Katia Chirkova

Between Tibetan and Chinese: Identity and language in Chinese South-West

Keywords: Xùmǐ, Báimǎ, Tibetans, ethnic identity, linguistic attitudes

This paper focuses on the ongoing process of changing local ethnic identities in the South-West of China. The analysis is based on a comparison of two ethnic groups: Xùmǐ and Báimǎ, both officially classified as Tibetans. While the experienced ethnic identity of the Xùmǐ is in conformity with their official classification, the Báimǎ feel their own identity as being distinct from the classification imposed by the state. The related changes in local identities are examined with special emphasis on the role played by the Xùmǐ and Báimǎ languages in creating and assessing group identity. I conclude that for both groups, albeit for various reasons, language is neither the major constructive element of ethnic boundaries nor an exclusive marker of ethnicity, which confirms Haarmaan's (1986) postulation that language is not a necessary criterion of ethnicity.

Introduction: Ethnicity, identity and language

The People's Republic of China (PRC) officially recognizes 56 nationalities, including the Hàn Chinese nationality which accounts for 91.5% of the total population. The current framework of nationalities was established in the 1950s in the course of the PRC's State Ethnic Classification Project. Over 400 groups applied for a nationality status, but only 55 have been eventually granted the official status of 'national minority', in addition to the Hàn majority.

In the wake of international efforts of endangered language preservation and research, the PRC has in recent years recognized that it hosts a considerable number of endangered languages, in fact far exceeding the number of officially recognized ethnic groups, and has launched several programs aimed at endangered language description and maintenance. 1 Thus, according to recent official Chinese estimations, China hosts 128 languages, of which two have recently become extinct. 2 The actual number of distinct languages in China may be even higher. For example, David Bradley notes that within the PRC there are at least forty-one more endangered languages, which are not on the official list. 3 In his recent article, Sūn Hóngkāi does not exclude the possibility that more languages will be discovered in the PRC in the coming years. 4 Language is the weightiest of the four Stalinist's principles on which the PRC State Ethnic Classification Project in the 1950s was essentially based. As formulated by Stalin, a nation is a historically constituted, stable community of people, formed on the basis of a common language, territory, economic life, and psychological make-up manifested in a common culture. 5 For Stalin ethnic identity was presumably intrinsically connected with language.

Large linguistic surveys in the 1950s constituted an integral part of the Ethnic Classification Project. 6 In reality, however, decisions in each particular case rarely relied on Stalin's criteria in their literal reading. They were employed instead, in Stevan Harrell's formulation, 'to confirm or legitimate distinctions [that were] for the most part already there in Chinese folk categories and in the work of scholars who wrote before [the establishment of the PRC in 1949]'. 7 Thus the actual or presumed historical relatedness of groups played a decisive role in the process of ethnic identifications, pushing Stalin's principles into the background.

There appears to be a general agreement among ethnologists working in China that the perception and classification of minorities in the 1950s largely stems from the process of national history construction during the late Qīng dynasty (1644-1911) and the Republican period (1911-1949), which created the concept of one Chinese nation (Zhōnghuá mínzú), including all Hàn and non-Hàn people. 8 In this scheme, the legendary Yellow Emperor (Huángdì, trad. 2697-2597 BCE) is commemorated by Hàn Chinese as their common ancestor, while his brother, the equally legendary Flaming Emperor (Yándì), is reconstructed in national history as the ancestor of many non-Hàn nationalities. Thus the term 'descendants of the Flaming Emperor and the Yellow Emperor' (Yán Huáng zǐsūn) has entered the Chinese language as an umbrella concept for all Chinese people belonging to one Chinese nation. This construction of minority group identities and histories from Chinese historical sources was inevitably prone to controversy. The official categories and histories of separate groups were created by the Hàn Chinese, rather than by the ethnic groups involved, the new identities received by the latter in the PRC have at times come in conflict with their self-consciousness and their perceived differences from other groups.

In this article I use Harrell's (1990) three-way model of reciprocal interaction to define group identity. This model distinguishes between (1) a group that considers itself distinctive, (2) neighbouring groups from which the group distinguishes itself, and (3) the state, which establishes official categories of group identification and distributes benefits to the groups so identified, which together give rise to ethnic consciousness and identity. 9 A more or less universal pattern in present-day China is that the official state ethnic distinctions tend to gradually shape and partly solidify people's ethnic consciousness so that they become strongly invested in the categories originally imposed upon them from outside. Eventually, this turns ethnic identity into a recognized identity. 10 Conflicts of identity, ethnicity and the correlation between a group's selfconsciousness and official classification among Chinese minority groups is a well researched topic, discussed in a number of case studies. 11 Xùmǐ and the Báimǎ ethnic groups present a notable contrast as to the appreciation of their official classification as Tibetans: the Xùmǐ experience their identity as being in conformity with their official classification, whereas until recently the Báimǎ challenged their official status as conflicting with their self-awareness.

My major interest is in examining the linguistic attitudes of each group. Both speak languages which are highly distinct from the neighbouring varieties of Tibetan.

The Xùmǐ speak a language (known in linguistic literature as Shǐxīng), which belongs to the Qiangic branch of the Tibeto-Burman language group, i.e. a branch distinct from the Bodish languages of the same language group, to which Tibetan in turn belongs. The Báimǎ speak a language that is mutually intelligible with the Tibetan Amdo dialects in its vicinity, though the exact relationship between Báimǎ and Tibetan dialects is currently subject to debate. 12 The contrast in the appreciation by the groups of their official classification gives an opportunity to observe the role language plays in creating and asserting their identities. It also supports Haarmann's (1986) postulation that language is not fundamentally related to ethnicity: 13 If ethnicity provides the most elementary framework of human relations on which identity can be constructed, then language provides the most elementary means for fulfilling this task.

Language is always involved in ethnic relations as the most refined vehicle of interacting according to local behavioural traditions, of expressing attitudes and values, and of stereotyping culture. This elementary function of language in ethnicity, however, does not support the idea of language always being the major constructive element of ethnic boundaries or an exclusive marker of ethnicity.

On yet another level, a discussion of these two cases hopefully sheds some light on how present-day China deals with the discrepancy between the relatively small 12 Speaking of the phylogenetic relationships between the Tibeto-Burman languages of the PRC, I use Sūn's classification, which recognizes five distinct branches within this language group: (1) Bodisch (or Tibetan), (2) Qiangic, (3) Yí (Loloish), (4) Burmish (or Burmese) and (5) Jǐngpō, see Sūn Hóngkāi 孙宏开, 'Shì Lùn Zhōngguó Jìngnèi Zàngmiányǔ de Pǔxì Fēnlèi' 《试论中国境内藏缅语的谱系分 类》 [A Classification Of Tibeto-Burman Languages in China], in Paul K. Eguchi (ed.), Languages and History in East Asia: Festschrift for Tatsuo Nishida On the Occasion of His 60th Birthday (Kyoto: Shokado, 1988), pp.61-73. 13 Harold Haarmann, Language in Ethnicity: A View of Basic Ecological Relations (Berlin, New York, Amsterdam: Mouton de Gruyter, 1986); Harold Haarmann, 'History', in Joshua A. Fishman (ed.), Handbook of Language and Ethnic Identity (New York, Oxford: Oxford University Press, 1999), p.63. number of recognized ethnic groups and the much larger number of distinct languages spoken in the PRC.

The official line on this issue (see, for example, the recent article by Sūn Hóngkāi on the scientific foundations of minority language classification) is that one minority can use several languages, which might even belong to genetically unrelated language groups, due to historical reasons. 14 As an example, Sūn quotes the language of Língāo County (Hǎinán Province), which genetically belongs to the Kam-Tai (Tai-Kadai) language group, but is spoken by a group of ethnic Hàn Chinese. 15 Another striking example is the Kǎzhuó language of Sìchuān Province, which is related to the Yí languages of South-eastern China, but is spoken by an ethnic group officially classified as Mongolians, as they historically stem from a Mongol army stationed in the area during the Yuán dynasty (1271-1368). 16 Thus, the Mongol nationality in China speaks languages of at least two distinct linguistic stocks: Mongolic (Altaic) and Yí (Tibeto-Burman). 17 In sum, ethnic identification in China cuts at times in quite an unparalleled fashion across established linguistic groupings.

As appears to be the case with one of the groups discussed in this article, this unusual situation reflects acceptance of loose standards of linguistic unity and prevailing tolerance to linguistic diversity within one ethnic group. 14 The common designation of these groups in the later Chinese historiographic tradition is also fān 'inhabitants of outer regions of the empire, barbarians' or xīfān 'western barbarians'. Overall, the xīfān people form a heterogeneous group comprising, besides the Tibetans, ethnic groups speaking various Qiangic languages, such as Nàmùyì, Minyak (Mùyǎ), Ěrgōng, Ěrsū, rGyal-rong (Jiāróng), Prinmi (Púmǐ) and Guìqióng.

Chinese South-West and the Tǔbō people of the Chinese western frontiers

After the establishment of the PRC in 1949, most of the fān people were incorporated into the Tibetan nationality, including the two groups discussed presently, Xùmǐ and Báimǎ.

Xùmǐ Tibetans and the Shǐxīng language

The Xùmǐ Tibetans of Shuǐluò Township, Mùlǐ County, are a relatively little studied group. They are mostly known in academic literature for their language, discovered in 1980 by Sūn Hóngkāi. 19 Sūn named the language and subsequently also the ethnic group that speaks it 'Shǐxīng', which is a Chinese transliteration of the autonym of the 18 See Louis Bazin and James Hamilton, 'L'origine du nom Tibet', in Ernst Steinkellner (ed.), Tibetan History and Language: Studies Dedicated to Uray Géza On His Seventieth Birthday (Wien: Arbeitskreis für Tibetische und Buddhistische Studien, Universität Wien, 1991), pp.11-7. 19 20 The name Shǐxīng is currently widely used in scholarly discourse, but is generally unheard of in Mùlǐ, where the group resides, and the Xùmǐ Tibetans do not identify themselves with this label.

The group is locally known under the Chinese name Xùmǐ or Sùmǔ, which are renderings of the group's autonym 'people of the Shu river' in local languages. 21 Hereafter I will use 'Xùmǐ' when speaking about the ethnic group, as this name was the preferred self-denomination used by my Xùmǐ informants during my fieldwork in Mùlǐ in 2005-2006, and 'Shǐxīng' when speaking about their language, as customary in linguistic literature.

The Xùmǐ group counts approximately 1,800 people, compactly residing in Shuǐluò Township. Due to its geographical isolation, the cultural integrity and the language of the group are relatively well preserved, even though the language has come under increasing pressure from Mandarin Chinese as a result of education and mass media in recent years. The Shǐxīng language is classified as belonging to the Qiangic branch of Tibeto-Burman. It has received little academic attention to date and is poorly understood.

The Xùmǐ people have all along resided in complex multi-ethnic and multilinguistic surroundings. Present-day Mùlǐ County hosts 18 ethnic groups-including Chinese, Tibetans, Yí, Nàxī, Miáo, Zhuàng, Bùyī and Lìsù, with Tibetans accounting for the majority of the population-and has therefore received the status of an autonomous Tibetan County (Zàngzú zìzhìxiàn). However, the composition of Tibetans of Mùlǐ is diverse and includes, apart from ethnic Tibetans, several groups speaking a variety of Qiangic languages. For example, based on their language and self-designation, the Tibetans of Shuǐluò County alone can be further subdivided into four separate groups, only one of which speaks a Tibetan dialect proper (that is to say, 20 Subsequent fieldwork on this language by Huáng Bùfán in 1987 and myself in 2005-2006 shows that it displays some dialectal variation, and the autonym of this ethnic group is pronounced as [ʂu 55 hĩ5 5 ] in the dialect of the upper reaches of the Shu river and, allegedly, [ʂɿ 55 hĩ5 5 ] in that of the lower reaches, investigated by Sūn. 21 a dialect related to the Tibetan dialect continuum on linguistic grounds): 22 (1) Khams Tibetans (speaking a dialect belonging to Khams Tibetan group); (2) Prinmi Tibetans (speaking the Prinmi (Qiangic) language); (3) Xùmǐ Tibetans (Shǐxīng, Qiangic) and (4) Bùláng Tibetans (speaking the western dialect of the Ěrsū (Qiangic) language). Multilingualism has always been the norm for the Xùmǐ people, who even perceive their own language as a non-standard variety and an amalgam of the neighbouring languages. 23 Overall, all different Tibetan groups of Mùlǐ share, despite this linguistic diversity as well as some minor cultural variations, one Tibetan lifestyle. 24 This is manifest in housing, diet, culture and, most importantly, religion. Tibetan Buddhism is the fundamental building block of Tibetan national consciousness. Thus, the Xùmǐ group by and large identifies with Tibetans and appears to have successfully developed not only bilingualism in the local Tibetan variety, but also a bicultural identity, which combines their own Xùmǐ ethnic identity with a larger, primarily religious, Tibetan identity. Their official status as Tibetan is therefore in conformity with their religious identification.

Moreover, the linguistic diversity of the area (and of Tibet at large) is locally perceived as natural. This tolerance to linguistic multiplicity is crystallized in a local adage that each valley in Tibet has its own dialect (which is not necessarily mutually intelligible with the dialects around it). In this view, the Tibetan language is perceived as a broad range of discrete varieties, to which virtually anything can belong.

Therefore, linguistic distinctiveness does not play a decisive role in the self-awareness of local groups.

In sum, in the case of Xùmǐ Tibetans, there is no overt or active conflict centering on the contradicting Xùmǐ-Tibetan identities. Religion is the major marker of Xùmǐ ethnic identity, more important than language.

Báimǎ Tibetans

Overview

The group known under the name 'Báimǎ Tibetans' number approximately 10,000 people residing in three counties in Sìchuān Province (Jiǔzhàigōu, Sōngpān (Zungchu), Píngwǔ) and one county in Gānsù Province (Wénxiàn). They immediately border the Qiāng (to their South-West), Chinese (East and South) and Tibetan ethnic groups (West and North). Despite the relatively large area of residence, all Báimǎ in different counties identify themselves as one ethnic group with a common selfappellation, common language, culture and history. The autonym of the Báimǎ people is [pe 53 ] and they are known as Dwags-po in Tibetan.

The Báimǎ appear to be traditionally monolingual, but they are currently by and large bilingual in Báimǎ (language of interpersonal communication in Báimǎ villages) and Mandarin Chinese (language of education and communication with neighbouring ethnic groups).

In contrast to the Xùmǐ group, which largely shares a Tibetan lifestyle, the Báimǎ historically gravitate towards Chinese standards of living. Furthermore, Báimǎ lifestyle appears to be, in the opinion of the group, considerably dissimilar from that of grassland Tibetans. For example, one notable difference is that the Báimǎ do not drink milk or use milk products, which are essential to the Tibetan diet. The Lama Buddhism present in the area at some period of time in the past and now widely practised by neighbouring Tibetan groups, never replaced the indigenous animist religious tradition of the Báimǎ, who revere a local mountain as their ancestor and god-protector. 25 Thus, the Báimǎ do not identify themselves with the religious (Buddhist) Tibetan identity.

Ethnic identification (the 1960s-80s)

25 Notably, in their rituals, local priests, [pe 13 mbu 53 ] (in all probability related to bon-po in Tibetan), make use of hand-written Tibetan books (mostly on divination), which appear to belong to the Bon tradition (Henk Blezer, personal communication, January 2005). This indigenous religion of Tibet was partly absorbed by the Buddhist tradition introduced in Tibet in the 8th century. The legendary founder of the Bon religion, sTon-pa gshen-rab mi-bo, is also a prominent character of Báimǎ folklore.

The Báimǎ were classified as Tibetans in 1951, an identification which was made based on the opinion of local representatives rather than on an in-depth study. [START_REF] Hóngkāi | Lìshǐ Shang de Dīzú hé Chuān Gān Dìqū de Báimǎrén' 《历史上的氐族和川 甘地区的白马人》 [On the Historical Minority Dī and Báimǎ People Of Sìchuān and Gānsù Provinces[END_REF] They questioned this conclusion on numerous occasions in the 1960s and 1970s and, pointing out their differences from Tibetans, they repeatedly demanded reclassification. This appeal was granted in the late 1970s, when a group of PRC researchers conducted two surveys in the Báimǎ areas. As a result of surveys conducted between 1978 and 1979, the research group published two collections of papers, in which the majority of researchers argued that the Báimǎ were descendents of the ancient Dī tribe (probably related to the Qiāng ethnic group), which set up influential kingdoms in the 3rd through the 6th centuries CE in the areas currently inhabited by the Báimǎ. [START_REF]Báimǎ Zàngrén Zúshǔ Wèntí Tǎolùnjí 《白马藏人族属问题讨论集》 [A Collection Of Discussions On the Question Of the Ethnic Classification Of Báimǎ Tibetans[END_REF] The designation of the group as 'Báimǎ' probably also stems from these surveys. 'Báimǎ' is known from the Historical Records by the father of Chinese historiography Sīmǎ Qiān (c. 145-90 BCE) to be the largest of the Dī tribes, with whom present-day Báimǎ claimed to be related in the 1970s. This tribe is also known to have historically resided in the neighbourhood of the present-day Báimǎ group.

Despite the conclusion that the Báimǎ people constitute a distinct ethnic group rather than a branch of Tibetans, they were never officially reclassified. According to Harrell, the general word among ethnologists working in China is that the reclassification of the Báimǎ as a separate nationality was blocked by the tenth Panchen Lama. 28 Overall, reclassification of ethnic groups listed as Tibetans remains a sensitive issue in the PRC and is considered by many Tibetans as an attack by the Chinese government on the Tibetan identity. In the minds of many Báimǎ and Chinese, on the other hand, it was a 1986 resolution of the State Ethnic Affairs Commission, which served as the final word on the reclassification issue, stipulating that the temporary preservation of the current classification of the Báimǎ group as Tibetans is in the interest of stability and unity. 29 Since then, the established designation of this group is 'Báimǎ Tibetans', asserting in one name their official ethnic affiliation to Tibetans, but recognizing at the same time their distinctiveness as Báimǎ.

On the whole, since the mid-1980s, the group advanced no more claims of independency and in my experience, now generally accepts their Tibetan-ness and routinely speak of themselves as Tibetans and of their language as Tibetan. The Tibetan-ness of the latter was after all supposedly not difficult to accept in view of the broad understanding of the Tibetan language as embracing many highly distinct varieties, as discussed above. In fact, the linguistic distinctiveness of the Báimǎ language was always contested by Tibetan intellectuals as a valid argument for seeing the Báimǎ as a separate ethnic group. [START_REF]See, for example, dMu dge bSam gtan's 'On the question of the 'Dwags po" nationality': 'although the language of all Tibet has one root, there have developed differences in pronunciation due to the large area of language and the great dispersion of settlements [in Tibet]' (quoted from Janet L. Upton, 'Notes Toward a Native Tibetan Ethnology: Dmu-dge bSan gtan's "Dwags-po (Baima)[END_REF] Paradoxically, even though the group considers the issue finalized and the link between the Báimǎ and the Tibetans is no longer openly questioned, a vivid discussion on the origins of the Báimǎ people in Chinese scholarly circles continues.

Moreover, recent publications on the 'Báimǎ Tibetans' indicate that their descent from the ancient Dī people is now generally seen as a well established fact. 31

Dī-Báimǎ relationship

Given the controversy surrounding the ethnic classification of the Báimǎ in the 1960s through the 1980s, research on the Báimǎ language mostly focused on the-by and large non-linguistic-issue whether the Báimǎ language should be considered a dialect of Tibetan or an independent language. An interesting detail here is that even convinced supporters of the Báimǎ language being a Tibetan dialect acknowledge that the Báimǎ language exhibits a number of distinctly non-Tibetan features in its lexicon, morphology and syntax. Huáng and Zhāng argue that these features are the result of substratum interference of the original language of the Dī people, whose descendants the Báimǎ were claimed to be during the Báimǎ surveys in the late 1970s. [START_REF] Huáng | Báimǎhuà Zhīshǔ Wèntí Yánjiū' 《白马话 支属问题研究》 [A Study Of the Genetic Affiliation Of Báimǎ[END_REF] But what do we know about the Dī and their language? Records on the Dī, believed to ultimately stem from the legendary Flaming Emperor, appear in Chinese sources early (the Dī are for instance mentioned in the Book of Odes, a collection of poems dating from the Zhōu dynasty (1022-256 BCE), but consist rather of legendary accounts based on hearsay. In the Hàn dynasty (206 BCE-220 AD), the Dī were primarily associated with and mostly mentioned together with the Qiāng group and used to refer to a vast range of people living mainly along the eastern edges of the Tibetan Plateau (i.e. present-day Gānsù, Shǎnxi and Sìchuān provinces), in other words, the western edges of the Hàn empire. [START_REF] Edwin | Central Asia and Non-Chinese Peoples Of Ancient China[END_REF] Despite frequent mentions (as a rule of military encounters) in Chinese historical sources between the Hàn and Northern Wèi (386-534 AD) dynasties, there is little concrete information on the group, the most detailed account being a meagre 375-character long description of the Dī by Yú Huàn in the chapter on the people of the Chinese western frontiers in the Wèi chronicles (Wèilüè). This report contains the autonym of the group, names of Dī chieftains, and some fairly general remarks on Dī customs and language. Altogether, it is so general that the Dī can in fact be linked to almost any ethnic group in presentday Sìchuān and Gānsù. 34 Nonetheless the link between the Dī of Chinese historical sources and a group with a disputed ethnic status brought forward in the surveys in the late 1970s is most likely anything but accidental. When in the course of the surveys, the group was found distinct enough from Tibetans to be granted an independent nationality status, the Dī, reconstructed from China's historical memories, presented themselves a convenient construct. This incorporated the newly fledged nationality into one Chinese nation descending from the Yellow and Flaming emperors.

Needless to say, this reconstruction of history from Chinese historical sources has little to do with the actual history or self-awareness of the group involved. In my experience, many older people who are regarded Báimǎ have never heard of either the Dī or the Báimǎ and are largely unaware of their alleged connection to these groups.

In sum, Báimǎ Tibetans have faced a conflict between contradicting local and Tibetan identities. The distinctiveness of their language from the neighbouring Tibetan dialects played an important role in their feeling of separateness, but even together with many other factors, it did not result in a change of status.

Conclusions

How important is language in creating and asserting group identity in the Chinese Southwest? And how many distinct languages can one nationality have without disturbing its unity? For the Xùmǐ group, linguistic distinctiveness plays no essential role in identity formation. Given the multilingualism of the area of its distribution, Xùmǐ take linguistic diversity for granted and consider general leniency towards linguistic and cultural diversity within one, Tibetan, group as the norm. In fact, this situation mirrors that of the Hàn nationality, which similarly treats a host of mutually unintelligible linguistic varieties (Sinitic languages) as dialects of one Chinese language, giving 34 For a discussion, including an examination of linguistic evidence, see Katia Chirkova, 'Báimǎ Zàngzú Wéi Dīzú Shuō Xīntàn' 《白马藏族为氐族说新探》 [On the Relationship Between the Báimǎ and the Dī], in Pān Wùyún 潘悟云, William Baxter and Laurent Sagart (eds), Hànyǔ Shànggǔ Yīn Gòunǐ Guójì Xuéshù Yántǎohuì Lùnwénjí 《汉语上古音构拟国际学术研讨会论文集》 [Proceedings Of the International Symposium On Old Chinese] (Běijīng 北京: Zhōnghuá Shūjú 中华 书局, fc 2007). them one single identity, which is generally not contested by the groups involved. 35 Thus, there appear to be no limits to the number of distinct languages one nationality, descending from one common origin, can have. Historical reasoning here outweighs linguistic interpretation.

For the historically largely monolingual Báimǎ, the distinctiveness of their language played a significant role in their feeling of separateness from other groups.

The group was provided with new identities: Firstly, descendants of the Dī, which was constructed through selecting and remodelling historical and ethnographical information. Secondly, the group was considered Tibetan. This discouraged the group to promote its own identity, and it is now in the process of turning its original ethnic identity into these recognized identities.

Hence for both groups, albeit for various reasons, language is neither the major constructive element of ethnic boundaries nor an exclusive marker of ethnicity. This confirms Haarmann's (1986) conclusion that language is not a necessary criterion of ethnicity.

Historically positioned in the realm of Tibetan influence, the Xùmǐ and Báimǎ groups have chosen separate strategies of contact with this dominant majority. The Xùmǐ have chosen integration while maintaining some degree of cultural integrity, whereas the Báimǎ have chosen separation, explicitly rejecting some features of Tibetan culture. In view of recent globalisation trends, however, both groups are probably inevitably heading towards assimilation. The Xùmǐ into the Tibetan group and the Báimǎ into the Chinese group, but it is a question what kind of identity-their local, Tibetan, Dī or Chinese-they will develop then. 35 For a discussion, see Dory Poa and Randy J. LaPolla, 'Minority Languages Of China', in Osahito Miyaoka and Michael E. Krauss (eds), The Vanishing Languages Of the Pacific (Oxford: Oxford University Press, fc 2007).

  In fact, in addition to Mongolian and Kǎzhuó, the Mongol nationality of China is known to speak also a Turkic language Túwǎ, closely related to Kazak. Turkic and Mongolic languages are separate branches of the putative Altaic language family. The two groups that this study focuses on are both situated in the South-West of China: Xùmǐ in the South-West of Sìchuān Province, bordering the Yúnnán Province; and Báimǎ in the North of Sìchuān, at the border of Sìchuān and Gānsù provinces. These areas traditionally fall under the realm of Tibetan influence. They originally constituted part of the Tibetan Yarlung Empire (7th-9th centuries), and the influence of Tibetan culture persists there until the present day. This is largely due to the pervasive spread of Lama Buddhism in this zone during the Yuán (1280-1368) and Míng (1368-1644) dynasties. Since the 18th century, when Sìchuān and Yúnnán provinces were adjoined to the Chinese empire by the Qīng emperor Yǒngzhèng (reigned 1722-1735), they belonged to border areas with mixed Tibetan and Hàn influences. The Hàn presence there has considerably intensified since the establishment of the PRC.The ethnic groups of this zone are associated in China with Tibetans, and from the Sòng dynasty (960-1279) onwards, are commonly referred to as Tǔbō or Tǔfān, a Chinese transcription of a Turkic word for Tibet *Töpän/*Töpüt 'peaks, heights'.18

4

  Sūn Hóngkāi 孙宏开, 'Yòng Kēxué de Yǎnguāng Kàndài Wǒ Guó de Yǔyán Shíbié Wèntí' 《用科 学的眼光看待我国的语言识别问题》 [Scientific Principles Of Linguistic Classification Of Chinese Minority Languages], in Yǔyán Wénzi Yìngyòng Yánjiū 《语言文字应用研究》 [Applied Linguistics], Vol.3 (2005), p.28. 5 Iosif Vissarionovič Stalin, 'Marxism and the National Question', Joseph V. Stalin: Reference Archive (Internet edition), [http://www.marxists.org/reference/archive/stalin/, transcribed by Karl Kavanagh, accessed 6 Mar. 2007], first published in Prosveshcheniye nos. 3-5 (March-May 1913). 6 See, for example, Hǎo Shíyuǎn 郝时远 (ed.), Tiányě Diàochá Shílù: Mínzú Diàochá Huíyì 《田野调 查实录--民族调查回忆》 [Records On the Linguistic Survey: Memories Of the State Ethnic Classification Project] (Běijīng 北京: Shèhuì Kēxué Wénxiàn Chūbǎnshè 社会科学文献出版社, 1999). 7 Stevan Harrell (ed.), Cultural Encounters On China's Ethnic Frontiers (Seattle and London: University of Washington Press, 1995), p.66. 8 See, for instance, Wáng, Míng-kē 王明珂, 'From the Qiang Barbarians to the Qiang Nationality: The Making Of a New Chinese Boundary', in Huang Shu-min and Hsu Cheng-kuang (eds), Imaging China: Regional Division and National Unity (Taipei: Institute of Ethnology, 2000), pp.43-80, and Wáng, Míng-kē 王明珂, 'Lùn Pānfù: Jìndài Yán Huáng Zǐsūn Guózú Jiàngòu de Gǔdài Jīchǔ' 《论攀附:近 代炎黄子孙国族建构的古代基础》 [On Mimicry: the Ancient Foundations Of the Modern Nation-Building in China Through the Identity Of 'the Offspring of Yellow/Yan Emperors'], in Lìshǐ Yǔyán Yánjiūsuǒ Jíkān 《历史语言研究所集刊》/ Bulletin Of the Institute Of History and Philology, Vol.73, no.3 (2002), pp.583-624.

  See Sūn Hóngkāi 孙宏开, 'Yòng Kēxué de Yǎnguāng Kàndài Wǒ Guó de Yǔyán Shíbié Wèntí' 《用科学的眼光看待我国的语言识别问题》 [Scientific Principles Of Linguistic Classification Of Chinese Minority Languages], in Yǔyán Wénzi Yìngyòng Yánjiū 《语言文字应用研究》 [Applied Linguistics], Vol.3 (2005), pp.26-8. This reasoning put forward by Sūn is very far indeed from Stalin's postulate that '[t]here is no nation which at one and the same time speaks several languages'. 15 Ibid., p.27. On the history of the group, see also Liáng Mǐn 梁敏 and Zhāng Jūnrú 张均如, Língāoyǔ Yánjiū 《临高语研究》 [A Study Of Língāo] (Shànghǎi 上海: Shànghǎi Yuǎndōng Chūbǎnshè 上海 远东出版社, 1997), pp.4-16. Tai-Kadai languages are considered to be part of the Sino-Tibetan language family in Chinese linguistic tradition, but are seen as unrelated to Sino-Tibetan languages in Sūn, ibid., p.26. For the history of the Kǎzhuó language, see also Mù Shìhuá 木仕华, Kǎzhuóyǔ Yánjiū 《卡卓语研究》 [A Study Of Kǎzhuó] (Běijīng 北京: Mínzú Chūbǎnshè 民族出版社, 2002), pp.2-8. 17

	Benedict's (1972), Shafer's (1974) and Matisoff's (2003) classifications. See Paul K. Benedict
	(Contributing editor James A. Matisoff), Sino-Tibetan: A Conspectus (New York: Cambridge
	University Press, 1972); Robert Shafer, Introduction to Sino-Tibetan (Wiesbaden: Otto Harrassowitz,
	1974); and James A. Matisoff, Handbook of Proto-Tibeto-Burman (Berkeley, Los Angeles, London:
	University of California Press, 2003).

16

  Sūn Hóngkāi 孙宏开, 'Liùjiāng Liúyù de Mínzú Yǔyán jí qí Xìshǔ Fēnlèi' 《六江流域的民族语言 及其系属分类》 [Minority Languages Of the Six River Valley and Their Genetic Classification], in Mínzú Xuébào 《民族学报》 [Scholarly Journal Of Nationalities], Vol.3 (1983), pp.196-213. group recorded by Sūn as [ʂɿ 55 hĩ5 5 ].

  See Huáng Bùfán 黄布凡, 'Shǐxīngyǔ' 《史兴语》 [The Shǐxīng Language], in Dài Qìngxià 戴庆夏, Huáng Bùfán 黄布凡, Fù Àilán 傅爱兰, Rénzēng Wāngmǔ 仁增旺姆, Liú Júhuáng 刘菊黄 (eds), Zàng-Miányǔ Shíwú Zhǒng《藏缅语十五种》/ Fifteen Tibeto-Burman Languages (Peking 北京: Běijīng Yànshān Chūbǎnshè 北京燕山出版社, 1991), p.174; Mùlǐ Zàngzú Zìzhìxiàn Zhì Biānzuǎn Wěiyuánhuì 木里藏族自治县志编纂委员会编纂 (eds), Mùlǐ Zàngzú Zìzhìxiàn Zhì《木里藏族自治县 志》 [Gazetteers Of Mùlǐ Tibetan Autonomous County] (Chéngdū 成都: Sìchuān Rénmín Chūbǎnshè 四川人民出版社, 1995), p.133; Sūn, ibid., p.196.

The most notable projects carried out by the Institute of Minorities of the Chinese Academy of Social Sciences since the 1990s are Zhōngguó xīn fāxiàn yǔyán yánjiū cóngshū [New found minority languages in China series], Zhōngguó shǎoshù mínzú xìliè cídiǎn cóngshū [Dictionary series of China's minority languages] and Zhōngguó shǎoshù mínzú yǔyán fāngyán yánjiū cóngshū [Dialects of minority languages of China series].

See Sūn Hóngkāi 孙宏开, 'Guānyú Bīnwēi Yǔyán Wèntí' 《关于濒危语言问题》 [On the

The term 'dialect continuum' refers to a series of historically related varieties which differ from one another with respect to one or more features.

According to Xūmǐ legend, languages were distributed to various ethnic groups at the beginning of the world. The representative of the Xūmǐ group came too late, when all languages were already given away. Left without a language, he then put together the Xūmǐ language from words and sentences borrowed from languages of other groups. Also, Sūn observes that the Shǐxīng language is perceived by its speakers as a mixture of Tibetan and Nàxī, see Sūn, ibid., p.196.

For example, the Xūmǐ group and local Tibetans have different legends of origin: the Xūmǐ believe to have arrived to their current place of residence from what is present-day Shānxī Province via Dàlǐ in Yúnnán.

Stevan Harrell, 'The Nationalities Question and the Prmi Problem', in Melissa J. Brown (ed.), Negotiating Ethnicities in China and Taiwan (Berkeley and Los Angeles: University of California Press, 1996), pp.285-6 note 6.

See Zēng Wéiyì 曾维益, 'Báimǎ Zàngzú jí qí Yánjiū Zōngshù' 《白马藏族及其研究综述》


Haydar G Öral
email: hgoral@gmail.com

ALGEBRAIC NUMBERS WITH ELEMENTS OF SMALL HEIGHT

Keywords: 1991 Mathematics Subject Classification. 12L12, 11G50, 03C45 . height, model theory, salem numbers

Introduction

The logarithmic height function measures the arithmetic complexity of an algebraic number and it plays a fundamental role in algebra and diophantine geometry. Before defining the logarithmic height function, we first define the Mahler measure of a polynomial over C. For a non-zero polynomial

f (x) = a d (X -α 1 ) • • • (X -α d ) ∈ C[X]
, its Mahler measure is defined as the finite product

m(f ) = |a d | |α j |≥1 |α j |.
By convention, the Mahler measure of zero is defined to be 1. Let Q be the field of algebraic numbers. For a non-zero α in Q with irreducible polynomial f (x) ∈ Z[X], we define its Mahler measure as m(α) = m(f ). The absolute non-logarithmic height of α is defined by H(α) = m(α) 1/d where d = deg f. Then the logarithmic height of α is defined as

h(α) = log H(α) = log m(α) d .
A central question in diophantine geometry is Lehmer's conjecture, which is still open and it states that there exists an absolute constant c > 1 such that if m(α) > 1 then m(α) ≥ c. This question was posed by D. Lehmer [START_REF] Lehmer | Factorization of certain cyclotomic functions[END_REF] around 1933, who claimed that the polynomial

X 10 + X 9 -X 7 -X 6 -X 5 -X 4 -X 3 + X + 1
has the smallest Mahler measure, which is approximately 1.17628 and this is still the smallest known Mahler measure of an algebraic number. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. This is a special case of Lehmer's conjecture since for a Salem number α we have m(α) = α, and Lehmer's conjecture for Salem numbers states that 1 is not a limit point of Salem numbers. For a detailed exposition on Mahler measure and Lehmer's problem, see [START_REF] Smyth | The Mahler Measure of Algebraic Numbers: A survey[END_REF].

Classification theory is an extensive project in model theory, emerged in the work of M. Morley in the 1960's and S. Shelah in the 1970's, pursuing the categorization of first-order theories. The question of how many models a theory can have has been at the center of model theory. Stable theories are important to classify their models. If a theory is not stable then its models are difficult to handle and various to classify, and arithmetic fits in this case. A theory T is said to be stable if there is no first-order formula in T which has the order property. Stable theories are equipped with a notion of independence, namely the forking independence, with a well-defined set of properties, reminiscent of the algebraic independence in fields and the linear independence in vector spaces. For instance, the theory of algebraically closed fields is stable. Conversely, stability also yields some structural properties of the models and a wellknown result is that a field is ω-stable if and only if it is algebraically closed [START_REF] Macintyre | ω 1 -categorical fields[END_REF]. A theory is simple, if the forking independence is symmetric and it is known that stable theories are simple. There are many fruitful interactions between model theory and algebra. The main examples are Ax-Grothendieck theorem via compactness [START_REF] Ax | Injective endomorphisms of varieties and schemes[END_REF] and Chevalley's constructibility theorem via quantifier elimination. More recently, other two important examples are the group configuration theorem by Hrushovski [START_REF] Hrushovski | Contributions to stable model theory[END_REF], which is a generalization of Weil's group chunk theorem, and Hrushovski's proof of the Mordell-Lang conjecture for function fields in all characteristics [START_REF] Hrushovski | The Mordell-Lang Conjecture for function fields[END_REF].

In this paper, we also study the interaction between model theory and diophantine geometry via the set of algebraic numbers with elements of small height. The results in this article are from the PhD thesis of the author [START_REF]Model Theory of Fields and Heights[END_REF].

The model theoretic properties of Q are well understood: it is strongly minimal and has quantifier elimination in the language of rings L r = {+, -, •, 0, 1}. In this paper, we mainly focus on the pair Q together with a predicate, in particular our predicate will be the elements of height less than a given positive real number. Moreover, we study these pairs in terms of stability theory. The first motivation came from B. Zilber. In 1990, B. Zilber [START_REF] Zilber | A note on the model theory of the compex field with roots of unity[END_REF] showed that the pair (C, µ) ≡ (Q, µ) is ω-stable where µ is the group of complex roots of unity, and his pattern was based on the result of H. Mann [START_REF] Mann | On linear relations between roots of unity[END_REF]. The result of B. Zilber [START_REF] Zilber | A note on the model theory of the compex field with roots of unity[END_REF] was generalized by L. van den Dries and A. Günaydın [START_REF] Van Den Dries | The Fields of Real and Complex Numbers with a Small Multiplicative Group[END_REF] to algebraically closed fields with a multiplicative subgroup satisfying a certain diophantine condition. In this paper our predicate, namely the elements of small height, contains the group µ. Moreover, the theory of Q with this predicate will be a proper expansion of the theory of (Q, µ). The model theory of pairs has attracted a lot of attention recently. More generally, stable theories with a predicate were analyzed in the paper of E. Casanovas and M. Ziegler [START_REF] Casanovas | Stable theories with a new predicate[END_REF], where they gave criteria for a pair to be stable. Their result in [START_REF] Casanovas | Stable theories with a new predicate[END_REF] implies the result of B. Zilber [START_REF] Zilber | A note on the model theory of the compex field with roots of unity[END_REF] and also B. Poizat's result on the ω-stability of the theory of pairs of algebraically closed fields [START_REF] Poizat | Paires de structures stables[END_REF]. Now we state Kronecker's theorem from diophantine geometry with no proof, as the details can be found in the book [2, 1.5.9]. Theorem 1.1. (Kronecker) Let α in Q be a non-zero algebraic number. Then h(α) = 0 if and only if it is a root of unity. Combining Zilber's result [START_REF] Zilber | A note on the model theory of the compex field with roots of unity[END_REF] with Kronecker's theorem, we see that the pair

(Q, {a ∈ Q : h(a) = 0}) is ω-stable.
Throughout this paper, the language L m will denote the language {1, •} where the binary operation • is the usual multiplication. Let S be the set of all Salem numbers. We put The pairs (Q, S b ) and (Q, S ε ) can be seen as an L r (U ) = L r ∪ {U } structures where U is a unary relation symbol whose interpretations are S b and S ε respectively. We call S ε the set of algebraic numbers of small height. Note that the Mahler measure of an algebraic number is again an algebraic number. Put also S = {a ∈ Q : h(a) < 1} = S 1 , as there is no algebraic number whose height is 1 by Lindemann's theorem.

P b := {a ∈ Q × : m(a)
Let T ε be the theory T h(Q, S ε ). We focus on the model theory of (Q, S ε ) in the language L m (U ) = L m ∪ {U } and we prove a result which shows that small perturbations of the property of being a root of unity changes drastically the stability properties of the ambient structure. This means that T ε is model-theoretically very complicated. We also relate the simplicity of a certain pair with Lehmer's conjecture and give a model-theoretic characterization of a special case of Lehmer's conjecture. We prove the following theorem (see Theorem (3.4) and Theorem (4.2)).

Main Theorem. The theory of (Q, S ε ) is not simple and has the independence property in the language L m (U ). Moreover, the theory of (Q, S b ) is simple for some b > 1 in L m (U ) if and only if Lehmer's conjecture holds for all Salem numbers. Furthermore, the simplicity of the pair (Q, P b ) for some b > 1 in L m (U ) implies that Lehmer's conjecture holds for all Salem numbers. • For all α in Q, we have h(α n ) = nh(α) where n ∈ N,

Preliminaries and Height Lemmas

• For all α and β in Q, we have h(α

+ β) ≤ h(α) + h(β) + log 2, • For all α and β in Q, we have h(αβ) ≤ h(α) + h(β), • For all non-zero α in Q, we have h(1/α) = h(α).
2.2. Nonstandard Analysis. Next we define nonstandard extensions which will be used later on.

Definition 2.1. (Nonstandard Extension of a Structure) Let M be a nonempty structure in a countable language L. A nonstandard extension * M of M is an ultrapower of M with respect to a non-principal ultrafilter on N.

Now let * M be a nonstandard extension of M with respect to a non-principal ultrafilter D on N. Note that the elements of * M are of the form (x n ) n /D where (x n ) n is a sequence in M. We identify each element x of M with the class of the constant sequence (x) n /D of * M. In this way, the structure M becomes an elementary substructure of * M. Ultraproduct of structures automatically become ℵ 1 -saturated. For a subset A of M, the set * A is defined to be the set

{(a n ) n /D : {n : a n ∈ A} ∈ D}.
Observe that * A contains A as mentioned above. The following sets * N, * Z, * Q, * R are called hypernatural numbers, hyperintegers, hyperrational numbers and hyperreals respectively. The elements * R \ R are called nonstandard real numbers. Let

R f in = {x ∈ * R : |x| < n for some n ∈ N}
be the finite numbers. The elements in * R \ R f in are called infinite. Note that the notion of a nonstandard extension and its properties can be extended to many-sorted structures. For more on nonstandard analysis, we direct the reader to [START_REF] Goldblatt | Lectures on the Hyperreals, A Introduction to Nonstandard Analysis[END_REF][START_REF] Henson | Foundation of Nonstandard Analysis, A Gentle Introduction to Nonstandard Extensions[END_REF].

Height Lemmas.

In this subsection, we compute heights of certain algebraic numbers which will be crucial in the proof of the Main Theorem. Lemma 2.2. For any non-zero algebraic number α which is not a root of unity, the set A(α) = {h(α q ) : q ∈ Q} is dense in the positive real numbers.

Proof. Let α be a non-zero algebraic number which is not a root of unity. Then by Theorem (1.1), we know that h(α) > 0. Furthermore, by the properties of the logarithmic height function given in Section 2.1, we have that h(α q ) = |q|h(α) for any q ∈ Q. Given an interval (a, b) where 0 ≤ a < b; choose r ∈ Q such that a/h(α) < r < b/h(α). Then we conclude that a < h(α r ) < b.

Lemma 2.3. Let p 1 , ..., p k and m 1 , ..., m k be positive natural numbers. Then

h(p 1 1/m 1 • • • p k 1/m k ) = h(p 1 1/m 1 ) + • • • + h(p 1 1/m k ). Proof. Put m = m 1 • • • m k and n i = m/m i .
As before, we know that for any algebraic number α and rational number q we have that h(α q ) = |q|h(α). Therefore, we see that

h(p 1 1/m 1 • • • p k 1/m k ) = h(p 1 n 1 • • • p k n k ) m .
Since for a natural number n ≥ 1 we have h

(n) = log n, we conclude that h(p 1 n 1 • • • p k n k ) m = h(p 1 n 1 ) + • • • + h(p k n k ) m = h(p 1 1/m 1 ) + • • • + h(p 1 1/m k ).
Lemma 2.4. Let p 1 , ..., p k , p k+1 and m 1 , ..., m k , m k+1 be natural numbers such that the product p 1 • • • p k and p k+1 are coprime. Then we have

h p 1 1/m 1 • • • p k 1/m k p k+1 1/m k+1 = max{h(p 1 1/m 1 • • • p k 1/m k ), h(p k+1 1/m k+1 )}. Proof. Put m = m 1 • • • m k m k+1 and n i = m/m i .
Recall that for coprime integers a and b, we have h(a/b) = max{log |a|, log |b|} = max{h(a), h(b)}. Thus by the properties of the logarithmic height function in Section 2.1, we have

h p 1 1/m 1 • • • p k 1/m k p k+1 1/m k+1 = 1 m h p 1 n 1 • • • p k n k p k+1 n k+1 = 1 m max{h(p 1 n 1 • • • p k n k ), h(p k+1 n k+1 )},
and the lemma follows from the properties of the logarithmic height function again.

Recall that µ is the set of complex roots of unity. Utilizing Kronecker's theorem, we can define µ uniformly in the pair (Q, S ε ) for any positive ε. Lemma 2.5. Let φ(x) be the formula x = 0 ∧ ∀y(U (y) → U (xy)) in the language L m (U ). Then for any positive ε, the formula φ(x) defines µ in the pair (Q, S ε ). In particular, the theory of (Q, S ε ) expands the theory of (Q, µ).

Proof. Let ε be a positive real number. A root of the unity satisfies the formula since the height function satisfies the inequality h(xy) ≤ h(x) + h(y) and the height of a root of unity is 0. Now suppose α satisfies the formula. If we take y = 1 in the formula, we see that h(α) ≤ ε and so we have α ∈ S ε . Then letting y = α, we get h(α) ≤ ε/2 and thus α 2 ∈ S ε . So taking powers of α, we conclude that h(α) ≤ ε/n for all n ≥ 1 and hence h(α) = 0. The lemma follows from Kronecker's theorem (1.1).

Elements of Small Height

In this section, we turn our attention to the model-theoretic properties of the field of algebraic numbers expanded by a predicate to denote elements of small height. We have all the tools to work on these pairs. 3.1. Simplicity and Independence Property. In this subsection we give two definitions for theories, namely simple theories and NIP, which play a significant role in the classification theory introduced by Shelah [START_REF] Shelah | Classification Theory[END_REF]. Then we prove the first part of the Main Theorem. Definition 3.1. A formula φ(x, y) has the tree property (TP) if there is a tree of parameters (a s : ∅ = s ∈ ω <ω ) such that (i) For all s ∈ ω <ω , (φ(x, a si ) : i < ω) is 2-inconsistent.

(ii) For all σ ∈ ω ω , (φ(x, a s ) : ∅ = s ⊂ σ) is consistent. We say that T is simple if no formula has the tree property.

Example 3.2. The theory of dense linear orders without end points (DLO) is not simple because the formula ϕ(x, y, z) : y < x < z has tree property. Let I be the set of rational numbers in the interval (0, 1) which is a model of DLO. We fix a tree of parameters (q s : ∅ = s ∈ ω <ω ) in this model for the formula ϕ above which we will need in order to prove the first part of our Main Theorem.

For more on simplicity we refer the reader to [START_REF] Tent | A Course in Model Theory: Lecture Notes in Logic[END_REF]Chapter 7]. Definition 3.3. Let T be a complete L-theory where L is a language. An L-formula φ(x, y) is said to have the independence property (IP ) if in every model M of T there is for each n a family of tuples b 1 , ..., b n , such that for each of the 2 n subsets I of {1, ..., n} there is a tuple a I in M for which

M |= φ(a I , b i ) ⇐⇒ i ∈ I. A theory T is called N IP if no formula has IP.
In other words, a theory T has the independence property, if there is a first-order formula in some model of T that can code any given subset of an arbitrarily large finite set. It is known that stable theories are NIP. Moreover T is stable if and only if T is simple and NIP. For more on the subject; see [START_REF] Shelah | Classification Theory[END_REF]. Now we are ready to prove the first part of our main theorem using lemmas (2.2), (2.3) and (2.4). Note that S ε is not a group under multiplication (or addition). In lemma (2.5), we proved that the theory of (Q, S ε ) expands the theory of (Q, µ) which is ω-stable. On the contrary, the theory of (Q, S ε ) is very complicated.

Theorem 3.4. The theory of (Q, S ε ) is not simple and has the independence property in the language L m (U ).

Proof. Recall that the Mahler measure of an algebraic number is again an algebraic number. Therefore by Lindemann's theorem, the logarithmic height of an algebraic number α is transcendental if h(α) = 0. First we prove that the theory is not simple. In order to show this, we exhibit a formula which has the tree property. Actually, the argument from Case 2 below also works in Case 1, and Case 1 is included to show the main idea of the proof in a more transparent way.

Case 1:

The element ε is in the range of the logarithmic height function, that is to say

ε = h(α) for some α ∈ Q. Now put φ(x, y, z, t) : U ty x ∧ U tx z .
Observe that for any rational number r, s ∈ (0, 1),

(1)

h α α r α s ≤ ε ⇐⇒ r ≤ s.
Furthermore if (r 1 , s 1 ) and (r 2 , s 2 ) are disjoint intervals of (0, 1), where r 1 < s 1 < r 2 < s 2 ∈ Q, then we cannot have

(Q, S ε ) |= ∃x(φ(x, α r 1 , α s 1 , α) ∧ φ(x, α r 2 , α s 2 , α)),
otherwise we have

h α x α s 1 ≤ ε and h α α r 2 x ≤ ε.
Therefore, if we multiply these elements, we obtain that h α 2 α r 2 α s 1 ≤ 2ε. This is a contradiction, since s 1 < r 2 and h(α 2 ) = 2ε. For a tuple β = (β 1 , ..., β m ) ∈ Q m , we put α β = (α β 1 , ..., α βm ).

Combining [START_REF] Ax | Injective endomorphisms of varieties and schemes[END_REF] and the previous argument, we see that the formula φ(x, y, z, t) has the tree property if we take the parameters (α qs , α) where the parameters (q s : ∅ = s ∈ ω <ω ) are as given in example (3.2).

Case 2: The element ε is not in the range of the logarithmic height function. Without loss of generality, we can assume that ε = 1 i.e S ε = S as 1 is not transcendental. This time we use the fact that the range of the logarithmic height function is dense in the positive reals. Again we set φ(x, y, z, t) : U ty x ∧ U tx z .

We will show that this formula has the tree property by finding some parameters in some model of T ε . Let * M be a nonstandard extension of the many-sorted structure

M = (Q, +, -, •, 0, 1, h, R ≥0 , 2 x , <, Q).
Then the logarithmic height function extends to * Q and it takes values in positive hyperreal numbers. We also denote this extension as h. Then the pair ( * Q, * S) is an elementary extension of (Q, S) in L m (U ). Note that * S is the set of hyperalgebraic numbers whose heights are less than 1. Let st(a) denote the standard part of a finite hyperreal number. By lemma (2.2), we know that the sequence {h(2 k )} is dense in positive real numbers where k is a rational number. In particular, there is a hyperrational number q such that q < 1/ log 2 and st(h(2 q )) = 1. In other words, q < 1/ log 2 and it is infinitely close to the real number 1/ log 2. Observe again that for any rational number r, s ∈ (0, 1),

h 2 q 2 r 2 s ≤ 1 ⇐⇒ r ≤ s.
Furthermore if (r 1 , s 1 ) and (r 2 , s 2 ) are disjoint intervals of (0, 1), where

r 1 < s 1 < r 2 < s 2 ∈ Q, then we cannot have ( * Q, * S) |= ∃x(φ(x, 2 r 1 , 2 s 1 , 2 q ) ∧ φ(x, 2 r 2 , 2 s 2 , 2 q )),
otherwise we have

h 2 q x 2 s 1 ≤ 1 and h 2 q 2 r 2 x ≤ 1.
Therefore, if we multiply these elements, we obtain that

h 2 2q 2 r 2 2 s 1 ≤ 2.
Similarly this is a contradiction, as s 1 < r 2 and st(h(2 2q ) = 2 imply

h 2 2q 2 r 2 2 s 1 > 2.
Thus the formula φ(x, y, z, t) has the tree property if we take the parameters (2 qs , 2 q ) where the parameters (q s : ∅ = s ∈ ω <ω ) are as given in example (3.2), and the tuple 2 qs is defined as in Case 1. Hence T ε is not simple. Now we show that T ε has the independence property. For simplicity, we assume that ε = 1. Let φ(x, y) be the formula U (x/y).

We will show that this formula has IP. Let n ≥ 1 be given. Let p 1 , ..., p n be distinct prime numbers. Put b m = p km m for m = 1, ..., n, where k m ∈ Q will be chosen later. Now let I be a subset of {1, ..., n} and r be the size of I. For i ∈ I, by lemma (2.2) we can choose

e i ∈ Q such that 1 r < h(p i e i ) ≤ 1 r + 1 (n + 1) 2 and we set a I = i∈I p i e i .
By lemma (2.3), we have

1 < h(a I ) ≤ 1 + 1 n + 1 . Now by lemma (2.2), choose k m such that 1 n + 1 < h(b m ) < 1.
Finally by lemma (2.4) and the properties of the height function, we get φ(a I , b i ) if and only if i ∈ I. This yields that φ(x, y) has the independence property and hence T ε is not N IP.

Salem Numbers and Lehmer's conjecture

Lehmer's conjecture (1933): There exists an absolute constant c > 1 such that if α is not a root of unity nor zero, then m(α) ≥ c. This conjecture stands still open. The best known result is due to E. Dobrowolski [START_REF] Dobrowolski | On a question of Lehmer and the number of irreducible factors of a polynomial[END_REF] in 1979, who proved that for d ≥ 2, m(α) > 1 + 1 1200 (log log d) 3 log d ,

where d = deg(α). A polynomial p(x) of degree d is called reciprocal if x d p(1/x) = ±p(x).
Furthermore, an algebraic number α is reciprocal if it is Galois conjugate to 1/α, in other words its minimal polynomial is reciprocal. In 1971, C. Smyth [START_REF] Smyth | On the product of the conjugates outside the unit circle of an algebraic integer[END_REF] showed that for α a nonreciprocal algebraic number m(α) ≥ m(x 3 -x -1) = θ ≈ 1.3247, where θ is the real root of x So a Salem number is reciprocal. Observe that for a Salem number α and a positive integer n, the element α n is also a Salem number, and we have

m(α n ) = m(α) n = α n .
In general this is not true for an arbitrary algebraic number. It is an open question whether 1 is a limit point of Salem numbers. This is a special case of Lehmer's conjecture. The smallest known Salem number α was given by D. Lehmer [START_REF] Lehmer | Factorization of certain cyclotomic functions[END_REF] which is the root of the polynomial X 10 + X 9 -X 7 -X 6 -X 5 -X 4 -X 3 + X + 1 with α ≈ 1.17628 and not many Salem numbers are known in the interval (1, 1.3). The list of small Salem numbers can be found in the link [START_REF]Small Salem numbers[END_REF]. For more about Lehmer's conjecture and Salem numbers, we direct the reader to [START_REF] Smyth | The Mahler Measure of Algebraic Numbers: A survey[END_REF][START_REF] Smyth | Seventy Years of Salem Numbers: A Survey[END_REF]. The rest of the paper is devoted to Lehmer's conjecture via model theory. Let S be the set of all Salem numbers. Recall that as before. Observe that for any rational numbers r and s in (0, 1), we have

P b := {a ∈ Q × : m(
m β N β [N r] β [N s] ≤ b ⇐⇒ r ≤ s
since N is infinite. Similar to the proof of Theorem (3.4), the formula φ(x, y, z, t) has the tree property by taking the parameters a s = (β [N qs] , β N ) which are coming from example (3.2). Hence (Q, S b ) is not simple, a contradiction. The proof of the second part follows from the methodology for proving the first part of the theorem. One way of showing the simplicity is to find a notion of independence which is symmetric and satisfying the axioms of non-forking [START_REF] Tent | A Course in Model Theory: Lecture Notes in Logic[END_REF]Chapter 7]. In [START_REF]Model Theory of Fields and Heights[END_REF][START_REF]Tame Expansions of ω-Stable Theories and Definable Groups[END_REF], the author characterized the independence in a sufficiently saturated model (U, G) of (Q, µ) in the language L r (U ). More precisely, let A and B be two algebraically closed sets in the pair (U, G). Then Hence in a sufficiently saturated model of (Q, P b ) for a small b, if the theory is simple then the independence should be the independence given above. When 1 < b < 1.3, by [START_REF] Smyth | The Mahler Measure of Algebraic Numbers: A survey[END_REF] we know that every element α in P b is reciprocal. By [START_REF] Casanovas | Stable theories with a new predicate[END_REF][START_REF] Pillay | The model-theoretic contents of Lang's conjecture[END_REF], every formula in the pair (Q × , P b ) is bounded, which means every formula in the pair is equivalent to a formula of the form Qα ∈ P b φ(x, α) where Q is a tuple of quantifiers and φ(x, α) is an L m -formula. Moreover by [START_REF] Casanovas | Stable theories with a new predicate[END_REF][START_REF] Pillay | The model-theoretic contents of Lang's conjecture[END_REF] again, if the induced structure on P b is stable or does not have the finite cover property, then the pair (Q × , P b ) is stable. Hence, understanding the induced structure on P b and how P b (a set of reciprocal algebraic integers) intersects a set of the form {(x 1 , ..., x n ) : x 1 a 1 • • • x n an = 1, x i ∈ Q × , a i ∈ Z} will lead to the stability of the pair, thus to Lehmer's conjecture. Another direction is first to show a weaker result that the pair (Q

× , P b )
is NIP in L m (U ). By the paper of A. Chernikov and P. Simon [START_REF] Chernikov | Externally definable sets and dependent pairs[END_REF], again we need to understand the induced structure on P b . If the induced structure on P b is NIP, then so is the pair (Q × , P b ).

We end the paper by posing the following question which is also related to Lehmer's conjecture.

• Is µ definable in (Q, P b ) for some b > 1?

  ≤ b} and S b = P b ∩ S where b ≥ 1, and we set S ε = {a ∈ Q : h(a) ≤ ε} where ε > 0. Both P b and S ε contain the group of roots of unity µ. In other words, Lehmer's conjecture and its version for Salem numbers state that there exists b > 1 such that P b = P 1 = µ and S b = S 1 = ∅ respectively.

2. 1 .

 1 The Logarithmic Height Function. Details of this subsection can be found in [2, Chapter 1] and [11, Part B, B.7]. The logarithmic height function behaves well under arithmetic operations and it satisfies the following properties: • h(0) = h(1) = 0, • For a non-zero rational number a/b where a and b are coprime, h(a/b) = max{log |a|, log |b|},

  a) ≤ b} and S b = P b ∩ S where b ≥ 1. Note that S b is the set of Salem numbers ≤ b. Lehmer's conjecture for Salem numbers is equivalent of there exists b > 1 such that S b = S 1 = ∅. The next Theorem relates the simplicity of the pair (Q, S b ) with Lehmer's conjecture and it is the other part of the Main Theorem.In particular, it states that the simplicity of the pair (Q, S b ) for some b > 1, which is a weaker statement than ω-stability, is equivalent of Lehmer's conjecture for Salem numbers.

Theorem 4 . 2 .

 42 The theory of (Q, S b ) is simple for some b > 1 in L m (U ) if and only if Lehmer's conjecture holds for all Salem numbers. Moreover, the simplicity of the pair (Q, P b ) for some b > 1 in L m (U ) implies that Lehmer's conjecture holds for all Salem numbers.Proof. If Lehmer's conjecture is true for Salem numbers, then we know that the pair (Q, S b ) = (Q, ∅) is ω-stable for some b > 1. Now suppose that (Q, S b ) is simple for some b > 1. Assume that 1 is a limit point of Salem numbers in order to get a contradiction. Then we can choose a Salem number α and a positive integer n such that α n is very close to b. Denote the integer part of a as[a].Let * M = ( * Q, +, -, •, 0, 1, m, * R ≥1 , * Q, * N, <, g, [•]) be a nonstandard extension of the many-sorted structure M = (Q, +, -, •, 0, 1, m, R ≥1 , Q, N, <, g, [•])where g is the binary function from Q × N to Q defined as g(a, b) = a b . The Mahler measure m extends to * Q and it takes values in positive hyperreal numbers ≥ 1. We also denote this extension by m. Now the pair ( * Q, * S b ) is an elementary extension of (Q, S b ) in L m (U ). Note that * S b is the set of hyperalgebraic Salem numbers that are less than b. Then there is a nonstandard Salem number β > 1 which is infinitely close to 1, and an infinite nonstandard natural number N such that β N < b and st(β N ) = b. Put φ(x, y, z, t) : U ty x ∧ U tx z

Remark 4 . 3 .

 43 We call an algebraic number α multiplicative if m(α n ) = m(α) n for all natural numbers n ≥ 1. Let D b be the set of multiplicative algebraic numbers whose Mahler measure is less than b. Note that D b contains S b . Then the same proof of Theorem (4.2) shows that the simplicity of the pair (Q, D b ) for some b > 1 in L m (U ) is equivalent to Lehmer's conjecture for multiplicative algebraic numbers.The following conjecture is the model-theoretic characterization of Lehmer's conjecture.

Conjecture 4 . 4 .

 44 The theory of (Q, P b ) is stable for some b > 1 in L m (U ). Moreover the stability of the pair (Q, P b ) in L m (U ) for some b > 1 implies Lehmer's conjecture.

  3 -x -1. Thus the remaining and the difficult case for Lehmer's conjecture is the set of reciprocal algebraic numbers. The next definition plays a central role in Lehmer's conjecture. Definition 4.1. A real algebraic integer α > 1 is called a Salem number if α and 1/α are Galois conjugate and all others Galois conjugates of α lie on the unit circle.

Partially supported by ValCoMo (ANR-13-BS01-0006) and MALOA (PITN-GA-2009-238381).