary:

Corollary 2.10. Let ; "; "⁰2 ð0;1Þ be such that "^1þ₁< "⁰<1. Let x;y;z be linearly dependent vectors inA",with X,Y,Z large enough in terms of,","⁰and .We assume

X;Y4Z; Z4ðXYÞ and detðxÞ;detðyÞ;detðzÞ 6¼0:

Then the bracket u¼ ½x;y;zbelongs to A"⁰.

Proof. Let the notation be as in the Proposition 2.9. Then we have

ZðXYÞ^"1 ðXYÞ^þ"1<1 since"þ <1, andXYLðzÞ XYZ¹5ðXYÞ¹>1

since detðzÞ 6¼0 (using Lemma 2.8). ThereforeU XYLðzÞandLðuÞ , hence U^1"0LðuÞ ðXYÞ^""0Z"þ"⁰""⁰4ðXYÞð1þÞ"ð1Þ"⁰

:

Sincex,yandzhave non-zero determinants,uis not zero; this concludes the proof of Corollary 2.10.

3. Davenport-Schmidt’s Minimal Points

In this section, we recall in paragraph 3.1 the definition and classical properties of the sequence of minimal points, introduced by Davenport and Schmidt ([5] and [6]). Then we move to new results on this sequence (Sections 3.2 to 3.4), that are of independent interest and may be used to study numberssuch that₁ðÞ<2 but not necessarily 0ðÞ<2. The most important results are Proposition 3.6 (which states that any sufficiently precise approximation is collinear to a minimal point) and Proposition 3.7 (which deals with linear independence of approximants).

We shall make a frequent use of the notation and results of Section 2.2. Recall that for any"such that 0< " <1, we let

A"¼ fx2Z³nf0g; LðxÞ4X^ð1"Þg;

withX¼maxðjx0j;jx1j;jx2jÞ, andA"¼Z³nf0gfor"51. Moreover, we identify a pointx¼ ðx0;x₁;x₂Þ 2Z³ with the symmetric matrix

hx0 x1

x1 x2

i .

Throughout this Section, we letdenote an irrational non-quadratic real num- ber such that₁ðÞ<2. We do not assume anything about₀ðÞ.

3.1. Definition and Notation.Letbe an irrational non-quadratic real number.

For any realX51, the set of alla¼ ða0;a₁;a₂Þ 2Z³ such that 14a₀4X and LðaÞ41 is finite, and contains exactly one element for which L is minimal.

Following [5] and [6], we call this element minimal point corresponding to X.

We denote byða_iÞ_i51the sequence of all minimal points, in such a way thata_ibe a minimal point corresponding to allXsuch thata_i;04X<a_iþ1;0. By definition, any a_iis a primitive point ofZ³(that is,ai;0,ai;1 andai;2are globally coprime), and any two distinct minimal points are always linearly independent overZ. As usual, we let Ai ¼maxðjai;0j;jai;1j;jai;2jÞ51. For anyi sufficiently large, we haveLða_iÞ<1=2 so thatai;1 (resp. ai;2) is the closest integer toai;0(resp. ai;0²), andAi<A_iþ1.

From now on, we assume₁ðÞ<2. Then Lemma 2 of [6] implies detða_iÞ 6¼0 fori sufficiently large, hence

1AiLða_iÞ ð10Þ

using Lemma 2.8.

Let I₀ denote the set of all indices i such that a_i1, a_i and a_iþ1 are linearly independent. This set is infinite (see [6]); we denote byðikÞthe sequence of all elements ofI₀, in increasing order. We let

d_k ¼a_ik:

We denote by Vk the intersection with Z³ of the sub-Q-vector space of Q³ spanned byd_ketd_kþ1, and byHðVkÞits height (defined in Section 2.2). It follows from [6] thatHðVkÞtends to infinity asktends to infinity. Moreover, Lemma 4.1 of [12] (see also Lemma 2 of [5]) is the following result:

Lemma 3.1.For any k and i sufficiently large such that ik4i<ikþ1,the points a_iand a_iþ1make up a basis of theZ-moduleVkand we have Aiþ1LðaiÞ HðVkÞ.

Let"2 ð0;1, andðu_iÞ_i51be the sequence of all minimal points inA", ordered according to their normsU_i. Then, we have

_"ðÞ ¼lim sup

i!1

logU_iþ1

logLðu_iÞ: ð11Þ This fact immediately follows from the definition of _"ðÞ (see [6]); the point is that in Eq. (1) we may assume thatx is a minimal point.

3.2. Estimates for the Height ofVk.In the following estimates, we assume ₁ðÞ<2 and use repeatedly the following consequence of Eq. (11): for any > ₁ðÞwe have

A_iþ1 Lða_iÞ andLða_iÞ A¹⁼ _iþ1 : ð12Þ Lemma 3.2. Let"2 ð0;1 and i be such that a_i2A".Then we have

A_iþ1 HðVkÞA^1"_i where k is the only integer such that i_k4i<i_kþ1.

Proof. We have HðVkÞ A_iþ1Lða_iÞ4A_iþ1A^1þ"_i , thanks to Lemma 3.1 and by definition ofA".

Lemma 3.3. For any > ₁ðÞwe have HðVkÞ D² _kþ1:

Proof.Leti¼i_kþ1; then the following inequalities hold thanks to Lemmas 2.8 and 3.1:

1 jdetða_i1;a_i;a_iþ1Þj Aiþ1Lða_iÞLða_i1Þ Lða_iÞ¹ A¹_i HðVkÞ A ²_i HðVkÞ:

This proves the Lemma.

Lemma 3.4. For any > ₁ðÞwe have HðVkÞ D¹⁼ _k : Proof.Let i¼ik. The following inequalities hold:

1 jdetða_i1;a_i;a_iþ1Þj A_iþ1Lða_iÞLða_i1Þ HðVkÞLða_i1Þ HðVkÞA¹⁼ _i and prove the Lemma.

Remark 3.5.Lemmas 3.3 and 3.4 are optimal whenis the number constructed by Roy [11] from the Fibonacci word. Actually, for this number we have HðVkÞ D²_kþ1 D¹⁼_k . Now, let us assume thatis, more generally, constructed (as in [1] and [3]) from a characteristic Sturmian wordw. Then Lemma 3.3 is still optimal for some values ofk; actually, ifk is such that A_ikþ1þ1 ¼Lðd_kþ1Þ then the estimates proved by Bugeaud and Laurent imply HðVkÞ D² _kþ1. However, Lemma 3.4 is not optimal, as can already be seen from the proof: the upper bound Lða_ik1Þ D¹⁼ _k can be sharp only if a_ik1 ¼d_k1, and there are words w for which this never happens.

3.3. Importance of Minimal Points. By definition of 1ðÞ, for any" >1 1=1ðÞall minimal points belong toA"(except for a finite number of exceptions, depending on"). This means that any minimal point is a rather good simultaneous approximant to and². The following Proposition is a converse statement, and proves that the approximations provided by the palindromic prefix method (see Section 5.2) are minimal points (up to proportionality).

Proposition 3.6.Let" >0be such that" <2₁ðÞ.Let x2A" with X suffi- ciently large(in terms of"and ).Then x is collinear to some minimal point a_i.

Question:Is it possible, in Proposition 3.6, to weaken the assumption on" to

" <11=₁ðÞ?

A positive answer to this question would be very satisfactory, since the assumption on"would be optimal. Indeed, letbe any real number such that₁ðÞ<2, and let"

be such that 11=₁ðÞ< " <1=2. Let satisfy" >11= , with₁ðÞ< <2.

Then we have Lða_iÞ4Lða_i1Þ A¹⁼ _i hence jdetða_i1;a_i;a_iþ1Þj A_iþ1A²⁼ _i . Fori2I0, this impliesAiþ1A²⁼ _i henceaiþ1;0>2ai;0 fori sufficiently large. Let x_i¼a_iþa_i1, fori2I₀. Thenx_iis primitive (otherwise Span_Qða_i;a_i1Þ \Z³would strictly contain the subgroup generated bya_ianda_i1), anda_i;0<x_i;0<2a_i;0<a_iþ1;0. Therefore x_i is not collinear to any minimal point; however it satisfies Lðx_iÞ4 2Lða_i1Þ A¹⁼ _i X¹⁼ _i hencex_i2A"ifi2I₀is sufficiently large.

Proof of Proposition 3.6. We may assume x₀>0. Let us denote by a_i the minimal point corresponding to x₀; we have a_i;04x₀<a_iþ1;0 and Lða_iÞ4LðxÞ.

Let > ₁ðÞ be such that þ" <2. Using Eq. (12) we see that XLða_iÞLða_iþ1Þ A_iþ1Lða_iÞ²Lða_iÞ² and

A_iþ1Lða_iÞLðxÞ Lða_iÞ¹ A^"1_i A ^þ"2_i :

Therefore Lemma 2.8 proves that the integer detðx;a_i;a_iþ1Þ is zero for i suffi- ciently large in terms of"and. This impliesx2Vk, wherek is the index such thati_k4i<i_kþ1.

Let us assume thatxis not collinear toa_i. Then we have HðVkÞ4kx^a_ik XLðxÞ A^"_iþ1 HðVkÞ^{"=ð2 Þ}

using Eq. (7) and Lemma 3.3. As þ" <2, this is impossible fork sufficiently large (in terms of " and ). We obtain in this way x¼a_i for some non-zero integer, which concludes the proof.

3.4. Linear Independence Properties.The following proposition is very useful as soon as_"ðÞ<2 for some"(see Eq. (11) above):

Proposition 3.7.Letðu_iÞbe a sequence of minimal points, ordered according to their norms U_i,such that

lim sup

i!1

logU_iþ1 logLðu_iÞ<2:

Then:

1. Up to a finite number of terms,ðd_kÞ is a subsequence ofðu_iÞ.

2. For any i sufficiently large,u_i is among the d_k if,and only if,u_i1,u_i and u_iþ1 are linearly independent.

To prove this proposition, we shall use the following lemma:

Lemma 3.8. Let ðu_iÞ be as in Proposition 3.7, and i be sufficiently large.

Denote by n and m the integers such that u_i¼a_n and u_iþ1 ¼a_m.Then the vectors a_n,a_nþ1;. . .;a_m1,a_m belong to a common plane.

Proof of Lemma 3.8.Letbe such that lim sup_log^logU_Lðu^iþ1

iÞ < <2, andt be such thatn<t<m. Then we have

jdetða_t1;a_t;a_tþ1Þj A_tþ1Lða_t1Þ²4U_iþ1Lðu_iÞ² Lðu_iÞ²

so the integer detða_t1;a_t;a_tþ1Þis zero ifiis sufficiently large. This concludes the proof of Lemma 3.8.

Proof of Proposition 3.7. Letkbe sufficiently large, with d_k¼a_p. Ifd_k were not among the u_i, Lemma 3.8 would apply with n<p<m: a_p1, a_p and a_pþ1 would be linearly dependent. This is impossible, so the first part of Proposition 3.7 is proved.

Let us prove the second part. Ifu_i¼a_p is not among thed_k, withisufficiently large, thena_p1,a_panda_pþ1belong to a common plane. Lemma 3.8 (applied twice) proves that this plane contains alsou_i1 andu_iþ1. To prove the converse statement, assume there is a plane that containsu_i1,u_iandu_iþ1, withu_i¼a_pandisufficiently large. Then this plane contains alsoa_p1anda_pþ1, by applying Lemma 3.8 twice; so u_i is not among thed_k. This concludes the proof of Proposition 3.7.

4. The Main Diophantine Result

This section is devoted to the statement (Section 4.1) and proof (Section 4.2) of our main diophantine result. This theorem will be the key point in the proofs of Section 5.

4.1. Statement of the Results.Throughout this Section, we define "₁ by

"1 ¼ ð21ðÞÞð21ðÞ þ ð20ðÞÞ1ðÞÞ: ð13Þ Since ₁ðÞ4₀ðÞ, we have "₁5ð2₀ðÞÞ²ð1þ₀ðÞÞ. For instance, if ₀ðÞ ¼1þ ffiffi

p2

2 (denoted by₂ in the Introduction) then"₁50:232.

This is the number "1 referred to in Theorem 1.3 stated in the Introduction, which is contained in the following result (recall that FandR were defined in Section 2.1, andA"in Section 2.2):

Theorem 4.1.Let be an irrational non-quadratic real number.

(a) Suppose first₀ðÞ<2.Choose a real number"such that0< " < "₁,and letðu_iÞ_i51 be the sequence of minimal points in A", ordered according to their norms U_i.Then,we have

Lðu_iÞ ¼U_i^1þoð1Þ; lim inf

i!1

logU_iþ1

logU_i >1 and ₀ðÞ ¼_"ðÞ ¼lim sup

i!1

logU_iþ1 logU_i : Moreover,there exists a function 2Fsuch that½ui;u_i;u_iþ1is collinear to u _ðiÞ for any i sufficiently large. At last,ðuiÞ(modulo a finite number of terms) and (moduloR) are independent from the choice of".

(b) Conversely,letðv_iÞ_i51be any sequence of primitive points inZ³such that:

The sequence of first coordinatesðvi;0Þ_i51 is positive and increasing, As i! 1, Lðv_iÞ ¼V_i^1þoð1Þ,

The real number¼lim sup_i!1ðlogV_iþ1Þ=ðlogV_iÞsatisfies <2, For any i sufficiently large, ifv_i1,v_i andv_iþ1 are linearly dependent then

½v_i;v_i;v_i1is collinear tov_iþ1.

Then ₀ðÞ ¼ <2, and ðv_iÞ is exactly (up to a finite number of terms) the sequenceðu_iÞ of Part(a).

In Part ðaÞ, the assertion on Roy’s bracket is equivalent to u_iþ1 collinear to

½u_i;u_i;u _ðiÞ thanks to (8). Therefore knowing enables one to construct the sequence ðu_iÞ by induction. On the other hand, it follows from Proposition 5.1 (proved in Section 5.1 below) that we have

0ðÞ ¼ð Þ;

whereð Þwas defined in Section 2.1. However, knowing₀ðÞdoes not deter- mine (modulo R). Indeed, there are functions and ⁰, distinct modulo R, such that ð Þ ¼ð ⁰Þ (see the end of Section 2.1). The numbers , ⁰ con- structed from them using the palindromic prefix method are such that ₀ðÞ ¼ ₀ð⁰Þ (thanks to Theorem 5.4), but their approximants satisfy completely dif- ferent recurrence relations.

In PartðbÞ, the last assumption on ðv_iÞis necessary. Indeed, let us consider the real numberconstructed using the palindromic prefix method (as in [1] and [3]) from the characteristic Sturmian word with slope ½0;3;3;3;. . .. Then 0ðÞ ¼ 1:767. . .<2, but the sequence ðv_iÞ consisting in all points denoted by d_k and e_k in Section 4.2 below satisfies the first three assumptions with¼1:868. . ..

Theorem 4.1 can be generalized to the case where we assume _"ðÞ<2 for some very small". This leads to Theeoreeme 2.2 announced in [8]; the proof follows the same lines as the one given here.

4.2. Proof of Theorem 4.1.The proof falls into eight steps. We letbe a real number such that½QðÞ:Q53 and₀ðÞ<2. These are the only assumptions throughout the first seven steps. The notation of Theorem 4.1 comes into the play only in Step 8.

Recall thatða_iÞdenotes Davenport-Schmidt’s sequence of minimal points con- structed from , and ðd_kÞ is the subsequence consisting in all a_i such thata_i1, a_i and a_iþ1 are linearly independent (see Section 3.1). As in Section 2.2, for

"2 ð0;1 we let A" be the set of all x2Z³nf0g such that LðxÞ4X^"1, where X¼maxðjx0j;jx1j;jx2jÞ.

The sketch of the proof is as follows. We first construct a sequence ðe_kÞ of minimal points, such that (essentially) e_k is the first very precise minimal point afterd_k. We have Dk<Ek4D_kþ1, and we show howd_k, e_k and HðVkÞ are in- terrelated. Then we construct a sequenceðw_tÞof minimal points, which turns out to be the sequence of ‘‘all’’ very precise approximants; ðd_kÞ andðe_kÞ are subse- quences of this sequence, and other points between e_k and d_kþ1 are constructed thanks to an interpolation procedure using Roy’s bracket. The key point of the proof is the definition and properties of this sequenceðw_tÞ, and of the function associated with it. Then Theorem 4.1 follows easily: the sequencesðu_iÞandðv_iÞin Theorem 4.1 are proved to coincide (up to a finite number of terms) withðw_tÞ.

Step 1. Construction of the sequence (e_k). Let us fix a real number"₂such that 0< "₂< "₁, where "₁ is defined by Eq. (13). Let ðb_tÞ_t51 be the sequence of all minimal points in A"2, ordered according to their norms B_t. Since _"2ðÞ4 ₀ðÞ<2, Proposition 3.7 applies (thanks to (11)) and shows that for anyksuffi- ciently larged_k is equal to someb_t; then we lete_k¼b_tþ1.

The sequenceðe_kÞ_k51defined in this way (by choosinge_karbitrarily for small values of k) seems to depend on the choice of "₂, but actually this dependence concerns only finitely many terms, as the following lemma shows.

Lemma 4.2.Let"be such that0< "4"2.Then for any k sufficiently large(in terms of"),d_kand e_k are consecutive elements of the sequence of minimal points inA".

It is possible to state more precise versions of this lemma (see Lemma 4.5 below). We shall prove the following one now:

Lemma 4.3.Let_V2 ð1;2Þ,andðv_iÞ_i51be any sequence of minimal points in

A"2,ordered according to their norms V_i, such that

lim sup

i!1

logV_iþ1

logLðv_iÞ4_V: Assume that for some"⁰>0with

"⁰<ð2₁ðÞÞð2₁ðÞ þ ð2_VÞ1ðÞÞ;

we have e_k2A"⁰ when k is sufficiently large.Then,for any k sufficiently large,d_k and e_k are consecutive elements of the sequenceðv_iÞ_i51.

To deduce Lemma 4.2, it suffices to apply Lemma 4.3 to the sequence ðv_iÞ of all minimal points in A", with "⁰ ¼"₂< "₁ and _V ¼_"ðÞ4₀ðÞ thanks to Eq. (11). The full generality of Lemma 4.3 will be useful in Step 8, since it turns out that e_k2A"⁰ for any "⁰>0 (as soon as kis sufficiently large in terms of "⁰).

Proof of Lemma 4.3. Let k be sufficiently large. Proposition 3.7 provides an integerisuch thatd_k ¼v_i. Sinceðv_iÞis a subsequence ofðb_tÞ, we haveV_iþ15Ek. Let us assume that V_iþ1>Ek. Since d_kþ1 is among the v_i, we have Dk<Ek<V_iþ14D_kþ1. Choosing 2 ðV;2Þ, we have V_iþ1 Lðv_iÞ hence, by Lemma 3.1:

HðVkÞ EkLðd_kÞ andHðVkÞ V_iþ1Lðe_kÞ Lðd_kÞLðe_kÞ:

The product of these relations yields, by choosing 2 ð1ðÞ;2Þ and using Lemmas 3.3 and 3.4:

HðVkÞ² Lðd_kÞ¹E^"_k⁰

D¹_k D^"_kþ1⁰ HðVkÞ ^{ð1Þþ2 "0}

hence"⁰5ð2 Þð2 ð1ÞÞ. This contradicts the assumption on"⁰ if and are close enough to₁ðÞ and_V.

Step 2. Relations between d_k and e_k. The following properties will be used many times in the proof of Theorem 4.1, sometimes without reference:

Lðd_kÞ ¼D^1þoð1Þ_k ; Lðe_kÞ ¼E_k^1þoð1Þ ð14Þ Dk<Ek4D_kþ1; Ek4D_k ð15Þ

HðVkÞD^1þoð1Þ_k 4Ek ð16Þ

D¹⁼ _k 4HðVkÞ; D² _kþ1 4HðVkÞ: ð17Þ In these formulas, andare chosen such that1ðÞ< <2 and0ðÞ< <2;

in the applications, they are assumed to be sufficiently close to1ðÞ and0ðÞ.

The inequalities hold forklarge enough (in terms of and). The symboloð1Þ denotes a sequence (possibly depending on and) that tends to zero asktends to infinity.

Equations (14) and (15) follow immediately from Lemma 4.2. Thanks to Lemma 3.1, they imply (16). At last, Eq. (17) follows from Lemmas 3.3 and 3.4.

We will use repeatedly the following consequence of these relations:

D^oð1Þ_k ¼E^oð1Þ_k ¼D^oð1Þ_kþ1 ¼HðVkÞ^oð1Þ:

Step 3.Construction of a sequence (w_t) of primitive points. Letk be a suffi- ciently large integer, say k5k₀. We define now an integer s_k51, and points n^ðkÞ₀ ;. . .;n^ðkÞ_s

k 2Z³, as follows. To begin with, let n^ðkÞ₀ ¼d_kþ1. For any 50 such that n^ðkÞ is defined and N^ðkÞ5Ek, we let n^ðkÞ_þ1 be the primitive element of Z³, with non-negative first coordinate, collinear to½n^ðkÞ ;d_k;e_k. At last, we lets_kbe the greatest integerfor whichn^ðkÞ is defined (ands_k¼ þ1ifn^ðkÞ is defined for any).

We letðw_tÞ_t51be the sequence of all pointsn^ðkÞ , withk5k₀and 044s_k, ordered according to their normsW_t. Since detðd_kÞand detðe_kÞare non-zero fork sufficiently large, we have detðw_tÞ 6¼0 fort sufficiently large.

Step 4.First properties of the sequence (w_t). In this step, we prove the follow- ing lemma:

Lemma 4.4. We have

sk4 1þoð1Þ

2₁ðÞ as k! 1; ð18Þ

Lðn^ðkÞ Þ ¼N^{ðkÞ1þoð1Þ}; ð19Þ and

all n^ðkÞ are minimal points; with D_kþ1 ¼N₀^ðkÞ> >N_s^ðkÞ_k ¼D_k; in particular n^ðkÞ_sk ¼d_k and n^ðkÞ_sk1¼e_k:

)

ð20Þ

Accordingly,s_k is finite.

Proof. Thanks to Eq. (14), Proposition 2.9 yields, fork5k₀and2 f1;. . .;s_kg:

N^ðkÞ4N₁^ðkÞ D_kE_k^1þoð1Þ and Lðn^ðkÞ Þ4Lðn^ðkÞ₁ÞD^1þoð1Þ_k E_k: ð21Þ By induction on, this implies

N^ðkÞ4D_kþ1

D_kE^1þoð1Þ_k

ð22Þ and

Lðn^ðkÞ Þ4D^1þoð1Þ_kþ1

D^1þoð1Þ_k Ek

ð23Þ for2 f0;. . .;s_kg, where the sequences involved in the notationoð1Þtend to 0 as ktends to infinity, uniformly with respect to. SinceLðn^ðkÞ ÞN^ðkÞ 1, it turns out that (22) and (23) are actually equalities.

Moreover, (16), (17) and (21) imply

N^ðkÞ<N₁^ðkÞ ð24Þ

ifkis large enough and 144s_k.

Let us prove (18) now. Letk5k₀and2 f0;. . .;s_k1g. Using Eqs. (17) and (22), and the fact thatN^ðkÞ5Ek, we obtain:

HðVkÞ^{1=ð2 Þ}D_kþ15D_k Eðþ1Þð1þoð1ÞÞ

k :

Now Eq. (16) yields

HðVkÞ^{1=ð2 Þ}5HðVkÞðþ1Þð1þoð1ÞÞ

D^1þoð1Þ_k : ð25Þ

If (18) fails to hold then for infinitely many such then we have ðþ1Þð1þ oð1ÞÞ>1=ð2 Þ(if is chosen small enough), hence (25) yields 1þoð1Þ<0 and (using Eq. (17))

1

2 5ðþ1Þð1þoð1ÞÞ þ ð1þoð1ÞÞ;

which implies

ðþ1Þð1þoð1ÞÞ4 1

2 :

This contradiction concludes the proof of (18). In particular, is bounded uni- formly with respect tok, henceoð1Þ ¼oð1Þ. Therefore the following inequality follows from (17), (22), and (23):

N^ðkÞLðn^ðkÞ Þ4HðVkÞ^oð1Þ for any2 f0;. . .;s_kg: ð26Þ NowHðVkÞ4E_kD^1þoð1Þ_k 4E_k4N^ðkÞif2 f0;. . .;s_k1g. In this case, Eq. (26) and Proposition 3.6 imply thatn^ðkÞ is a minimal point. Let us conclude the proof of (20) now.

We know that n^ðkÞ_s

k1 is a minimal point, with N^ðkÞ_s

k15E_k hence Lðn^ðkÞ_sk1Þ4 Lðe_kÞ. Eqs. (14), (15) and (21) yield

Lðn^ðkÞ_sk Þ4D^1þoð1Þ_k : ð27Þ

If N_s^ðkÞ

k <D_k then (27) gives n^ðkÞ_s

k 2A_oð1Þ; otherwise we have N_s^ðkÞ

k 5D_k

E_k¹⁼²5HðVkÞ¹⁼² and (26) impliesn^ðkÞ_s

k 2A_oð1Þ. Therefore Eq. (19) holds in both cases. Then Proposition 3.6 proves thatn^ðkÞ_s

k is a minimal point, saya_i. Ifi4i_k1 (that is,N_s^ðkÞ

k <D_k) then Eqs. (27) and (17) yield

D² _k HðVk1Þ D_kLða_ik1Þ4D_kLðn^ðkÞ_sk Þ4D^oð1Þ_k which is impossible. ThereforeN_s^ðkÞ

k 5D_k; but we have alsoN_s^ðkÞ

k <E_kby definition of sk. So n^ðkÞ_sk is a minimal point in Aoð1Þ between d_k and e_k, distinct from e_k. Lemma 4.2 proves thatn^ðkÞ_s

k ¼d_k.

By definition of n^ðkÞ_sk , this proves that d_k is collinear to

n^ðkÞ_sk1;d_k;e_k

. Eq. (8) implies that e_k is collinear to

n^ðkÞ_sk1;d_k;d_k

, hence to n^ðkÞ_sk1 thanks to (9). This concludes the proof of Lemma 4.4.

Step 5. Connection between (w_t) andA". The following result is a stronger version of Lemma 4.2:

Lemma 4.5.Let "2 ð0; "1Þ,andðu_iÞbe the sequence of all minimal points in

A",ordered according to their norms U_i.Then the sequencesðu_iÞandðw_tÞcoin-

cide up to a finite number of terms.

Proof. Eq. (19) meansLðw_tÞ ¼W_t^1þoð1Þ, and shows thatw_tbelongs toA"for tsufficiently large in terms of". Assume there is a minimal pointa_jinA", withj arbitrarily large, such that Wt<Aj<W_tþ1. Then we can write w_t ¼n^ðkÞ_þ1 and w_tþ1 ¼n^ðkÞ for somekand some 2 f0;. . .;sk1g. Lemma 3.1 yields

HðVkÞ AjL n^ðkÞ_þ1

and HðVkÞ N^ðkÞLðajÞ

hence, by choosing 2 ð1ðÞ;2Þand2 ð0ðÞ;2Þand using (22), (23), (15), and (17):

HðVkÞ² A^"_jEkD^1þoð1Þ_k D^"_kþ1D_k^1þoð1ÞHðVkÞ"

2 þ ð1Þþoð1Þ: Since" < "₁, this gives a contradiction when andand sufficiently small, andk is sufficiently large. This concludes the proof of Lemma 4.5.

Corollary 4.6.We have

Lðw_tÞ ¼W_t^1þoð1Þ and

0ðÞ ¼"ðÞ ¼lim sup

t!1

logW_tþ1 logWt

for any"2 ð0; "1Þ.

Proof. The first equality is nothing but Eq. (19). The second one follows from it, thanks to Eq. (11) and Lemma 4.5.

Step 6.Linear dependence between consecutive terms of the sequence (w_t).

Lemma 4.7.Let t be a sufficiently large integer such that w_t1,w_tand w_tþ1are linearly dependent. Then the bracket ½w_t;w_t;w_tþ1 is collinear to w_t1, so that

½w_t;w_t;w_t1is collinear to w_tþ1.

Proof. Thanks to Corollary 4.6, Proposition 3.7 applies (since ₀ðÞ<2) and provides an integerksuch that D_k4W_t1<W_t<W_tþ14D_kþ1. By construction, there exist non-zero integers,⁰ such that

w_t¼ ½w_tþ1;d_k;e_k ¼ w_tþ1Je_kJd_k and

⁰w_t1 ¼ ½w_t;d_k;e_k ¼ w_tJe_kJd_k: This implies

½w_t;w_t;w_tþ1 ¼ w_tJw_tþ1Jw_t

¼¹w_tðJw_tþ1Þ²Je_kJd_k

¼detðw_tþ1Þ¹⁰w_t1

since ðJxÞ² ¼ detðxÞId for anyx2Z³ identified with a symmetric matrix. This concludes the proof of the first statement of Lemma 4.7; the second one immedi- ately follows from Eq. (8).

Step 7.Construction and properties of . Letbe such that0ðÞ< <2, and ¼=2. Corollary 2.10 proves, thanks to Corollary 4.6, that½w_t;w_t;w_tþ1 2Aoð1Þ. By Proposition 3.6, this bracket is collinear to a minimal point fort sufficiently large; and by Lemma 4.5 this minimal point belongs to the sequenceðw_tÞ.

Therefore for any sufficiently large integer t, there exists a unique integer, denoted by ðtÞ, such that½w_t;w_t;w_tþ1is collinear tow _ðtÞ. Iftis not sufficiently large, we let ðtÞ ¼t1.

In this definition, depends on the choice of a threshold for determining when tis ‘‘sufficiently large’’, on the integerk₀chosen at the beginning of Step 3 (in the sequenceðw_tÞ, the indextis shifted if the choice ofk₀is modified), and on finitely many arbitrary choices of initial values (see Step 1). Actually this dependence is very mild, as shows the following lemma.

Lemma 4.8.

The function belongs to the setFdefined in Section2.1,and up toRit depends only upon.

For any t sufficiently large,the following assertions are equivalent:

(i) ðtÞ ¼t1.

(ii) w_t1,w_t and w_tþ1 are linearly dependent.

(iii) w_t is not among the d_k.

Proof. Let us start with the second statement. Thanks to Corollary 4.6, Proposition 3.7 applies and proves that ðiiÞ , ðiiiÞ. Since w_t, w_tþ1 and

½w_t;w_t;w_tþ1are always linearly dependent, the implicationðiÞ ) ðiiÞis obvious.

At last, the implicationðiiÞ ) ðiÞfollows from Lemma 4.7.