Yann Heuzé
email: heuze@cict.fr

José Braga

Application of non-adult Bayesian dental age assessment methods to skeletal remains: the Spitalfields collection

Keywords: Age estimation, Probabilistic methods, Incomplete dentition, Archaeology, Legal medicine

The aim of this study was to test with non-adult skeletons the quality of a recently published dental age assessment method based on Bayes' theorem. The test sample consisted of dental x-rays from 16 individuals (range 2.6 to 15.1 years) recovered from the crypt of Christ Church, Spitalfields, London, for which age-at-death and sex were known. The training sample consisted of dental x-rays from 1,346 girls and 1,040 boys (range 1.9 to 16.0 years). Bayesian dental age assessment method was alternatively used with dependent and independent assumptions; the dependent Bayesian method (DBM) and independent Bayesian method (IBM), respectively. Moreover, both DBM and IBM were used alternatively with sex-specific standards and with non-sex-specific standard.

Due to the training sample age distribution, both methods are efficient between 5.0 and 14.0 years. Whatever the method or the sex standard used, when dental age was different from chronological age, the discrepancy did not exceed one age group of one year. DBM produced tighter dental age 100% confidence intervals than IBM for the majority of assessments. The observed stability between results produced according to the type of sex standard used (i.e. sex-specific or non-sex-specific), and the number of teeth studied (range 4 -7), is worthy of note for both archaeological and forensic contexts.

Introduction

In archaeology and forensic sciences, x-ray studies of dental remains represent one means to assess age-at-death of non-adults. The majority of non-adult dental age assessment methods have been developed with biological samples, i.e. x-ray samples of living children (e.g. [START_REF] Gleiser | The permanent mandibular first molar: its calcification, eruption and decay[END_REF][START_REF] Garn | The sex difference in tooth calcification[END_REF][START_REF] Nolla | The development of the permanent teeth[END_REF][START_REF] Fanning | A longitudinal study of tooth formation and root resorption[END_REF]Moorrees et al.1963a,b;[START_REF] Liliequist | Skeletal and tooth development. A methodologic investigation[END_REF][START_REF] Demirjian | A new system of dental age assessment[END_REF][START_REF] Anderson | Age of attainment of mineralization stages of the permanent dentition[END_REF][START_REF] Proy | Maturation dentaire : constitution de tables[END_REF]Teivens and Mörnstad 2001;[START_REF] Chaillet | Dental maturity in south France: a comparison between Demirjian's method and polynomial functions[END_REF]). Before its application in archaeological or forensic contexts, such methods should be validated using dental remains of skeletons of known age in order to test its accuracy and reliability (i.e. its quality). Presenting easily interpretative results, this type of study contributes effectively to the estimation of non-adult age, which remains a central question in our discipline.

The aim of this paper was to test a Bayesian dental age assessment method of non-adults recently published by [START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF]. To do so, the remains recovered from the crypt of Christ Church, Spitalfields, London, for which age-at-death was known, were used. The 16 individuals suitable for this methodological test were studied according to both dependent and independent Bayesian methods [START_REF] Heuzé | Chronologie et étiologie de la maturation macrostructurale des dents définitives[END_REF][START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF] alternatively used with sex-specific and non-sex-specific standards (i.e. girls and boys in a same training sample). The use of non-sex-specific standards is linked to the difficulty to determine sex from skeletal remains of non-adults. Our results were compared to those presented by [START_REF] Liversidge | Dental maturation of 18th and 19th century British children using Demirjian's method[END_REF] who tested Demirjian's method [START_REF] Demirjian | A new system of dental age assessment[END_REF] with the Spitalfields collection concluding that if many of the younger children were dentally delayed, it was probably due to the dental age assessment method which is not entirely suitable for younger children. Contrary to Demirjian's method [START_REF] Demirjian | A new system of dental age assessment[END_REF], which needs all left permanent mandibular teeth (i.e. 31, 31, 33, 34, 35, 36 and 37 or I1, I2, C, P3, P4, M1 and M2), the two Bayesian dental age assessment methods (i.e. dependent and independent) allow the possibility to work with incomplete permanent mandibular dental remains (e.g. with only 1, 2, 3, 4, 5 or 6 permanent mandibular teeth).

Materials and methods

Training sample

The training sample consisted of dental panoramic radiographs (Table 1) which had been collected from 2,386 healthy children aged 1.9 to 16.0 years (1,346 girls and 1,040 boys). Nevertheless, the training sample contained only 8 children between the ages of 1.9 and 3.0 years. As [START_REF] Liversidge | Dental maturation of 18th and 19th century British children using Demirjian's method[END_REF] Montpellier, Toulouse). The panoramic x-rays had been taken for diagnostic purposes and none of the children had undergone orthodontic treatment before taking the selected radiograph. Non pathologic cases (i.e. for which a surgical operation was necessary) were selected. Only children who were free of anomalies in tooth numbers were selected. These descriptive data were based on the growth of children living in normal environments in which they experienced normal levels of infectious diseases and were not protected from environmental insults, corresponding to the definition of a reference sample [START_REF] Cameron | Methods of human growth research[END_REF], often called a training sample. Our study was ethically approved by a French institutional board (Commission Nationale de l'Informatique et des Libertés) and no regulations exist in Morocco, Iran or the Ivory Coast. The choice to use a non-specific geographic standard was supported by (i) the difficulty to assess the geographic origin of an individual from skeletal remains and (ii) the quality of age assessment which does not seem to depend predominantly on the use of geographicspecific standards [START_REF] Liversidge | Dental maturation in British children: Are Demirjian's standards applicable?[END_REF][START_REF] Liversidge | Growth of permanent mandibular teeth of British children aged 4 to 9 years[END_REF][START_REF] Liversidge | Variation in modern human dental development[END_REF][START_REF] Heuzé | Chronologie et étiologie de la maturation macrostructurale des dents définitives[END_REF][START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF][START_REF] Maber | Accuracy of age estimation of radiographic methods using developing teeth[END_REF][START_REF] Liversidge | Timing of Demirjian's tooth formation stages[END_REF]. This assertion (ii) supported by the authors does not create a consensus in our community. Some authors reported significant influences of geographic origin on teeth formation chronology or dental age (e.g. [START_REF] Hägg | Dental maturity as an indicator of chronological age: tha accuracy and precision of three methods[END_REF][START_REF] Staaf | Age estimation based on tooth development: a test of reliability and validity[END_REF][START_REF] Davis | The accuracy and precision of the Demirjian system when used for age determination in Chinese children[END_REF][START_REF] Koshy | Dental age assessment: the applicability of Demirjian s method in south Indian children[END_REF]. Nevertheless, as several authors [START_REF] Staaf | Age estimation based on tooth development: a test of reliability and validity[END_REF][START_REF] Liversidge | Accuracy of age estimation from developing teeth of a population of known age (0-5.4 years)[END_REF][START_REF] Scheuer | Developmental juvenile osteology[END_REF][START_REF] Liversidge | Variation in modern human dental development[END_REF] we think that some of these "geographical origin effects" reported so far are actually more linked to methodological (statistic, sampling) bias than use of inappropriate geographic standards.

Our point of view is well summarized by Liversidge (2003, p 101) in an exhaustive comparison of the different studies focusing on this point; "…if population differences in cross-sectional studies do exist, they are small".

Test sample

The material was an eighteenth and nineteenth centuries coffin-buried collection from Spitalfields in London [START_REF] Molleson | The anthropology: the middling sort[END_REF]. Due to the age distribution of the training samples of the Bayesian methods (Table 1), only radiographs of children with the incisors and/or the first molar at or after Demirjian et al.'s (1973) stage B were used.

Moreover, individuals who presented all their suitable permanent teeth at Demirjian et al.'s stage H (except the third molar) were not assessed in this study, since they could be aged between 14 and 99 years, or more. This way, developing permanent mandibular teeth from 16 individuals of recorded age-at-death were studied. It is important to note that for each child, three types of data were available in order to determine the age-atdeath: (i) the baptism record, (ii) the burial record and (iii) the coffin plate information.

For 12 out of 16 children, these data gave the same age-at-death. For the other children, conflicts existed. Concerning individual 2845, the coffin plate read "3 years 4 months 15 days", but this differed with the baptism and burial records ("4.28 years" and "4 years 6 months", respectively). We have chosen to use the baptism/burial age-at-death (i.e. "4.11 years"). Concerning individual 2283, both baptism and burial records read "3.53 years", but the coffin plate read "3 years 6 weeks". Nevertheless, the 1-year age interval remained the same (i.e. 3-4) and the baptism/burial age-at-death was used.

Concerning individual 2677, the burial record described her as 12 years (12-13 in modern terms) and the baptism record read "in her 12th year" (implying 11-12 years).

Here, the use of an interval of 11-13 years appeared more reasonable. Finally, concerning individual 2721, the baptism record read "14.98 years", the burial record read "13 years 4 months", and the coffin plate read "13.98 years". The use of an interval of 13-15 years interval seemed more appropriate also. [START_REF] Cox | Life and death in Spitalfields 1700 to 1850[END_REF] reported that parish records reveal that the individuals interred in the crypt were the middle-to upper-class members of the congregation. Nevertheless, they experienced the effects of overcrowding, air pollution, and inadequate sewage disposal.

Regardless of the relative wealth of families, the industrial environment contributed to deaths in early childhood and to growth retardation, exacerbated by the high prevalence of rickets and scurry [START_REF] Lewis | Impact of industrialization: comparative study of child health in four sites from medieval and post medieval England (A.D. 850-1859)[END_REF]. Moreover, the high frequency of linear enamel hypoplasias widely used as indicator of systemic growth disturbance indicated systemic physiological stress [START_REF] King | Linear enamel hypoplasias as indicators of systemic physiological stress: evidence from two known age-at-death and sex populations from postmedieval London[END_REF].

Methods

The dental stage assessment method

The degree of formation of the crowns and roots for the seven permanent teeth of the mandible were assessed according to the eight stages defined by [START_REF] Demirjian | A new system of dental age assessment[END_REF]. Stage 0 is added for the so-called "stage of the crypt" (the bone crypt is visible but dental mineralization has not yet begun); stages "1" -"8" corresponding to Demirjian et al.'s stages A to H. All orthopantomographs were assessed twice and the agreement was greater than 95%. When a difference existed, it was never greater than one stage.

Dependent Bayesian method (DBM)

The principle of a Bayesian dental age assessment method (dependent or independent) is to produce for each individual, according to his dental mineralization status, a distribution of posterior probability to belong to a particular age interval. The age interval with the highest posterior probability is the "dental age interval" of the tested individual. In order to know the 100% confidence interval (100% CI) of the estimation, it is necessary to consider all the age intervals for which the posterior probability is different from zero. Considering the following posterior probability distribution obtained with individual 2139 and presenting the dental mineralization sequence (DMS): 8_ 7 76 76 (dash corresponds to absence of information for I2) (Table 2), we can read: (i) "≤10-11< years" as the dental age interval and (ii) "≤9-14< years" as 100% CI. In this study, 1-year age intervals were preferred to 6-month age intervals due to the recent archaeological context of the Spitalfields collection.

After having assessed dental stages, the dependent Bayesian method (DBM) described by [START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF] using uniform priors was applied. The specificity of this method is to consider the seven permanent mandibular teeth studied as dependent attributes. Indeed, a DMS is the result of a dynamic pattern corresponding to sequenced, interrelated mechanisms and processes between developing teeth. A DMS is the macrostructural event sequence observed on the individual developing teeth at a precise moment of his life (i.e. a formula with seven numbers, e.g. 88 8 87 87 if I1, I2, C, P3, P4, M1 and M2 are Demirjian et al.'s stages H, H, H, H, G, H, G, respectively). A limited variability (but not necessary a low variability) in DMS is observed because teeth are topographically, developmentally and functionally associated to each other and essentially grow as a unit [START_REF] Heuzé | Chronologie et étiologie de la maturation macrostructurale des dents définitives[END_REF][START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF]. Calculations are based on the posterior probability for a DMS to be observed at a precise age interval. Each DMS is characterized by a posterior probability distribution. The inconvenience of DBM concerns the age assessment of children with DMS not represented in the training sample. Because of calculation features, the dental age of these children cannot be assessed (for more details on calculations see [START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF].

In order to validate our reference sample, we used cross-validation [START_REF] Tukey | Exploratory data analysis[END_REF] which offers the possibility to use the same sample alternatively as training and as test sample. Due to the inconvenience of DBM, a DMS represented only once in the reference sample led to no dental age estimation when the cross-validation technique is used.

Independent Bayesian method (IBM)

Although the consideration of teeth as independent attributes does not correspond to the biological reality of the dental development process, some researchers have developed an independent Bayesian method (IBM) associated with non-uniform priors [START_REF] Lucy | A Bayesian approach to adult human age estimation from dental observations by Johanson's age changes[END_REF][START_REF] Chaillet | Applications anthropologiques de l'approche Bayésienne dans la détermination de l'âge des immatures[END_REF][START_REF] Foti | New forensic approach to age determination in children based on tooth eruption[END_REF] or with uniform priors [START_REF] Heuzé | Chronologie et étiologie de la maturation macrostructurale des dents définitives[END_REF]). The last one is used in this paper. This method offers the possibility to produce dental age estimation for more children than DBM, because calculations are made tooth by tooth and not DMS by DMS. Indeed, the probability to find at least in one age interval of the training sample a I1 at the same stage as the I1 of the tested individual, a I2 at the same stage as the I2 of the tested individual, …, and a M2 at the same stage as the M2 of the tested individual, is always greater or equal to the probability of finding the DMS of the tested individual (for more details on calculations see [START_REF] Lucy | A Bayesian approach to adult human age estimation from dental observations by Johanson's age changes[END_REF][START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF]. Nevertheless, if none of the age intervals of the training sample presents the seven stages of the seven teeth of the child tested, the dental age of this child cannot be assessed. The validation of the training sample was also made by crossvalidation [START_REF] Tukey | Exploratory data analysis[END_REF].

Assessing quality of results

The percentage of response corresponds to the number of children for whom the method gives a dental age. With DBM, the percentage of response indicates the number of tested children with a DMS represented at least once in the training sample. With IBM, the percentage of response corresponds to the number of tested children for whom at least one age interval of the training sample presents a child with I1 at the same stage of the I1 of the child tested, the same child or another with I2 at the same stage of the I2 of the child tested,..., the same child or an other with M2 at the same stage of the M2 of the child tested.

Each child was assigned to 1 of 16 chronological age groups representing 1-year intervals. For all individuals tested, each dental age assessment was compared to the chronological age assignment and subsequently was given a rank (chronological agedental age). For example, if the child dental age assessment fell in the chronological age interval, it was classified as rank 0 (for more details see [START_REF] Braga | Non-adult dental age assessment: correspondence analysis and linear regression versus Bayesian predictions[END_REF]. If the rank value is positive, it corresponds to a delay in dental age, and if it is negative, it corresponds to an advance in dental age. The performance corresponds to the percentage of children classified into the rank 0 (i.e. best accuracy: ±6 months). In order to provide the confidence interval associated to a reliability of at least 95%, we added the necessary number of ranks. Indeed, if we had to sum up rank -3 to rank +3 in order to obtain 95% of the sample, we deal with a confidence interval of 84 months (7×12 months). The symmetry to rank 0 of the rank distribution allowed us to study the relative proportions of both negative and positive ranks (i.e. advance or delay in dental age respectively). If the sample is derived from a normal population following a Gaussian shape as close as possible and if the quality of the method is high, there will be no significant difference between both positive and negative rank proportions.

Results

Training sample

In order to discuss the results obtained with the Spitalfields children, the quality of assessments produced during the training samples validation step are presented (Table 3). With DBM, the percentage of response increased with the number of individuals contained in the training sample used. When non-sex-specific standards were considered, a significant majority of negative ranks (i.e. advance in dental age) was observed (Table 3). This was due to the significant majority of girls in the non-sexspecific training sample. Numerous examples in the literature reported sexual dimorphism in permanent teeth emergence [START_REF] Hurme | Ranges of normalcy in the eruption of the permanent teeth[END_REF][START_REF] Dahlberg | Emergence of the permanent teeth in Pima Indian children[END_REF][START_REF] Garn | Variability of tooth formation[END_REF] were the firstsee [START_REF] Demirjian | Dentition[END_REF] for more details) and formation; girls reaching most stages ahead of boys [START_REF] Haavikko | The formation and alveolar and clinical eruption of the permanent teeth, an orthopantomograph study[END_REF] reported difference between 0.3 and 1.7 years). Recent work on chronology of dental macrostructural mineralization using DBM and 6-month age intervals confirmed this dimorphism [START_REF] Heuzé | Chronologie et étiologie de la maturation macrostructurale des dents définitives[END_REF]). Indeed, 68.9% of negative ranks were due to girls but girls represented only 56.4% of the training sample.

With IBM, the percentages of response were maximal (i.e. 100%) and the 95% confidence intervals were significantly better (i.e. tighter) for both girls and boys than those obtained with DBM, except when non-sex-specific standards were used. The performances produced with IBM were significantly better for both girls and boys than those produced with DBM, but only when sex-specific standards were used. With IBM, rank distributions did not present significant asymmetry to the rank 0 (Table 3).

Test sample: Spitalfields children

Results from dental remains of 16 individuals from the Spitalfields collection are presented in Tables 4 and5 (with sex-specific and non-sex-specific standards, respectively). Because of their uncertain age-at-death (two different age intervals), individuals 2677 and 2721 were studied twice. Whatever the method used, there was no result for 2 out of the 16 children (2456 and 2845). These 2 children presented the same incomplete DMS: 55 4 1_ 4_ (dashes correspond to absence of information for P4 and M2). This was due to the lack of variability expressed by our training sample for the younger children.

IBM produced more dental age assessments than DBM (Tables 4 and5) but with IBM, the use of non-sex-specific standards allows the number of age assessments to be increased (Tables 4 and5). This is due to the higher variability expressed by the nonsex-specific training sample because of its greater number of individuals. When IBM was used alternatively with sex-specific and non-sex-specific standards, the rank values remained the same for all the children for whom the comparison was possible. With DBM 3 individuals (2302, 2730 and 2677) presented different ranks according to the training sample used (i.e. sex-specific or non-sex-specific). Nevertheless, these rank value changes were never greater than one age interval.

Whatever the method used, when the dental age interval was different from the age-at-death interval, this difference was never greater than one age interval (i.e. ranks -1 or ranks +1). The only exceptions concerned the individuals 2677 and 2721 for whom the precise age-at-death was unknown. In these two cases, when the assumed older ageat-death was considered, there was a difference of 2 age intervals between dental age and age-at-death (i.e. ranks +2).

In order to know if the Spitalfields children were dentally delayed, the proportions of positive and non-positive ranks (i.e. ranks 0 and negative ranks) between training and Spitalfields samples were compared (Table 6). With both DBM and IBM, the Spitalfields girls presented significant dental age delays (i.e. a significant greater proportion of positive ranks than the training sample) when the female standards were used. With IBM, the Spitalfields children also presented dental age delays when the non-sex-specific training sample was used. This was not the case when DBM was used with the non-sex-specific standards.

Regarding the 100% confidence intervals (i.e. the totality of age intervals with a posterior probability different from 0), DBM produced tighter confidence intervals than IBM for the majority of assessments. However, problems existed with the Spitalfields children younger than 5.0 years: sometimes the 100% CI did not contain the age-atdeath of the tested child (2625, 2302, 2621 when DBM was used with sex-specific standards; 2365, when IBM was used with sex-specific standards; 2625, 2621 when DBM was used with non-sex-specific standards; 2730 when IBM was used with nonsex-specific standards) (Tables 4 and5).

For the older children, the 100% CI indicated "≤8-16 or more", "≤9-16 or more" or "≤10-16 or more" because for these children the posterior probability of the 15-16 age interval was not different from zero. Consequently, we have to consider that these dental mineralization statuses can exist after 16 years and this is why the mention "or more" was reported.

Discussion

The application of both Bayesian dental age assessment methods to the Spitalfields children aged 2.56 -15.07 years shows that in the present state of the age distribution of our biological training sample (Table 1), the method is satisfactory only between 5.0 and 14.0 years. This situation could be solved on one hand by increasing the training sample size with older children. Unfortunately, on the other hand, the possibility to increase the number of younger children of the training sample is problematic, because panoramic x-rays of living children younger than 3.0 years are not numerous and it is impossible to make this type of examination only for dental age assessment studies. Due to the legislation, a study like the growth records of the Fels Research Institute in Yellow Springs, Ohio (USA), used by Moorrees et al. (1963a,b), is hard to develop in France. Other types of examinations have to be considered.

The ranks produced with the non-sex-specific standards are the same than those produced with the sex-specific standards, except for three children with DBM. This is an interesting result from archaeological and forensic perspectives, because sex determination of non-adults remains problematic and associated to low reliability (except when DNA analysis is possible). [START_REF] Saunders | Accuracy tests of tooth formation age estimations for human skeletal remains[END_REF] reported the same observation applying a different dental age assessment method (Moorrees et al. 1963a,b) to dental remains of 17 non-adult skeletons of known sex and age-at-death from St Thomas' Church cemetery, Ontario, Canada. Nevertheless, with both DBM and IBM, 100% CIs were tighter when sex-specific standards were used than when non-sexspecific standards were used. These results should not to be interpreted as a stronger sensitivity to sexual dimorphism of DBM compared to IBM. Indeed, when we tested the boys of the training sample, no significant difference (p=0.5273) was observed between the proportions of rank value changes according to the sex standards used (27.6% with IBM and 28.9% with DBM). Testing the girls of the training sample, IBM appeared to be more sensitive (p=0.0019) to sexual dimorphism than DBM (28.0% of rank value changes with IBM and 22.6% with DBM).

The observed stability of results according to the number of teeth available (range 4 -7 this study) is appealing for both archaeological and forensic fields.

Nevertheless, the teeth that complete their development later (i.e. mainly P4, M2 and secondly P3 and C) are more "powerful" (i.e. more informative) in age assessment than teeth that complete their development earlier. This tendency is illustrated by the following simulation: dental age of the 2,386 children from the reference sample is estimated by cross validation using DBM and non-sex-specific standard with only four teeth; I1, I2, C and M1 on the one hand, P3, P4, M1, and M2 on the other hand. The two rank distributions (Fig. 1) showed significant differences (χ²=116.1, p=8.46x10 -32 , df=8). Using P3, P4, M1, and M2 allowed to produce tighter and more symmetric to rank 0 distribution and better performance (i.e. number of rank 0).

Applying Demirjian et al.'s method (1973) to the Spitalfields children, [START_REF] Liversidge | Dental maturation of 18th and 19th century British children using Demirjian's method[END_REF] reported a tendency for the younger ones to be delayed in dental age and interpreted these delays as a methodological bias. With both Bayesian methods, the children delayed in dental age (i.e. with positive ranks) were not only the younger ones (Tables 4 and5). With both DBM and IBM, the Spitalfields girls presented significant dental age delays (i.e. a significant greater proportion of positive ranks than the training sample) when the female training sample was used. With IBM, the Spitalfields children presented dental age delays when the non-sex-specific training sample was used. These delays could be linked to the industrial environment and its consequences as systemic physiological stress [START_REF] Lewis | Impact of industrialization: comparative study of child health in four sites from medieval and post medieval England (A.D. 850-1859)[END_REF][START_REF] King | Linear enamel hypoplasias as indicators of systemic physiological stress: evidence from two known age-at-death and sex populations from postmedieval London[END_REF]). Another explanation could be the use of non-specific geographic standards and the resulting potential reduction of the age estimations accuracy. On the other hand, with IBM and DBM, the Spitalfields boys did not present significant delays when the male training sample was used. Finally, when DBM was used with the non-sex-specific training sample, Spitalfields children did not present significant delays (Tables 6). Significant dental age delays or advances seem to depend predominantly on both method and training sample used. Consequently we should be cautious. According to the observed variability in dental mineralization chronology, it appears to be difficult to generalize interpretations in terms of advance or delay in dental age based on results concerning less than 16 children. The delay or advance observed in dental age makes sense solely for the individual concerned.

Finally, it is important to remember the way to interpret the results. In fact, we are dealing with individual age assessment (the term "individual" is very important): each child presents a particular posterior probability distribution according to its dental mineralization status (i.e. DMS). The child rank represents the difference between the age-at-death interval and the most probable estimated age interval (i.e. the dental age interval). Ranks form a necessary tool in order to produce interpretable results when samples are treated. But rank is not a perfect tool. Considering the following posterior probability distribution obtained with individual 2139 presenting the DMS: 8_ 7 76 76 (Table 2), the dental age interval was 10-11 years (p=0.435) and the rank was 0. But the posterior probability of belonging 11-12 years (p=0.426) is almost equal. Presenting this boy with the probability of 0.861 of belonging to the 10-12 years group appears more realistic. This is precisely one of the most important advantages proposed by both dependent and independent Bayesian methods: the flexibility offered to the user for the interpretation of the posterior probability distribution according to the aims and restraints. Producing a dental age interval of 1 year using, for example, a M1 at Demirjian et al.'s (1973) stage G is not realistic when we know that this stage of mineralization for a M1 is observed in our non-sex-specific training sample from 5.1 to 15.3 years.

Conclusions

According to the small sample studied, this study cannot represent a validation of both dependent and independent Bayesian dental age assessment methods of nonadults in archaeological or forensic contexts. The 16 children of the Spitalfields collection studied here give only indications about the quality of these two methods when skeletal material is considered. The same comments could be made for the studies of [START_REF] Saunders | Accuracy tests of tooth formation age estimations for human skeletal remains[END_REF] or [START_REF] Liversidge | Dental maturation of 18th and 19th century British children using Demirjian's method[END_REF] based on 17 and 15 children, respectively. Results produced by both DBM and IBM are comparable to those presented by [START_REF] Liversidge | Dental maturation of 18th and 19th century British children using Demirjian's method[END_REF] with Demirjian et al.'s method (1973). Nevertheless, both Bayesian methods present two important advantages over Demirjian et al.'s method (1973): (i) they can be applied with incomplete dentition and (ii) the results produced (i.e. posterior probability distribution) allow a flexible and adaptable interpretation by the user. The observed stability between results according to the type of sex standard used (i.e. sex-specific or non-sex-specific) and the number of teeth studied (range 4 -7) is worthy of note in both archaeological and forensic contexts.

According to the best confidence intervals produced and the biological reality of dental development, we recommend the priority use of DBM when the DMS of the child is represented in the training sample (if it is not the case, IBM has to be applied).

  mentioned the method of Demirjian et al. (1973) is probably not suitable for the younger Spitalfields children studied (i.e. 7 children under 4.5 years) because of the lack of children younger than 4.0 years in the Demirjian et al.'s (1973) original reference sample. Due to difficulties linked to the availability of x-rays for children younger than 4.0 years, our training sample presented the same problem. Therefore, dental age assessment of the younger Spitalfields children would still remain problematic. All the children of our training sample lived in the areas of Abidjan (Ivory Coast), Shiraz (Iran), Marrakech (Morocco) or in southern France (Bordeaux,

Table 1

 1 Age and sex distribution of the training samples used for Bayesian methods SL of symmetry to rank 0 significance level of the difference between both positive and negative rank proportions, (-) significant majority of negative ranks Table4Dental age assessments of the Spitalfields children according to the method used with sex-specific standard

	Age interval (years) Girls Boys Combined sex
	≤0-1<		0	0		0
	≤1-2<		1	0		1
	≤2-3<		2	5		7
	≤3-4<		20	34		54
	≤4-5<		50	57	107
	≤5-6<		69	72	141
	≤6-7<		86	74	160
	≤7-8<		109	77	186
	≤8-9<		164 107	271
	≤9-10<		170 129	299
	≤10-11<		177 117	294
	≤11-12<		178 126	304
	≤12-13<		134 103	237
	≤13-14<		95	78	173
	≤14-15<		50	37		87
	≤15-16<		41	24		65
	Total		1346 1040	2386
	Table 2
	An example of posterior probability distribution: the DMS 8_ 7 76 76
	Number	SMD	≤8-9< ≤9-10< ≤10-11< ≤11-12< ≤12-13< ≤13-14< ≤14-15<
	2139	8_ 7 76 76	0	0.007	0.435	0.426	0.124	0.008	0
	Table 3
	Quality of age assessments with training samples according to the method used
				Dependent Bayesian method Independent Bayesian method
	Training sample		Girls Boys Combined sex Girls	Boys Combined sex
	N of individuals		1346 1040	2386	1346	1040	2386
	Percentage of response (%) 87.2	85.4	91.2	100.0 100.0	100.0
	95% CI (months)		84	84	84	60	60	84
	Performance (%)		25.9	33.4	34.9	40.1	38.8	36.7
	SL of symmetry to rank 0	0.18	0.09	0.00 (-)	0.25	0.93	0.40
	CI confidence interval,

N teeth number of teeth suitable, AAD age-at-death, Rank chronological age intervaldental age interval, DBM dependent Bayesian method, IBM independent Bayesian method, CI confidence interval

Table 5

 5 Dental age assessments of the Spitalfields children according to the method used with non-sex-specific standards Comparison of positive and non positive ranks proportions between training and tests samples (p level of Yates corrected χ²)

	Number Sex N teeth AAD DBM rank IBM rank DBM 100% CI IBM 100% CI
	2431 M	5	2.56	0	0 ≤2-3<	≤2-3<
	2520 F	5	2.81		0	≤2-3<
	2815 F	5	3.21		1	≤2-3<
	2365 M	7	3.34		1	≤2-3<
	2283 F	5	3.53		1	≤2-3<
	2456 F	5	3.63
	2845 M	5	4.11
	2625 M	7	4.17	1	1 ≤3-4<	≤3-5<
	2302 M	5	4.50	0	0 ≤3-5<	≤3-7<
	2621 M	7	4.55	1	1 ≤3-4<	≤3-5<
	2730 M	6	5.40	1	1 ≤4-6<	≤3-5<
	2266 M	7	7.88	0	0 ≤5-10<	≤5-11<
	2139 M	6 10.35	-1	0 ≤9-12<	≤8-16 or more
	2677 F	7 11-12	0	1 ≤9-12<	≤8-16 or more
	2677 F	7 12-13	1	2 ≤9-12<	≤8-16 or more
	2721 F	5 13-14	1	1 ≤9-16 or more ≤9-16 or more
	2721 F	5 14-15	2	2 ≤9-16 or more ≤9-16 or more
	2175 F	4 15.07	1	1 ≤10-16 or more ≤10-16 or more
			Female		Male	Combined sex
		N ranks >0 N ranks ≤0 N ranks >0 N ranks ≤0 N ranks >0 N ranks ≤0
	DBM
	Training sample	407	767	266	622	632	1543
	Test sample	5	0	3	4	7	5
	Comparison		0.0097		0.7431	0.0567
	IBM
	Training sample	381	965	320	720	732	1654
	Test sample	5	0	4	4	11	5
	Comparison		0.0023		0.4304	0.0026

N teeth number of teeth suitable, AAD age-at-death, Rank chronological age intervaldental age interval, DBM dependent Bayesian method, IBM independent Bayesian method, CI confidence interval

Table 6

Acknowledgments

Yann Heuzé was funded by a postdoctoral fellowship of the Foundation Fyssen and this study was supported by the EVAN project and the GDR 2152 of the CNRS. We thank L Humphrey of the Palaeontology Department (Natural History Museum, London) for allowing access to the collection and for advice. We also thank H Liversidge for interesting discussions.

DBM dependent Bayesian method, IBM independent Bayesian method


Thérèse Fallièro

A On

Fekete-Szeg

Theorem 2024 Hal-

T H Fekete-Szeg Ö Theorem

Èse Ér

Falliero

Keywords: Capacity, Fekete-Szegö theorem, calibration, Motzkin's theorem, Minkowski's theorem

Introduction

A classical theorem of Fekete and Szegö [START_REF] Fekete | On algebraic equations with integral coefficients whose roots belong to a given set[END_REF] says that, given a compact set K in the complex plane, then (a) If C(E) < 1, there is an open set U containing K such that there are only finitely many algebraic integers α having all of their conjugates in U , and (b) if C(E) ≥ 1, then every open U containing K contains infinitely many such α. Furthermore Fekete and Szegö proved also that if E is a compact set in the complex plane, stable under complex conjugation and having logarithmic capacity C(E) ≥ 1, then every neighborhood of E contains infinitely many conjugate sets of algebraic integers. In [START_REF] Ennola | Conjugate algebraic integers in an interval[END_REF] V. Ennola solved a question raised by R. M. Robinson that if ∆ is any real interval of length greater than 4, then for any sufficiently large n there exists an irreducible monic polynomial of degree n with integer coefficients all of whose zeros lie in ∆ .

We should emphasize that many diophantine inequalities are reduced to the existence of lattice points in some convex body [START_REF] Cassels | An introduction to the geometry of numbers[END_REF]Chapter III]. In this direction, it is remarkable that T. Chinburg [START_REF] Chinburg | Capacity theory on varieties[END_REF] reduces the proof of Fekete theorem to an application of Minkowski's Convex Body theorem [7, Chapter III] that we recall for later use.

This problem is actually related to a precise form of the Stone-Weierstrass theorem. This classical theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated by polynomials. A more general statement is if X is a compact Hausdorff topological space and if C(X) is the algebra of real-valued continuous functions f : X → R, then A subalgebra A ⊂ C(X) is dense if and only if it separates points.

The question is for which compact set continuous functions can be approximated by polynomials with integer coefficients? This question is a major one in approximation theory and the literature is very extensive [START_REF]Approximation by polynomials with integral coefficients[END_REF]. Let f be a continuous real-valued function defined on [0, 1], then the sequence of polynomials (p n ) defined by

p n (x) = n ν=0 f ν n n ν x ν (1 -x) n-ν
converges uniformly to f . This is therefore a constructive proof of the Stone-Weierstrass theorem.

It is due to Bernšte ȋn. We deduce from this result [START_REF] Ferguson | What can be approximated by polynomials with Integer coefficients[END_REF]Theorem 5] that for a continuous real-valued function f on the unit interval I = [0, 1] to be uniformly approximable by polynomials with integer coefficients it is necessary and sufficient that it be integer-valued at both 0 and 1.

As was pointed out in [14, Lemma 1], If q is a non constant polynomial with integer coefficients and I is an interval of length at least four, then q ≤ 2. Hence, clearly, the approximation by polynomials with integer coefficients on a set E is related to to the capacity of E. Consequently [START_REF] Ferguson | What can be approximated by polynomials with Integer coefficients[END_REF]Theorem 2 ] if the capacity C(I) ≥ 1 the only functions that are uniformly approximable in I by polynomials with integer coefficients are these polynomials themselves. On the other hand one can prove the surprising result that any f ∈ L 2 ([a, b]), b -a < 4 can be approximated on the interval [a, b] by polynomials with integer coefficients. Theorem 1.1 (Pál). If f is continuous on [-a, a], |a| < 1, and f (0) is an integer then f may be uniformly approximated thereon by polynomials with integer coefficients.

These problems are at the core of the approximation theory by polynomials with integer coefficients. It is a classical result, due to Chebyshev that for an interval [a, b]

inf Q Q(z) ∞ = 2 b -a 4 n
where Q(X) describes the set of monic polynomials of degree n. Hilbert showed in [START_REF] Hilbert | Ein Betrag zur Theorie des Legendreschen Polynoms[END_REF] that if we restrict to monic polynomials of degree n with integer coefficients, then

inf Q Q(z) L 2 ([a,b]) ≤ C √ n b -a 4 n/2
and Fekete showed in [START_REF] Fekete | Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten[END_REF] the more flexible

inf Q Q(z) ∞ ≤ 2 1-1 n+1 (n + 1) b -a 4 n/2
.

The main objective of this work is the following question suggested by J.P. Serre in [START_REF] Serre | Distribution asymptotique des valeurs propres des endomorphismes de Frobenius[END_REF]. If K is a compact of C, symmetric with respect to the real axis with C(K) > 1, and if U is an open containing K, there is a sequence (P n ) of monic polynomials with integer coefficients whose roots are in U and are such that the associated zeros counting measure δ Pn converge weakly to equilibrium µ K of K.

Organization of the paper: Very succinctly, we give the definitions in the first section, then we introduce the counting measures in the second section. Minkowski's theorem will be discussed in section three. The fourth section is devoted to certain approximations results and the fifth section, the longest, is devoted to the introduction of certain Riemann surfaces, in relation to certain Jacobi matrices and the solution of the Serre question.

Definitions

2.1. Approximation on intervals. If µ is a measure on C of compact support, its logarithm potential is defined by

Φ µ (z) = ln(|z -w| -1 ) dµ(w) .
This integral converges if z ∈ supp(dµ), and since dµ has compact support, ln(|z-w| -1 ) is uniformly bounded below for (z, w) ∈ supp(dµ) × supp(dµ), so the integral for each z ∈ supp(µ) either converges or diverges to +∞, in which case we set Φ µ (z) = +∞.

Potentials enter naturally in studying growth of polynomials as n → ∞. For if

P n (x) = n j=1 (x -x (n) j ) then 1 n ln |P n (x)| = -Φ νn (x)
where

ν n = 1 n n j=1 δ x (n) j
is the counting measure for the zeros x

(n) j . The function Φ µ (z) is bounded below on supp(µ), so

I(µ) = Φ µ (z) dµ(z) = K×K ln(|x -y| -1 )µ(x)µ(y)
is either finite or diverges to +∞. I(µ) is called the potential energy of µ or, for short, the energy of µ. Consider a compact K ⊂ C. We consider all probability measures M +,1 (K) on K. We say K has capacity zero if and only if I(µ) = ∞ for all µ ∈ M +,1 (K). We set

v(K) = inf µ I(µ)
where µ runs over all positive probability measures supported in K. Then the capacity of K is defined as C(K) = e -v(K) . Its logarithm v(K) is called the logarithm capacity of K.

For a compact K with non-zero capacity there exists a unique positive probability measure µ, such that I(µ) = ln C(K) -1 . This measure µ = µ K is called the equilibrium measure of K.

Let us also recall that if (µ n ) n and µ ∞ are probability measures on a compact Hausdorff space

X, (µ n ) n converges weakly µ ∞ if f dµ n → f dµ as n → ∞
for every f continuous on X and if f a real valued function defined on a topologic space E, f is lower semicontinuous at a if

f (a) = lim inf x→a f (x) .
We have the following property of the potential energy

µ -→ I(µ)
is weakly lower semicontinuous. The lower semi continuity means

µ n → µ ⇒ lim inf I(µ n ) ≥ I(µ) equivalently (1) I -1 ((-∞, a]) is closed for all a (2) I -1 ((a, ∞]) is open for all a.
Before continuing further, we wish to recall some definitions that will be need. Given a bounded set E in the complex plane, we denote by E(r) the r-neighborhood of E. Definition 2.1. If E 1 and E 2 are two bounded sets in C, the difference between E 1 and E 2 is the smallest r such that E 1 (r) contains E 2 and E 2 (r) contains E 1 .

We will denote this difference by δ(E 1 , E 2 ), it is small if and only if E 1 and E 2 are (almost) super imposable. We will say that E 1 is near E 2 (and reciprocally). If E is a compact set and (E ν ) a family of sets such that δ(E, E ν ) tends to zero as ν tends to ∞ we will say simply that (E ν ) tends to E.

We then deduce the lemma

Lemma 2.1. If δ(K n , K) → 0 and C(K n ) → C(K) then µ Kn → µ K .
Proof. As M +,1 (K) is compact let η be a weak limit point of µ Kn . By lower semicontinuity of the energy I,

I(η) ≤ lim inf I(µ Kn ) = lim ln(C(K n ) -1 ) = ln(C(K) -1 ), so η = µ K , that is, µ Kn → µ K .
Note that the inverse is not true δ Pn → µ K and C({zeros of P n }) = 0.

3. Algebraic integers with all conjugates in a given compact.

For a set V ⊂ C let Pol V be the set of monic polynomials in Z[X] of degree at least 1 such that all their roots lie in V . If z is an algebraic integer, a root of a polynomial P (X) ∈ Pol V , then all the conjugate of z are in V and we say that z is totally in V .

Let Irr V be the set of irreducible monic polynomials in Z[X] of degree at least 1 such that all their roots lie in V . for such a polynomial P (X) of degree g let µ P be the corresponding probability measure supported in its roots, µ P = 1 g g i=1 δ x i . Now let K ⊂ C be compact. There are two quite different cases [START_REF] Serre | Distribution asymptotique des valeurs propres des endomorphismes de Frobenius[END_REF], [START_REF] Rumely | Capacity Theory With Local Rationality: The Strong Fekete-szegö Theorem on Curves Mathematical Surveys and Monographs[END_REF], depending on the capacity of K.

(

) If C(K) < 1, then Irr K is finite. (2) If K ⊂ R is a union of finite number of segments and C(K) > 1, then Irr K is infinite. (3) If K ⊂ C is Gal( Q/Q)-stable and C(K) ≥ 1, 1
then for any open U, K ⊂ U , the set Irr U is infinite.

3.1. The case of a finite union of intervals. Let E ⊂ R be a finite union of segments with C(E) > 1 and µ E be its equilibrium measure. We have the following theorem ([28] Theorem 1.6.2).

Theorem 3.1 (Serre). There exists a sequence of polynomials

P n ∈ Pol E such that µ Pn → µ E .
Then from the Fekete-Szego theorem a natural question, suggested by J.P Serre, arises.

Question 3.1. Let K be a compact of C stable under complex conjugation, of capacity C(K) ≥ 1 and U an open set containing K. Is there a sequence of polynomials P n ∈ Pol U such that µ Pn → µ K ?

3.2. Precisions on the properties of K. Let U ⊂ C be a set and let U * be the so called "symmetric kernel" of U consisting of those points of U which belong, together with their conjugates to U . So U * is symmetric with respect to the real axis and naturally C(U * ) ≤ C(U ). M.Fekete [START_REF] Fekete | Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten[END_REF], [START_REF] Chinburg | Capacity theory on varieties[END_REF] proved that if K is a compact of C such that C(K * ) < 1, then there is only a finite number of irreducible algebraic equations with integer coefficients of the form z n + a 1 z n-1 + ... + a n-1 z + a n = 0 whose roots lie all in K * . Theorem 3.2 (Minkowski's theorem). Suppose K to be a symmetric, convex, bounded subset R d . If vol(K) > 2 d , then K contains at least one lattice point other than 0.

There is an extension to general lattices

Λ = Zu 1 ⊕ • • • ⊕ Zu d , where {u 1 , • • • , u d } is a basis of R d .
We define vol(Λ) as the volume of the parallelotope

d i=1 α i u i , 0 ≤ α i ≤ 1
Theorem 3.3 (Minkowski's theorem for general lattices). Suppose Λ to be a lattice and K to be a bounded symmetric convex subset in R d . If vol(K) > 2 d detΛ, then K contains at least a point of Λ different from 0.

We must perhaps insist that Minkowski's theorem as well as Motzkin's theorem on simultaneous approximation (which in turn depends on two theorems of Kronecker. The first one [20, p.159]) is at the heart of the diophantine approximation and then at the heart of the approximation by polynomials with integer coefficients. The second one [START_REF] Kronecker | Zwei Sätze über Gleichungen mit ganzzahligen Koeffizienten[END_REF] state that if an algebraic integer α and all of its conjugates are in the closed unit disk D := {z ∈ C : |z| ≤ 1}, then it is either α = 0 or it is root of unity. This is apparent at [7, Chap III], Feruguson [13, Theorem 1.1] and Chinburg [START_REF] Chinburg | Capacity theory on varieties[END_REF]. For the sake of completeness and in order to see how the different idea articulate we give an idea of the proof of the first part of Fekete-Szegö theorem. For a ∈ R n+1 we define the polynomial

f a (z) = a 0 + • • • a n z n . Let K be a compact such that C(K) < 1, then [8] then f n (E) = {a = (a 0 , a 1 , • • • , a n ) ∈ R n+1 \ {0}, | f a (E) |< 1.}
is an open symmetric convex set, with large volume for a large n. By Minkowski's theorem,

f n (E) contains some a = (a 0 , a 1 , • • • , a n ) ∈ Z n+1 \ 0}.
We fix a such a and we consider the open set

U = {z ∈ C, | f a (z) |< 1}.
If α and its conjugates are contained in U , then f a (α) and its conjugates are contained in the unit disk D. By the second Kronecker theorem α is one of the many roots of the polynomial f a (z).

Remark 3.1. The idea of using Minkowski's convex body theorem in this context goes back to Hilbert [START_REF] Hilbert | Ein Betrag zur Theorie des Legendreschen Polynoms[END_REF]. We can rephrase, in a classical way [START_REF] Cassels | An introduction to the geometry of numbers[END_REF], what we said on the proof of the first part of Fekete-Szegö theorem: Any convex body of volume at least 2 n contains at leasr 2 n + 1 integral points. In particular the system of linear inequalities

n k=1 a k,m x k ≤ b m , 1 ≤ m ≤ n with det (a k,m ) 1≤k,m≤n = 0, n m=1 b m ≥ det (a k,m ) 1≤k,m≤n has a nonzero integral solution. Now let K be a compact of C of capacity C(K) ≥ 1. Let U a neighborhood of K then U * is a neighborhood of K * (
with the convention that the empty set is the neighborhood of the empty set). If C(K * ) < 1 then by continuity of the capacity, there exists V a neighborhood of K such that C(V * ) < 1. From what precedes, there exists only a finite number of algebraic integers with all its conjugate in V . Then to show that there exists an infinity of algebraic integers totally in U , we can assume C(K * ) ≥ 1 and finally the hypothesis of K symmetric with respect to the real axis is natural.

K being a metric compact set, many assumptions can be made on ∂K.

Definition 3.1. We say that K has a continuous boundary when K is a non empty union of connected components non reduced to a point.

Recovering K with a finite number of small enough closed balls, we can assume that the boundary ∂K is continuous. From [4, Proposition p. 18 ] we can suppose K with regular boundary of class C ∞ . In fact let U be an open set containing K, there exists a

C ∞ function ϕ in R 2 such that (1) ϕ = 1 on K, (2) supp (ϕ) ⊂ U .
Then K ⊂ supp (ϕ) ⊂ U with a compact support and with regular C ∞ boundary. Let us recall that a Jordan analytic curve is a closed curve Γ in C which possesses a neighborhood V and a conformal map ξ from V on {α < |z| < β}, such that the image of Γ by ξ is the circle {|z| = r}, α < r < β. We can also suppose that ∂K is a set of analytic Jordan curves. To see this a first method is to use [START_REF] Fekete | On algebraic equations with integral coefficients whose roots belong to a given set[END_REF]Theorem G] . Let K(ρ) be the ρ-neighborhood of K (ρ > 0), there exists ρ sufficiently small so that K(ρ) ⊂ U . From the preceding theorem, there exists a domain defined by a lemniscate containing K and contained in K(ρ):

for n ≥ n 1 (ρ), {z, |w n (z)| ≤ ν n } where w n (z) = n k=1 (z -ζ (n) k )(z -ζ (n) k ), ν n = max K |w n (z)|.
In conclusion this compact set is invariant under complex conjugation and its boundary consists in analytic Jordan curves.

A second method can be found in [1, p.144

from U p onto a neighborhood V p of 0 in R 2 such that ϕ p (p) = 0, ϕ p (U p ∩ Ω) = V p ∩ {(x, y) ∈ R : x ≤ 0}
and the Jacobian determinant J(ϕ p ) is > 0 in U p .

We have (see for example [START_REF] Berenstein | Complex variables, An introduction[END_REF]Proposition p. 27]) that for Ω a relatively compact, open subset of C with piecewise regular boundary (of class C k , k ≥ 1), there is only a finite number of connected components of ∂Ω and each of them is a Jordan curve (piecewise C k ).

In conclusion we can always suppose that K is a compact with C ∞ boundary, then the number of connected components of ∂K is finite.

3.3.

A generalization of Serre's method, the case of a compact of R. We use [START_REF] Baker | Plane domains with hyperelliptic double[END_REF]Theorem 13] which characterizes such domains. In these domains are those symmetric with respect to the real axis and whose intersection with the open upper half plane is simply connected. We can also find domains such that Γ = (Γ 1 , Γ 2 ) or (Γ 1 , Γ 2 , Γ 3 ) which are invariants under conjugation.

We have to notice that a compact e of R of capacity C(e) ≥ 1 is an example of such domains. The fact that for all neighborhood U of e in R, there exists a sequence of monic polynomials of degree > 0, with coefficients in Z, whose all roots are in U , (P n ) such that lim µ Pn = µ e will be a consequence of [START_REF] Serre | Distribution asymptotique des valeurs propres des endomorphismes de Frobenius[END_REF]Theorem 1.6.2 ].

For the convenience of the reader we recall some facts (see for example [START_REF] Simon | Equilibrium measures and capacities in spectral theory[END_REF], Chapter 5).

Proposition 3.1. Let e ⊂ R be compact. Let ẽn = {x ∈ R, dist(x, e) ≤ 1 n } .
Then

j \e j | ≤ C 1 m -1 (1) C(e) ≤ C(e (m) ) ≤ C(e) + C 2 m -1 (2)
Then we have and each e n is the spectrum of some two-sided periodic Jacobi matrix. Moreover [START_REF] Ahlfors | Riemann Surfaces[END_REF]

ρ en → ρ e , ( 2
) C(e n ) → C(e).
Now, we know that e n is a finite union of calibrated intervals and then by [START_REF] Sebbar | Th. Capacities and Jacobi matrices[END_REF], [START_REF] Serre | Distribution asymptotique des valeurs propres des endomorphismes de Frobenius[END_REF], there exists a sequence of monic polynomials with coefficients in Z, P k,en , whose all roots are in e n , such that lim k µ P k,en = µ en . By the diagonal process µ e = lim j µ P k j ,en j .

Remark 3.2. In particular we can release the condition of countinuous boundary in the theorem 9.3 (Bilu, Rumely) of [START_REF] Autissier | Chapter VIII : Autour du théorème de Fekete-Szego[END_REF].

Some approximations

We have the followings theorem [START_REF] Bilu | Limit distribution of small points on algebraic tori[END_REF], [START_REF] Rumely | On Bilu's equidistribution theorem[END_REF]. Theorem 4.3. Given any positive Borel measure µ, 0 ≤ µ(C) ≤ 1, that is symmetric about real line, there is a sequence of complete sets of conjugate algebraic integers such that their counting measures τ n converge weakly to µ.

There exist many sequences of polynomials with their zeros in E, that verify µ n → µ E .

Let E ⊂ C be compact and infinite. An n point Fekete set is a set {z j , j = 1, ..., n} ⊂ E that maximizes

q n (z 1 , ..., z n ) = i =j |z i -z j | .
From the maximum principle for analytic functions, it follows that for all n, the Fekete sets lie on the outer boundary of E.

The normalized density of Fekete sets converges to dµ E , the equilibrium measure for E.

Remark 4.1. Let {P n } be any sequence of monic polynomials having all their zeros in E and such that the normalized zero counting measures for P n converge weakly to µ E . If ∂ ∞ E (the outer boundary of E, that is the boundary of the unbounded component of C\E) is regular (e.g., if it is connected), then (1) P n are asymptotically optimal for the Chebyshev problem:

lim n→∞ ||F n || 1/n = C(E) .
If C(E) > 0 (so that µ E is defined), then we also have: (2) Uniformly on compact subsets of the unbounded component of C\E,

lim n→∞ |F n (z)| 1/n = exp{-Φ µ E (z)} .
Now we recall two results of approximation. In [START_REF] Motzkin | From among n conjugate algebraic integers, n -1 can be approximatively given[END_REF], Theorem 3.2, p. 158 (see also [START_REF] Dubickas | Conjugate algebraic numbers close to a symmetric set[END_REF] for its effective version), it is proving that for every n given numbers z 1 , ..., z n , and every > 0, there exists an irreductible equation with complex integral coefficients α 1 , ..., α n+1 and with roots ζ k, such that |ζ k, -z k | < for k = 1, ..., n. The same is true for real integral coefficients provided that the numbers z 1 , ..., z n are symmetric to the real axis.

In [START_REF]Approximation by polynomials with integral coefficients[END_REF] (Theorem A.1., p. 147) we have that if α 1 , ..., α n are a complete set of conjugate algebraic integers over Q[i], any positive number, and z 2 , ...,z n any complex numbers. Then there is a polynomial q

∈ Z[i][z] such that |q(α j ) -z j | < , 2 ≤ j ≤ n.
We have then the following result ([21] p. 16). Let a sequence of polynomials (P n ) such that the corresponding normalized zero counting measure for P n , µ n converge to µ E . We can suppose deg(P n ) = n and denote its zeros by z 1 , ..., z n . Therefore, we can approximate this measure by a sequence of the counting measures τ n+1 for the complete set of conjugate algebraic integers ζ k = ζ k,1/n , k = 1, ..., n + 1 by using the theorem of Motzkin. For any n ∈ N, we approximate each point z k as close as we wish by one of the conjugate algebraic integers ζ k , 1 ≤ k ≤ n, obtained from Motzkin's theorem, while let the remaining (n + 1)th conjugate algebraic integer ζ n+1 → ∞ as n → ∞ (see [START_REF] Dubickas | Conjugate algebraic numbers close to a symmetric set[END_REF] p.160-161 for details). It follows that the resulting measures

τ n+1 = 1 n + 1 n+1 k=1 δ ζ k converge to µ E as n → ∞. In fact µ n = 1 n n k=1 δ z k → µ E , then ν n = 1 n n k=1 δ ζ k → µ E .
For f continous on E and n sufficiently large we obtain

f τ n+1 = f n n + 1 ν n = n n + 1 f ν n → µ E .
But as already seen, the polynomial Q n with roots the complete set of conjugate algebraic integers ζ k , has ζ n+1 not in E (for n sufficiently large). So the sequence (Q n ) doesn't answer Serre's suggestion (Question 3.1).

Demonstration of the theorem following Serre.

Let Ω be a plane domain. The boundary of Ω is denoted by ∂Ω = Γ = Γ 1 + Γ 2 + ... + Γ r . In the following the regularity of K is understood. Note first that which is important in K is the boundary of the outer component of C\∂K. Then K can be a compact whose boundaries of the bounded connected components are what ever you want.

Denoting by Ω the outer component of C\∂K, we know that the equilibrium measure of K is supported on ∂Ω. We denote by ∂Ω = (Γ 1 , ..., Γ r ) = Γ.

The double of a plane domain.

Let Ω be a plane domain. We have seen that we may assume that each boundary component of the boundary of Ω, denoted by ∂Ω = Γ = Γ 1 + Γ 2 + ... + Γ r , Γ j is a smooth analytic curve. Alternatively, one may think of Ω as a plane bordered Riemann surface. More precisely Definition 5.1. For each r = 1, 2, ... we shall denote by U r the class of plane domains whose boundary consists of r disjoint Jordan curves Γ 1 , Γ 2 , ..., Γ r which satisfy the following smoothness condition: with each Γ j there is associated a function z j (t) analytic and univalent in a neighborhood of Γ j which maps this neighborhood onto the circular ring 1 -δ < |z| < 1 + δ and the curve Γ j onto the circle |z| = 1.

Much of the functions theory on Ω is conveniently described in terms of the Schottky double Ω of Ω. This is the compact Riemann surface of genus g = r -1 obtained, when Ω has analytic boundary, as is henceforth assumed, by welding Ω along Γ, with an identical copy Ω. Thus, as a point set Ω = Ω ∪ Γ ∪ Ω. The "backside" Ω is provided with the opposite conformal structure. This means that if z ∈ Ω denotes the oppposite to z ∈ Ω then the map z -→ z is a holomorphic coordinate on Ω. The construction of this double generalizes to any bordered Riemann surface and the result is always a symmetric Riemann surface, i.e., a Riemann surface provided with an antiholomorphic involution. In our construction this is the map U : Ω -→ Ω which exchanges z and z and which keeps Γ pointwise fixed. The double of a plane domain Ω has a holomorphic atlas consisting of only two charts: the corresponding coordinate function are the identity map ϕ 1 : z -→ z on Ω and the map ϕ 2 : z -→ z on Ω. When Γ is analytic, both these maps extend analytically across Γ in Ω, hence their domains of definitions overlap and the union covers all Ω. Let S = ϕ 2 • ϕ -1 1 be the coordinate transition function. It is analytic and defined in a neighbourhood of Γ in C, and on Γ it satisfies S(z) = z (z ∈ Γ).

A function f on Ω is most conveniently described as a pair of functions f 1 , f 2 on Ω, continuously extendable to Γ, such that f 1 (z) = f 2 (z) (z ∈ Γ). The formal relations to f in terms of the coordinates functions ϕ 1 and ϕ 2 above are

   f 1 = f • ϕ -1 1 f 2 = c • f • ϕ -1 2 • c
, where c denotes complex conjugation. It follows, for example, that f is meromorphic if and only if f 1 and f 2 are meromorphic. It is useful to note that if two domains in the class U r are conformally equivalent, then so are their doubles conformally equivalent. Indeed, the Schwarz refection principle permits an extension of a conformal homeomorphism between plane domains of the class U r to a conformal homeomorphis of their doubles.

We will deal with domain Ω, in the class U r whose double is a hyperelliptic Riemann surface. This is equivalent to the fact that Ω can be mapped one-to-one conformally onto the exterior of a system of slits taken from the real axis ( [START_REF] Baker | Plane domains with hyperelliptic double[END_REF]). At first, notably to put on the notations, we recall some results on the double of the exterior of a system of slits taken from the real axis. Let E = r j=1 E j , E j = [e 2j-1 , e 2j ] ⊂ R, e 1 < e 2 < e 3 < ... < e 2r-1 < e 2r . The double of C\E, C\E, is the hyperelliptic curve of equation

w 2 = 2r i=1 (z -e i ) = q(z) ,
where we add two points at infinity ∞ + and ∞ -, caracterised by the fact that w z r-1 = 1 at ∞ + and -1 at ∞ -. It's a two-sheeted branched covering space of the sphere P 1 (C), branched at the 2r points e j . It's a topological covering space of C\E. The covering map π E : C\E -→ P 1 (C), π : (z, w) -→ z is a meromorphic function of order two on C\E whose only multiple points, each of multiplicity two, are located at the points (e j , q(e j )), j = 1, ..., 2r.

Let us denote by E * j the inverse image of E j in C\E and

E * = r j=1 E * j .
5.2. The notion of "calibrated". In this section we refer to [START_REF] Sebbar | Th. Capacities and Jacobi matrices[END_REF]. We will develop the case where C\E corresponds to a two-sided r-periodic real symmetric Jacobi matrix. First, we recall the following terminology. The meromorphic 1-forms on a compact Riemann surface are called abelian differentials. The abelian differentials which are holomorphic will be called of the first kind; while the meromorphic abelian differentials with zero residues will be called of the second kind. Finally, a general abelian differential (which may have residues) will be called of the third kind.

We denote by w ∞ + ,∞ -the differential of the third kind having a simple pole at ∞ + and ∞ -, with residue -1 and +1 respectively, normalized by

α j w ∞ + ,∞ -= 0, j = 1, • • • , 2r.
We recall now the link between w ∞ + ,∞ -and the Green function of C\E with pole at ∞ + , g(z):

(1) g(z) is harmonic in C\E,

(2) g(z) -ln |z| is harmonic in a neighborhood of ∞ +

(3) lim z→ζ g(z) = 0 for all z ∈ E. We shall denote by G(z) the multiple-valued function obtained by adding to g(z) its conjugate.

g(z) = Re G(z) = G(z) + G(z) 2 .
The derivative of this multi-valued function is clearly single-valued. Hence, G (z) is a holomorphic function in C\E.

The functions g and G extend to all of C\E by reflecting across E, since they vanish on E: G(T (z)) = -G(z) for z ∈ C\E and T the canonical anti-conformal involution of C\E. In particular dG = w ∞ + ,∞ -. It is an abelian differential with poles at ∞ + and ∞ -and zeros at the λ j (the zeros of G (z)) and λ * j ( the points of the copy of C\E corresponding to the λ j ∈ C\E).

w ∞ + ,∞ -= r-1 j=1 (λ -λ j ) q(λ) dλ = i dp ,
where dp is called the quasimomentum in [START_REF] Krichever | Nonlinear equations and elliptic curves[END_REF] 1 2π

E 2j+1 E 2j dp = m k N , m k ∈ N, Definition 5.2. A compact set E = [E 1 , E 2 ]∪[E 3 , E 4 ]∪• • •∪[E 2n-1 , E 2n
] of the real line is calibrated if the complex Green's function G(z) of C \ E with pole at infinity satisfies the conditions

E 2k+1 E 2k Q(t)dt = 0, k = 1, 2, • • • n -1 and E 2k E 2k-1 Q(t)dt = ± r k πi N , k = 1, 2, • • • n,
where

Q(t) = (t -c 1 ) 2 (t -c 2 ) 2 • • • (t -c n-1 ) 2 (t -E 1 )(t -E 2 ) • • • (t -E 2n-1 )(t -E 2n ) = h(t) 2 and r 1 , • • • r n , N ∈ N, r 1 + • • • + r n = N .
We observe that if a compact E is calibrated with respect to r 1 , r 2 , • • • r n , N it is also calibrated with respect to kr 1 , kr w ∞ + ,∞ -= iπ .

5.3.

Approximation of Γ by Γ v calibrated. Let K be a compact with C ∞ boundary and Ω be the unbounded component of C\K (i.e. the component of the complement of K which contains infinity). The boundary ∂Ω consists of r mutually exterior curves. We know that µ K fas support in ∂Ω. K and ∂Ω have same capacity, same equilibrium measure and same potential function (see for example [START_REF] Tsuji | Potential Theory in Modern Function Theory[END_REF], p.61).

Let f be a conformal application from Ω to C\E where E is the union of r segments taken from the real axis E = r j=1 E j , E j = [e 2j-1 , e 2j ] ⊂ R, e 1 < e 2 < e 3 < ... < e 2r-1 < e 2r . The map f is such that

f (∞) = ∞, 1 ≤ j ≤ r f (Γ j ) = E j
We know that C\E = Ker{C\E n } and E n calibrated. We chose E n such that for 1 ≤ j ≤ r we have E j ⊂ E j,n .

We begin by open each interval of E (see for example the "open up" process in Widom) on the model

ψ : [-1, 1] -→ {|s| = 1}, ψ(z) = z + (z 2 -1) 1/2 .
The complement of E, C\E, is conformal to a domain ∆, obtained by opening all the intervals E j , ∂∆ = (C 1 , ..., C r ). The boundary of ∆ is C ∞ and symmetric with respect to the real axis. We denote by Ψ the corresponding conformal map.

We construct in the same way an application from E j,n (E j ⊂ E j,n ) j = 1, ..., r on C j,n near C j , on the model

ξ : z ∈ [α, β] -→ z + [(z -α)(z -β)] 1/2 .
Consequently there exists a conformal application from C\E n to ∆ n where ∂∆ n = (C 1,n , ..., C r,n ). The boundary of ∆ n is C ∞ and symmetric with respect to the real axis.

Then let denote by Φ = Ψ • f : Ω → ∆.

In the following section we will denote again Γ v by Γ.

5.4.

How to obtain a polynomial with rationnal coefficients. The aim of this section is to obtain the following theorem Theorem 5.1. In the case Γ is calibrated and symmetric with respect to the real axis, the Chebychev polynomial of Γ ∩ H of degree r achieve its maximum at each extremity of Γ ∩ H and has precisely one zero in each of the r components of Γ ∩ H.

Moreover we can achieve that in each neigborhood of Γ ∩ H, we can find such a polynomial with coefficients in Q

[i].
From [START_REF] Van Moerbeke | The spectrum of difference operators and algebraic curves[END_REF], C\E corresponds to a two-sided r-periodic real symmetric Jacobi matrix, A, and an algebraic curve of equation {h + h -1 -∆(z) = 0} with ∆ a real polynomial of degree r, known as the discriminant. It is usually defined as the trace of the one-period transfer matrix. It is also the unique polynomial (with positive leading coefficient) such that

σ(A) = {x, ∆(x) ∈ [-2, 2]} ,
where σ(A) is the spectrum of the two-sided operator (acting on l 2 (Z), it is purely absolutely continuous and of multiplicity two). Let us make more precise the polynomial ∆. Let A be such a Jacobi matrix, it defines a difference operator of period r with a, b

∈ l ∞ R (Z), b n = 0, (3)
a n+r = a n , b n+r = b n , n ∈ Z.
To simplify the notations, we write the operator A as b n . We consider the solutions of the eigenvalue problem (4) A u = z u.

A =           . .
It defines a second-order difference euqtion for n = 1, 2, 3, ...,

u n+1 = b -1 n ((z -a n )u n -b n-1 u n-1 ) ; we rewrite u n u n+1 = U n (z) u n-1 u n ,
where

U n (z) = 1 b n 0 b n -b n-1 z -a n .
The monodromy matrix M r (z), called also the one-period transfer matrix, is defined by the fundamental matrix M r (z) = U r (z)...U 1 (z) . Finally, the Floquet discriminant ∆(z) is given by a trace [START_REF] Bilu | Limit distribution of small points on algebraic tori[END_REF] ∆(z) = T r(M r (z)).

Similarly, the Floquet multipliers m ± (z) are the eigenvalues of M r (z). They are given by

m ± (z) = ∆(z) 2 ± [∆(z) 2 -4] 1/2
2 and are independant of n. They verify

m + (z)m -(z) = 1 , m + (z) + m -(z) = ∆(z) .
{e j , j = 1, ..., 2r} are the points where ∆(λ) = ±2. Indeed, ∆(λ) = 2 at e 2r , e 2r-3 , e 2r-4 , e 2r-7 , ... and -2 at e 2r-1 , e 2r-2 , e 2r-5 , e 2r-6 , .... Clearly, ∆ is determined by the p points where it is +2 and one of the points where it is -2, showing the rigidity in possibilities of E. More precisely the Floquet discriminant ∆(z) given by ( 5) of a real r-periodic Jacobi matrix is a r-hyperelliptic real polynomial. More precisely, we have

∆(z) 2 -4 = 1 B 2 2r l=1 (z -e l ) ∆(z) ∓ 2 = 1 B r j=1 (z -e ± j ) . (6)
According to [START_REF] Van Moerbeke | The spectrum of Jacobi matrices[END_REF], [START_REF] Bulla | algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies[END_REF], the zeros {e + j } 1≤j≤r and {e - j } 1≤j≤r are respectively the eigenvalues of the periodic Jacobi matrix Ã+ and the anti-periodic Jacobi matrix Ãgiven by

ñ =       -a 1 b 1 ±b r b 1 -a 2 ... -a r-1 b r-1 ±b r b r-1 -a r      
If r is even, we have

e + 1 < e - 1 ≤ e - 2 < e + 2 ≤ e + 3 • • • ≤ e + r-1 < e - r-1 ≤ e - r < e + r . If r is odd e - 1 < e + 1 ≤ e + 2 < e - 2 ≤ e - 3 • • • ≤ e - r-2 < e + r-2 ≤ e + r-1 < e - r-1 ≤ e - r < e + r .
The set {e l } 1≤l≤2r coincides with the corresponding sequence (e ± j ). We need also to recall the existence of a regular divisor D on C\E. Then the divisor D = r-1 j=1 λ j has the property that D + D -∞ + -∞ -is the divisor of the differential dp on C\E, which is real positive on E. Then D is regular ( [START_REF] Van Moerbeke | The spectrum of difference operators and algebraic curves[END_REF], p. 120). Therefore the curve is hyperelliptic of genus g = r -1 with two points ∞ + and ∞ -at infinity; besides (h) = -r∞ + r∞ -. Moreover, the hyperelliptic involution coincides with (z, h) -→ (z, h -1 ). h(z) = e rG(z) . In this case, the fixed points for the anti-holomorphic involution are given by {p = (z, h), |h(p)| = 1} = E * (see [START_REF] Van Moerbeke | The spectrum of difference operators and algebraic curves[END_REF], p.122), and

π(E * ) = R -1 ([-2, 2]).
The fixed points for the hyperelliptic involution are given by the 2r points where h = ±1, equivalently ∆ = ±2, i.e. the branched points. A sufficient condition to be mapped one-to-one conformally onto the exterior of a system of slits taken from the real axis is that Ω is symmetric with respect to the real axis and Ω ∩ H is simply connected.

We will adopt this situation and denote by f a conformal application from Ω to C\E where E is the union of r segments taken from the real axis E =

f (∞) = ∞, 1 ≤ j ≤ r, f (Γ j ) = E j .
Hence the double of Ω, Ω is hyperelliptic ( [START_REF] Baker | Plane domains with hyperelliptic double[END_REF]). The covering map π Ω : Ω -→ P 1 (C) is a branched covering of order 2. Let us denote by Γ * j the mirror image of Γ j in Ω and Γ * = r j=1 Γ * j .

The conformal application from Ω to C\E f , can be extended to conformal homeomorphism of their double. Let us denote this conformal application by

f : Ω -→ C\E, p -→ f (p) .
The Weierstrass points of Ω and those of C\E are in one to one correspondence by f . We introduce α ± j = (e ± j , h(e ± j )), A ± j = f -1 (α ± j ) and define H(p) = h( f (p))). We will identify, with a small abuse of notations, Z with Z(p) and z with z( f (p)). The maps Z and H have the property stated in [START_REF] Van Moerbeke | The spectrum of difference operators and algebraic curves[END_REF]. In particular we have for the divisor (H) = -r∞ + +r∞ -.

We will denote in the same manner D and f -1 (D), the regular divisor corresponding to the preceding regular divisor D of the zeros of ∆ .

To simplify the notations, we write the operator C as corresponding to the hyperelliptic surface following [START_REF] Van Moerbeke | The spectrum of difference operators and algebraic curves[END_REF]

C =           . .
         with A i+r = A i , B i+r = B i , B i+r = B i , i ∈ Z. Note that being r-periodic, C is bounded. Following [19], det(C H -ZI) = (-1) r-1 Φ 0 H+Φ 1 (Z)+(-1) r-1 Φ 2 H -1 , where Φ 0 = Π r i=1 B i , Φ 2 = Π r i=1 B i and Φ 1 is a polynomial of degree r, Φ 1 (Z) = (-1) r Z r + ....
By an argument of connectivity, we deduce that, for exemple σ

(C) = P -1 1 ([-2, 2]) = (aΓ + b) ∩ (aH + b). Consider then the polynomial Q(Z) = P 1 (aZ + b). P -1 1 ([-2, 2]) = Q -1 ([-2, 2]) and 1 a r Q is the Chebychev polynomial of Γ ∩ H.
Approximating C by rational coefficients, we can construct C q as close of C as we please and so, corresponding P 1,q as close of P 1 as we please, with the coefficient of P

1,q in Q[i]. As σ(C q ) = P -1 1,q ([-2, 2]) = {Z 1 = aZ + b, P 1,q (Z 1 ) ∈ [-2, 2]}, if a τ and b τ , in Q[i], are closed to a and b respectively, then {Z 1 = a τ Z + b τ , P 1,q (Z 1 ) ∈ [-2, 2]} are closed to σ(C q ). Then the polynomial Q τ (Z) = P 1,q (a τ Z + b τ ) has its coefficients in Q[i].
On the other hand P

-1 1,q ([-2, 2]) = Q -1 τ ([-2, 2]
) and for some convenient

β ∈ Q[i], such that Q r (z) = 1 β z R +• • • , the polynomial βQ τ is the Chebychev polynomial of K τ = K 1,τ +K 2,τ +...+K r,τ , with K j,τ closed to the corresponding component in Γ ∩ H.
Let us denote by aA ± j,q + b the points P -1 1,q {±2}, they are closed to the aA ± j + b. Now βQ τ is the Chebychev polynomial of K τ = K 1,τ + K 2,τ + ... + K r,τ and it achieves its extrema at the extremity of K j,τ , denoted by a τ A ± j,q,τ + b τ . In summery it oscillates on K j,τ being extremum at the a τ A ± j,q,τ + b τ . 5.5. How to obtain integer coefficients. We start with the notations of the preceding section: The polynomial βQ τ is the Chebychev polynomial of K τ = K 1,τ + K 2,τ + ... + K r,τ with coefficients in Q[i] and it achieves its extrema at the end points of K j,τ , denoted by a τ A ± j,q,τ + b τ . Moreover Q r (K τ ) = [-2, 2] and since it is of degree r, Q r has an unique zero in each K l,τ , denoted by ξ l , l = 1, ..., r. Let T n the Chebychev polynomial of [-2, 2], then the polynomial P n defined by P n = β -n T n (Q r ) is the Chebychev polynomial of degree nr of K τ . In fact it verifies clearly for all z ∈ C\K except for z in a polar set. Then letting n → ∞ in 7, we get Φ η (z) ≥ Φ µ K for all z ∈ C\K except for z in a polar set. By continuity of Φ η and Φ µ K in C\K, we have Φ η ≥ Φ µ K for all z in C\K. By using [START_REF] Simon | Equilibrium measures and capacities in spectral theory[END_REF]Theorem A.21], we have η = µ K . Thus, µ K is the only limit point of the zeros, and so the limit is µ K .

K τ = β -n T n (Q r ) -1 [-2, 2], Q r (K τ ) = [-2, 2], Q r (a τ A ± j,q,τ + b τ ) = ±2. Let B k,n such that Q r (B k,n ) = 2 cos kπ n , then T n (B k,n ) = ( -
With the same proof we deduce that for any l ≥ 1, µ P l n → µ K . Now, following [12, p 164] and its notations we consider

P n (z) = z k + γ 1 z k-1 + γ 2 z k-2 + ... + γ k m , m ∈ N * , γ j ∈ Z[i].
Let us denote by b = a k , a ∈ N and by c = b! m b , c > b ≥ a, then (P n (z)) c = z kc + ... and the coefficients of z kc-1 , z kc-2 , ..., z kc-b are in Z[i]. We can then determinate complex numbers, λ

1 , ..., λ

k , λ

1 , ..., λ

|P n (z) c | ≤ 1 2 |P n (z) c | < |P n (z) c |
for a sufficiently large. By Rouché's theorem, in each K j,τ , Γ a has nc zeros near from the nc zeros of (P n (z)) c . After clarifying the earliest "near", we conclude that lim µ Γa = µ K .

5.6. End of the proof. We will need the following result Lemma 5.2. Let K be a compact set of C, then (1) If T n is the Chebychev polynomial of K, then T n is the Chebychev polynomial of K, (2) C(K) = C( K).

Proof. It is clear that the second point will follow from the first, but also from the fact that if P n a monic polynomial of degree n then

P n || ∞,K = ||P n ∞, K .
Again in this first part we withdraw the ν symbol. We are going to prove that

µ Kτ → µ Γ + ,
where Γ + = Γ ∩ H. For this we use lemma 2.1. By construction δ(K τ , Γ + ) → 0 . Now σ(C) = P -1 1 ([-2, 2]) = aΓ + + b and then C(σ(C)) = |a|C(Γ + ). We recall that for a r-periodic Jacobi matrix J, the capacity of its spectrum is the 1 r th power of the product of the diagonal coefficients. Since the rational coefficients of C q approximate those of C, we have C(σ(C q )) → C(σ(C)). Since C(σ(C q )) = C(aΓ q + b) = |a|C(Γ q ), then C(Γ q ) → C(Γ + ). Moreover

C(K τ ) = C( 1 a τ (aΓ q + b -b τ )) = |a| |a τ | C(Γ q ), hence C(K τ ) → C(Γ + ).
Hence we have found a sequence of monic polynomials with integer coefficients (Θ λ ) λ such that µ Θ λ → µ Kτ . By the diagonal process we conclude that µ Θ λ → µ Γ + .

Theorem 3 . 4 .

 34 Let e = l+1 j=1 e j be an l-gap set with e j = [α j , β j ], α j < β j < β j+1 . Then for all m large, there exist l-gap sets e (m) e j ⊂ e (m) j , (2) Each e (m) j has harmonic measure in e (m) equal to k (m) j /m with k (m) j ∈ {1, 2, ..., }, (3) For some positive constants C 1 , C 2 , |e (m)

Theorem 3 . 5 .

 35 Let e ⊂ R be compact. Then there exist e n so that e ⊂ ... ⊂ e n+1 ⊂ e n ⊂ ... ⊂ R and n ẽn = e holds, and e n ⊂ e int n-1

Theorem 4 . 1 (

 41 Rumely). Suppose a compact set E ⊂ C with continue boundary, has capacity C(E) = 1 and is stable under complex conjugation. Let (α n ) n≥1 a sequence of algebraic integers (α n = α m if n = m): for all open U containing K, there exists n 0 such that for all n ≥ n 0 , α n with all its conjugate, O(α n ), are in U . Let ∆ n the measure∆ n = 1 d αn β∈O(αn) δ β ,then the measures ∆ n converge weakly to the equilibrium measure of E, µ E .Theorem 4.2(Rumely, Bilu ). Suppose a compact set E ⊂ C has capacity C(E) = 1 and is stable under complex conjugation. If {α n } ⊂ Q is a sequence for which deg(α n ) → ∞ and h E (α n ) → 0, then the measures ∆ n converge weakly to the equilibrium measure of E.The following result is due to Pritsker[START_REF] Pritsker | Asymptotic distribution and symmetric means of algebraic numbers[END_REF] Theorem 2.3]

E

  j , E j = [e 2j-1 , e 2j ] ⊂ R, e 1 < e 2 < e 3 < ... < e 2r-1 < e 2r . The map f is such that

2 )

 2 k and |P n | achieves its maximum at a τ A ± j,q,τ + b τ and the B k,n . The polynomial P n has n zeros in each K j,τ . It enables us to prove the lemma Lemma 5.1. µ Pn converge to the equilibrium measure of K τ . Proof. The proof follows [29, Theorem B1]. To simplify the notations K τ is denoted by K. By the Bernstein-Walsh lemma we have for z ∈ C\K, µ K (z) with the remark that lim n ||P n || 1 nr K = C(K). Now let dη be a limit point of the normalized density of zeros of P n (z), dη n and let f be continuous with support contained in C\K, then f dη n = 0 because P n has its zeros in K and if dη n → dη, then f dη = 0 and supp(dη) ⊂ K. By the Upper Envelope Theorem [29, Theorem A7] lim inf n→∞ Φ ηn (z) = Φ η (z)

j

  z k-j (P n (z)) c-a-i (8) = (P n (z)) c + ∆ a (z)[START_REF] Dubickas | Conjugate algebraic numbers close to a symmetric set[END_REF] has its coefficients in Z[i]. We have|Γ a (z) -(P n (z)) c | ≤ 2 j |P n (z)| c-a-i ≤ 2 k j=1 |z| k-j |P n (z)| c-a -1 |P n (z)| -1 |Γ a (z) -(P n (z)) c | |(P n (z)) c | ≤ 2 |P n (z)| -a -|P n (z)| -c |P n (z)| -1 . On the lemniscate |P n (z)| = R 2 with R 2 > 1 for example R 2 = M 1 n n , then with M = max |Pn(z)|=R 2 k j=1 |z| k-j , |Γ a (z) -(P n (z)) c | ≤ 2M R a 2 (R 2 -1)

  Definition 3.2. Let Ω be an open subset of R 2 . We say that Ω has a regular boundary of class C k (k ≥ 1) if for every p ∈ ∂Ω there is a neighborhood U p of p and a diffeomorphism ϕ p of class C k

]. As an open set of C, U is a natural Riemann open surface, there exists a sequence of regular subregions (

U n ), such that U n ⊂ U n+1 and U = ∞ i=1 U n .

We recall the following definition (see for example

[4, p.22]

),

  [START_REF] Autissier | Chapter VIII : Autour du théorème de Fekete-Szego[END_REF]

We denote by Let f be an infinite column vector f = (...f -1 , f 0 , f 1 , ...). Let D acts on f as the shift Df k = f k+1 . C is assumed to be periodic of period r, this amounts to the commutation relation CS = SC, where S = D r .

Following [START_REF] Naȋman | On the theory of periodic and limit-periodic Jacobi matrices[END_REF], αS + (αS) -1 -P 1 (C) = and by the analytic functional calculus

).

Z and Z 1 are two meromorphic function of degree 2 then ([?]), then there exits complex numbers a, b, c, d such that ad -bc = 1 and

Looking at infinity, we have

The second step is to verify that lim µ Pn = µ Γ -where Γ -= Γ\Γ + . The third step is to remark that P n Pn is a monic polynomial with integer coefficients and that lim n δ Pn Pn = 1 2 (δ Γ + + δ Γ -). Which follows from the definition of δ Pn Pn and the previous results. The ultimate step is to verify that 1 2 (δ Γ + + δ Γ -) = δ Γ . For this we look at the energy. Let

From the beginning we have that C(Γ + ) = C(Γ -) = C(Γ). Moreover as Γ + and Γ -are disjoint we have that on Γ

and by definition of the equilibrium measure I( 1 2 (δ Γ + + δ Γ -)) = I(δ Γ ) .

Thérèse Falliero, Avignon Université, Laboratoire de mathématiques d'Avignon (EA 2151), F-84018 Avignon, France

Email address: therese.falliero@univ-avignon.fr