Warning Young Adults Against Tobacco Consumption Through Ad Parodies: its Effects on Cigarette Brands Attitude

Keywords: ad parodies, tobacco consumption, emotions, cognitive, Ab Track: Social Responsibility, Ethics and Consumer Protection

This paper compares the effects of anti-tobacco ad parodies and visual cigarette package warnings on emotional and cognitive responses of young adults. The findings indicate that graphic-only ad parodies can compete with warnings in their attempt to damage consumers' attitude toward tobacco brands through the health beliefs they lead consumers to associate to the brand. On the contrary, text-only ad parodies prove counterproductive and lead to a boomerang effect characterized by an increase in consumers' tobacco brand attitude.

In the attempt to reduce the number of smokers, anti-tobacco nonprofit groups use creative methods to voice their message, by creating ad parodies they spread thoughout websites [START_REF] Vanden Bergh | The multidimensional nature and brand impact of user-generated ad parodies in social media[END_REF]. However, studies mainly focus on consumer-created ad parodies undertaken by brand loyalists on behalf of the brand and left anti-brand ad parodies effects on persuasion largely unexplored [START_REF] Berthon | Ad lib: when customers create the ad[END_REF]. Clearly, research is lacking on whether anti-brand ad parodies could be a means of influencing the attitudes, perceptions and resulting behaviors. In the case of tobacco, it is now important for both anti-brand activists who create and spread anti-tobacco brand ad parodies, and the researchers who study them, to gain a better understanding of their relative efficiency compared to more classical anti-tobacco warnings, such as cigarette package warnings.

We thus study whether ad parodies can really impact consumer's attitude toward tobacco brands, and, if so, the mechanisms by which such an effect occurs. Such a focus on cigarette brands attitude as the dependent variable appears highly relevant since cigarette brands attitude has largely been shown to influence tobacco consumption behaviors as a means of self-expression and peers acceptance among young adults [START_REF] Pechmann | An experimental investigation of the joint effects of advertising and peers on adolescents' beliefs and intentions about cigarette consumption[END_REF]. Also, focusing on consumers' attitude toward specific cigarette brands and not toward smoking in general finds support in that anti-tobacco warnings appear relatively ineffective in influencing attitude toward smoking among young adults who already hold strong preconceptions on smoking's adverse health effects [START_REF] Pechmann | The effects of antismoking and cigarette advertising on young adolescents' perceptions of peers who smoke[END_REF]. However, they can clearly influence a more changing variable such as cigarette brands attitude. Thus, we focus more precisely on the following questions:

1. What is the influence of anti-tobacco brand ad parodies compared with cigarette package warnings on consumers' attitude about cigarette brands? 2. Do the effects of anti-tobacco brand ad parodies that parody only the ad text differ from those that parody both the ad text and graphics?

Anti-Brand Ad Parodies: An Overview

Anti-brand ad parodies refer to hijack actions on official brand ads, mixing part of those official ads' materials with new ones in a sarcastic way to make the original ad ridiculous. In this paper, we distinguish between "text-only ad parodies", only parodying the official ad text, and "graphic-only ad parodies", only parodying the official graphics. While ad parodies were previously the prerogatives of professionals, today, anyone with a computer and a statement to make can craft a professional looking ad parody [START_REF] Berthon | Ad lib: when customers create the ad[END_REF]. Spoofing cigarette brands ads on line, activists question cigarette brands advertising impact and legitimacy. In the long term, their objective is to counter the effects of official cigarette brands advertising and to warn people against their hazards. To do so, they target specific brands, such as Marlboro or Camel, because they symbolize the tobacco industry. Still, we do not know much on their precise effects in the short term on cigarette brands attitude. Vanden [START_REF] Vanden Bergh | The multidimensional nature and brand impact of user-generated ad parodies in social media[END_REF] suggest the brand may be harmed if the denunciation effect predominates and the original brand is its target. However, if the humorous effect predominates, humor might reinforce pleasant emotional associations with the brand, preventing it form being harmed. Alternatively, one would expect that the brand may not be harmed if people understand that the parody's purpose is to use the brand's meanings to attack something more general and not the targeted brand in particular. Our goal is thus to ensure that activists are right in targeting specific brands through humoristic ad parodies to warn people against hazards of smoking and to damage consumer's attitude toward the brand. The question remains how -that is through what specific route -ad parodies exert their persuasive effects.

Conceptual Background

The emotional route to persuasion

As warnings, anti-tobacco brand ad parodies and cigarette package warnings are likely to induce negative emotional reactions. However, the extent to which they do so may differ. Following the elaboration likelihood model [START_REF] Petty | Attitudes and persuasion: classic and contemporary approaches[END_REF] which suggests that peripheral elements have their main influence through the emotional route while central elements have their main influence through the cognitive route, the influence of ad parodies might more depend on the image they parody than on the text they parody. As text-only parodies display the positive official ad graphics, when graphic-only ad parodies display new striking materials, we expect graphic-only ad parodies to elicit more negative emotional reactions than text-only ad parodies. As cigarette package warnings feature shocking pictures while graphic-only ad parodies take their inspiration in official ads materials and therefore depict the dangers of tobacco use in a less concrete manner, we predict that cigarette package warnings will elicit more negative emotional reactions than graphic-only ad parodies. Thus: H1: Graphic-only ad parodies elicit more negative emotional reactions than text-only ad parodies (a) but less than cigarette package warnings (b).

Anti-tobacco brand ad parodies are consumer-created ad parodies designed to make laugh [START_REF] Vanden Bergh | The multidimensional nature and brand impact of user-generated ad parodies in social media[END_REF]. Therefore, ad parodies are likely to elicit more positive emotional reactions compared with cigarette package warnings. As caricatures, they have to create a disparity between the image and the realty of the object of caricature to make laugh. They surprise the viewer when they create confusion by incorporating elements that do not fit with his expectations [START_REF] Speck | The humorous message taxonomy: A framework for the study of humorous ads[END_REF]. Then, the viewer understands that the parody is indicating how the original ad might not have been telling the whole truth and experiences positive emotions such as release and humor. Text-only ad parodies display a strong discrepancy, mixing graphic elements favorable to cigarette brands and familiar to the viewer with an unfavorable text, while graphic-only ad parodies do not provide such gap. Therefore, textonly ad parodies might elicit more positive emotional reactions. Thus: H2: Graphic-only ad parodies elicit less positive emotional reactions than text-only ad parodies (a), but more than cigarette package warnings (b).

The cognitive route to persuasion

The cognitive route represents the product of a consumers' elaboration of the information presented in a persuasive message. Brand associations are crucial in the process of elaboration. We focus on the two strongest cigarette brands associations, namely their perceived risk and their symbolic image. Regarding perceived risk, visual warnings are more noticeable and easier to understand and thus more efficient in communicating health hazards [START_REF] Gallopel-Morvan | The use of visual warnings in social marketing: the case of tobacco[END_REF]. The same reasoning should play to compare text-only ad parodies with graphic-only ad parodies. The former communicate health hazards in an ambivalent way as it contains non-convergent elements when the latter depicts them in a clearer way. Therefore, the risks associated with cigarette brands are easier to understand in graphic-only ad parodies compared with text-only ad parodies. When comparing graphic-only ad parodies versus cigarette package warnings, cigarette packages are poorly branded, exhibiting only cigarette brands name and warnings. They thus could be perceived as boring and unattractive, making therefore health warning more noticeable. Thus:

H3: Subjects exposed to graphic-only ad parodies perceive cigarette brands as more risky than subjects exposed to text-only ad parodies (a), but less risky than subjects exposed to cigarette package warnings (b).

Cigarette brands official ads are crafted to address young adults' need for independence, self reliance, and freedom [START_REF] Pechmann | The effects of antismoking and cigarette advertising on young adolescents' perceptions of peers who smoke[END_REF]. When creating ad parodies, consumers create a viral object that might help in spreading the original ad's themes and imagery [START_REF] Vanden Bergh | The multidimensional nature and brand impact of user-generated ad parodies in social media[END_REF]. If anti-tobacco brand ad parodies present smokers as physically attractive, engaged in exciting activities, then they could run counter to messages that smoking is dangerous to one's health and reinforce perceptions that smoking is a normative consumption product [START_REF] Pechmann | An experimental investigation of the joint effects of advertising and peers on adolescents' beliefs and intentions about cigarette consumption[END_REF][START_REF] Devlin | Comparative study of young people's response to anti-smoking messages[END_REF]. Antitobacco activists might thus reactivate cigarette brands positive symbolic beliefs in consumers' mind and obtain the same results as official ads. As text-only ad parodies appropriate the graphic elements of official ads, they are more likely to enhance brands positive symbolic beliefs compared with other forms of anti-tobacco warnings. Thus: H4: Subjects exposed to graphic-only ad parodies associate less positive symbolic beliefs to cigarette brands than subjects exposed to text-only ad parodies (a) and cigarette package warnings (b).

The change in cigarette brands attitude

On the emotional route, negative emotions result in an increase in persuasiveness of [START_REF] Hammond | Health warning messages on tobacco products: a review[END_REF]. Besides, negative emotions are supposed to transfer to the object to which they are associated by mere association [START_REF] Mitchell | Are product attribute beliefs the only mediator of advertising effects on brand attitude[END_REF]. Following the same argument, anti-tobacco brand ad parodies that generate upbeat feelings might enhance cigarette brands attitude. The cognitive route provides an alternative explanation for the process of change in cigarette brands attitude. First, the Protection Motivation Model [START_REF] Rogers | A protection motivation theory of fear appeals and attitude change[END_REF] and the Health Belief Model [START_REF] Rosenstock | Historical origins of the health belief model[END_REF] posit that consumers' persuasion depends on the appraisal of the severity of the risks, such risk being likely to lead to brand avoidance. Second, the positive symbolic meanings associated with cigarette brands are likely to transfer to cigarette brands attitude [START_REF] Pechmann | The effects of antismoking and cigarette advertising on young adolescents' perceptions of peers who smoke[END_REF][START_REF] Pechmann | An experimental investigation of the joint effects of advertising and peers on adolescents' beliefs and intentions about cigarette consumption[END_REF]. It may thus be expected that the effect of anti-tobacco warnings (representing both ad parodies and cigarette package warnings) on Ab is mediated by the emotional reactions experienced by the subjects, and by their brand beliefs: H5: The effects of anti-tobacco warnings and ad parodies on Ab are mediated by emotional and cognitive reactions. Graphic-only ad parodies will lead to lower Ab because of their positive effect on negative emotions (fear and disgust) (H5a and H5b) and health beliefs (H5c). On the contrary, text-only ad parodies will lead to higher Ab because of their effect on positive emotions (upbeat feelings) (H5d) and on symbolic beliefs (H5e).

Method: Research Design, Stimulus Selection, Sample and Measures

We carried an experiment considering a text-only ad parody, a graphic-only ad parody and a cigarette package warning, all these stimuli being real materials. To control for familiarity, Marlboro was chosen as the brand in the experiment [START_REF] Hemdev | Marlboro -a mini case study[END_REF]. The textonly ad parody was the Marlboro picture presenting two cowboys riding into the sunset. The only single change from the original ad which had been made by the ad parody developers resulted from the humoristic text warning "I miss my lung Bob". The graphic-only ad parody used the original sentence "Welcome to Marlboro Country", the graphic change made by the ad parody developers consisting in depicting a graveyard in the background. The cigarette package warning condition included a graphic warning showing one healthy lung and one damaged, and a verbal warning "Smoking causes fatal lung cancer".

Participants were 139 university students (mean age = 22, 35% being men and 65% being women, 43% being smokers). The respondents were randomly assigned to one of the three experimental conditions. Chi-Square tests and a t-tests revealed that respondents in the three experimental conditions were similar in terms of gender (χ² Gender = .868, df= 2, p > .05) and tobacco consumption habits (χ² Smoker = 1.345, df = 2, p > .05).

Measures of upbeat feelings [START_REF] Mooradian | Personality and ad-evoked feelings: the case for extraversion and neuroticism[END_REF], fear [START_REF] Laroche | A cross-cultural study of the persuasive effect of fear appeal messages in cigarette advertising: China and Canada[END_REF], disgust [START_REF] Izard | Human Emotions[END_REF], health and symbolic beliefs associated to the brand [START_REF] Hemdev | Marlboro -a mini case study[END_REF], and Ab were adopted from previous research. Reliabilities ranged from 0.82 to 0.91. Ab was measured through "the extent to which they liked the Marlboro Brand". Each item was rated on a 7-point scale ranging from 1 (Not at all) to 7 (Very Strongly).

Results

Effects of ad parodies exposure on emotional responses

Findings indicate that the ad parodies versus cigarette package warning exposure manipulation had a significant effect on fear (F(4, 136) = 7.54; p < .05). Graphic-only ad parody exposure condition resulted in a significant increase in fear only when compared with text-only ad parody exposure condition (M Graphic = 4.06 vs. M Text = 2.72, p < .001). No significant difference between graphic-only ad parody exposure condition and cigarette package warning exposure condition was found (M Warning = 3.80, p > .05). Regarding disgust, the highest level is obtained in the cigarette package warning condition (M Warning = 4.84). Disgust in the graphic-only ad parody exposure condition is significantly higher than the ones obtained in the text-only ad parody exposure condition (M Graphic = 4.06 vs. M Text = 2.75, p < .001) but lower than in the cigarette package warning exposure condition (M Warning = 4.84, p < .05). These results offer support for H1(a) and partial support for H1(b).

Regarding positive emotions, our findings indicate that the ad parodies versus cigarette package warning exposure manipulation had a significant effect on upbeat feelings (F(2, 136) = 6.62 ; p < .01). Specifically, graphic-only ad parody exposure condition resulted in a significant decrease in upbeat feelings when compared with text-only ad parody exposure condition (M Graphic = 1.46 vs. M Text = 2.25, p < .01). However, the graphic-only ad parody exposure did not induce more upbeat feelings that the cigarette package warning exposure (M Warning = 1.55, p > .05). Thus, H2(a) is supported but not H2 (b).

Effects of ad parodies exposure on cognitive responses

Results indicate that the ad parodies and warning exposure manipulation has a marginally significant effect on health beliefs (F(2, 136) = 2.65 ; p < .10), and a significant effect on symbolic beliefs (F(2, 136) = 4.99; p < .01). Results revealed a marginally significant difference between graphic-only and text-only ad parodies exposure conditions (M Graphic = 5.49 vs. M Text = 4.96, p < .10), and a more significant difference when comparing the effect of graphic-only ad parody to the one of cigarette package warning (M Graphic = 5.49 vs. M Warning = 4.79, p < .05). Thus, these results offer support for H3(a), but not for H3(b).

Results regarding the effects of ad parodies and warnings on symbolic beliefs revealed that the highest level of symbolic beliefs text-only ad parodies (M Text = 3.64) is significantly higher than the one of individuals exposed to graphic-only ad parodies (M Text = 3.64 vs. M Graphic = 2.82, p < .05). Symbolic beliefs induced by exposure to graphic-only ad parodies did not differ to the ones elicited by exposure to cigarette package warnings (M Graphic = 2.82 vs. M Warning = 2.72, p > .05). These results offer support for H4(a) but not H4(b).

Tests of the mediating roles of emotions and cognitions

We tested emotions and cognitions as potential mediators of the ad parodies exposure effects on Ab following [START_REF] Zhao | Reconsidering Baron and Keeny: myths and truths about mediation analysis[END_REF] procedure. We compared the effects of graphic-only ad parodies to respectively the ones of text-only ad parodies, and then the ones of cigarette package warnings. The graphic-only ad parody condition was coded 1, so that a positive effect can be interpreted as the effect of the graphic-only ad parody exposure and a negative effect as the one caused by the other kind of warning. Smoking status was included in the analyses as a control variable.

When comparing the mechanism by which being exposed to graphic-only ad parodies versus text-only ad parodies lead to changes in Ab, the most striking result lies in the absence of mediating effects of emotions. Indeed, all indirect effects of fear, disgust and upbeat feelings are not significant, providing no support to H5 (a), H5(b) and H5 (c). On the contrary, results highlight that the only mediating variables is symbolic beliefs (a × b = -.12, the confidence interval excluding 0). The negative sign of the mediating effect reveals that increases in Ab may be due to increases in symbolic beliefs induced by text-only ad parodies. This result thus supports H5 (e). No mediating effect of health beliefs was found out.

When comparing the mechanism underlying the effects of being exposed to graphiconly ad parodies versus cigarette package warnings, the only indirect effect that was significant is the one of health beliefs (a b = -.18, the confidence interval excluding 0). Thus, here again, emotions do not play any mediating role in the influence of ad parodies versus warnings exposure on attitude, leaving the influence of ad parodies and warnings on Ab being explained by cognitions. Since graphic-only ad parodies exposure condition was coded 1, the negative sign of the mediating effect reveals that increases in Ab may be due to health beliefs induced by the graphic-only ad parodies exposure. This result supports H5 (b).

General Discussion

This research aimed at demonstrating that ad parodies may be a new way to take part in the public fight against smoking behavior by leading young adults to develop negative attitudes toward tobacco brands. Overall, our study contributes to a better understanding of the effects of ad parodies by first showing that cognitions mediate the comparative effects of ad parodies and warnings. Second, results show that graphic-only ad parodies can compete with warnings in their attempt to damage consumers' attitude toward tobacco brands through the health beliefs they lead consumers to associate to the brand, which decreases attitude. Furthermore and of much importance, due to the symbolic beliefs they induce, text-only ad parodies may prove counterproductive and lead to a boomerang effect characterized by an increase in consumers' tobacco brand attitude.

Since our main result relates to the mediating role of health beliefs, it may be suggested that directly showing how through graphic ad parodies the hazards related to tobacco consumption, public officers can lead consumers to develop negative attitude toward tobacco brands. For activists who create ad parodies to hurt tobacco brands, graphic-only ad parodies may represent creative ways of decreasing such attitude indirectly through health beliefs. Meanwhile, our results also suggest avoiding text-only spoof ads prime symbolic beliefs which in turn positively affect consumers' attitude toward the tobacco brand, the exact opposite reaction of the one that ad parodies developers aim at generating.


Javier Couto
email: jcouto@syllabs.com

Helena Blancafort
email: blancafort@syllabs.com

Somara Seng
email: seng@syllabs.com

Nicolas Kuchmann-Beauger

Anass Talby
email: talby@syllabs.com

Claude De Loupy

OAL 1 : A NLP Architecture to Improve the Development of Linguistic Resources for NLP

The performance of most NLP applications relies upon the quality of linguistic resources. The creation, maintenance and enrichment of those resources are a labour-intensive task. In this paper we present the NLP architecture OAL, designed to assist computational linguists in the whole process of the development of resources in an industrial context: from corpora compilation to quality assurance. To add new words more easily to the morphosyntactic lexica, a guesser that lemmatizes and assigns morphosyntactic tags as well as inflection paradigms to a new word has been developed. Moreover, different control mechanisms are set up to check the coherence and consistency of the resources. Today OAL manages resources in five European languages: French, English, Spanish, Italian and Polish. Chinese and Portuguese are in process. The development of OAL has followed an incremental strategy.

1 OAL stands for French -Outil d'aide au linguiste‖, this is, a tool to assist linguists

Introduction

One key issue in Natural Language Processing (from now on NLP) is related to the quality of the linguistic resources (e.g. morphosyntactic lexicons, semantic lexicons, specialized corpora) that are at the heart of NLP software. The creation, maintenance an enrichment of those linguistic resources are labour-intensive tasks, especially when no tools are available. Very often, in industrial NLP contexts, the linguist develops lexicographic resources without any specific tool but just using a text editor and several processing tools such as web crawlers, text content extractors (scrapers), tools to import and export information, tools to connect to resources such as GeoNames or Wikipedia or engines such as Yahoo! Boss, and scripts to generate inflections and to resolve text encoding problems. Thus, the information is often stored in textual format. Text file manipulation with Unix commands or scripts written in PERL or Python are a common practice.

To improve of the development of linguistic resources and provide quality assurance, Syllabs has designed OAL, an architecture to provide a user-friendly environment to develop lexicographic resources. The linguists are assisted during the whole process -from corpora compilation to the quality check of the resources.

Corpora may be acquired from the web by focused crawling or from selected RSS feeds. Quality assurance is provided by several quality control mechanisms such as regression tests, automatic and semi-automatic procedures to evaluate the consistency of each linguistic resource. Today OAL manages resources in five European languages: French, Esnglish, Spanish, Italian and Polish. Chinese and Portuguese are in process.

German will be included in the near future. From the beginning, we wanted the OAL platform to be language independent, even if we know that it is impossible to define a single formalism for any language in the world and that it will be necessary to add new features for new languages. The ongoing integration of Polish resources proves that the platform (i.e. the formalisms and tools developed) is not only useful for Romance languages or English. Chinese is a major challenge for OAL, in particular for the SylLex formalism (see section 5).

For now, the architecture and all the tools and resources were designed and developed for internal use at Syllabs. Thus, resources definitions and tools are proprietary. The design and development of OAL are carried out following an incremental strategy. For example, semantic lexica are at present being developed. This implies the extension of the SylLex formalism and the lexicon editor (see section 5). In addition, a named entities guesser is under construction, which involves the API enrichment of the linguistic resources server (see section 3).

In this paper we give an overview of the whole OAL architecture and describe in detail its use for morphosyntactic lexica management (formalism and tools). However, this is only a part of the system. The paper is organized as follows. First, some related works are presented. Then, the OAL architecture is described.

The fourth section briefly presents a morphosyntactic guesser, an example of an OAL component. The fifth section describes in detail the morphosyntactic lexicon editor, another example of an OAL component. Finally, we draw some conclusions and further work is presented.

Related Work

There are works related to each OAL component. For example, Lexicon Creator [START_REF] Fontenelle | Lexicon Creator: A Tool for Building Lexicons for Proofing Tools and Search Technologies[END_REF], TshwaneLex [START_REF] Joffe | TshwaneLex. A State-of-the-Art Dictionary Compilation Program‖[END_REF] and Word Manager [START_REF] Hacken | Word Formation and the Validation of Lexical Resources‖[END_REF][START_REF] Hacken | A Knowledge Acquisition and Management System for Morphological Dictionaries‖[END_REF]) are lexicon editors with powerful functionalities. Nevertheless, they are not integrated in a global processing architecture. An example of such architectures, we can mention the Victoria project [START_REF] Nicolas | Producción eficiente de recursos lingüísticos: el proyecto Victoria[END_REF]) and Nooj [START_REF] Silberztein | Nooj's Dictionaries‖[END_REF].The main difference of OAL is its client-server approach empowered by a rich lexicon editor, and the integration of a whole process that may start with the focused crawling of textual content from the Web and end by the automatic suggestion of term candidates to be validated by the linguist.

OAL Architecture

The OAL architecture (cf. figure 1) draws its inspiration from [START_REF] Loupy | Aide à la construction de lexiques morphosyntaxiques[END_REF].

Figure 1: OAL Architecture.

The main components of this client-server architecture are:

-an intelligent web crawler, -a processing pipeline, -a linguistic resources server, specialized editors for creating, maintaining and enriching linguistic resources.

The processing pipeline role is to extract from a raw corpus the elements that are potentially relevant for an existing resource from a raw corpus. When developing a morphosyntactic lexicon, for example, the pipeline cleans the input (deletion of boilerplate texts) and extracts unknown words (not in the lexicon). This is not a trivial task, as a large amount of noise is generated during this automatic procedure. This is why a guesser tool has been developed (cf. section 4). The corpora are collected in three different ways: 1) manually built, 2) crawling, or 3) RSS feed. The processing pipeline relies on several tools developed internally at Syllabs or distributed as freeware: tokenizers, guessers (see next section), POS taggers, KWIC, etc.

The linguists do not have direct access to results obtained from the focused crawler and the processing pipeline. These components transfer the results, via web services, to the linguistic resources server. The server offers API to integrate the results and to access the resources. Let us consider the development of a domain-specific lexicon in the field of economics, first economic news are crawled to compile a corpus in this domain. Then, the pipeline cleans the texts, extracts new terms and transfers the results to the server, which integrates them as suggestions to the server version of the economy lexicon. Next, the server distributes the suggestions between the lexicon versions of the users who are in charge of maintaining the resource. Finally, when an authorized user checks out the economy lexicon, he can manually validate the suggestions. Managing the resources implies managing the server and local user versions, allowing the merging of different user versions, resolving conflicts when merging, but also other aspects such as user permissions, logs and statistical information concerning the word frequencies run on lexica and corpora.

A specialized lexicon editor provides an easy-to-use environment to create, maintain and enrich lexica. The editor is a smart client that interacts with the linguistic resources server (cf. section 5).

Using a Morphosyntactic Guesser

Lexicon enrichment means adding new words. When dealing with morphosyntactic lexica, the procedure implies assigning an inflection paradigm to a lemma or running a program to generate all the inflections of a lemma with their corresponding morphosyntactic tag.

Usually, the linguist is the one who has to organize the words to be added in different files. Each file corresponds to an inflection class to associate to each word or list of words. It is possible of course to create sublists with words classified automatically depending on the word ending of the word. Suffix information may indicate the inflection class of a word, but some noise can be generated, as some suffixes may correspond to more than one word category In French, for instance, words ending with the suffix -ment can be an adverb or a noun, as the adverb joliment and the noun jugement. But still, suffixes are not enough information to associate automatically an inflection class to a word. This is why linguists do have to check the list manually.

A solution to speed up this process and make it more user-friendly is to integrate a guesser. Today, most PoS taggers include a guessing procedure to guess the PoS. However, a morphosyntactic lexicon needs more information. This is why we have developed a more advanced guesser to associate one or more triplet (lemma, inflection paradigm, confidence score) to an unknown word. The guesser uses a hybrid approach: a statistical model inspired from [START_REF] Mikheev | Automatic rule induction for unknown-word guessing‖[END_REF]) that computes probabilities from words endings and a set of heuristics.

The heuristics includes the use of prefixes, the detection of foreign words and the verification over Internet of the forms guessed (Kuchmann-Beauger 2009, Kuchmann-Beauger et al. 2010).

As the guesser has been trained with the inflection paradigms of our lexicon, for each word it suggests a lemma and an inflection paradigm corresponding to the inflections paradigms of our lexicon. To illustrate, for the word commercialization the guesser proposes the following suggestions:

RG Paradigm suggested Probability 1

N_Fem-s 0.679254835 2 N_Masc-s 0.041687166 3 N_Masc-NULL 0.002511341 4

N_MascFem-NULL-s-s 0.002426684 5 A-NULL 1.5267522 E-4 Table 1: paradigms suggested by the guesser for the word commercialization.

Editing the lexicon

In this section we present the morphosyntactic lexicon editor implemented into the OAL framework. As we said before, similar works exist [START_REF] Fontenelle | Lexicon Creator: A Tool for Building Lexicons for Proofing Tools and Search Technologies[END_REF][START_REF] Joffe | TshwaneLex. A State-of-the-Art Dictionary Compilation Program‖[END_REF][START_REF] Hacken | Word Formation and the Validation of Lexical Resources‖[END_REF]. Two main differences have to be stressed: (1) OAL editor is a rich client that communicates with a linguistics resources server and (2) the editor implements SylLex [START_REF] Blancafort | Morphosyntactic Lexica in the OAL Framework: Towards a Formalism to Handle Spelling Variants, Compounds and Multi-words[END_REF]), a morphosyntactic lexicon formalism.

SylLex organizes every morphosyntactic lexicon in three components: a list of lemmas, a set of inflection paradigms and a set of patterns. Patterns help the linguist to create inflection paradigms: they contain the information about all morphosyntactic tags, all possible inflection forms of a lemma and further morphological information (e.g., stems and suffix). All this information is needed for word inflection and is specified by the linguist will when he/she creates an inflection paradigm. Each language has its own SylLex model, thus its own set of patterns and flexion paradigms.

When a new lexicon is created, the patterns are first outlined and then inflection paradigms are defined by filling the corresponding fields: stems and suffixes for Romance Languages. For German, for instance, a further field -prefix‖ is required for verbal inflection for the inflectional prefix ge as well as a specific attribute -particle‖ for separable verbs. SylLex formalism models the notion of word variants (as spelling and geographical variants) and compound words. See [START_REF] Blancafort | Morphosyntactic Lexica in the OAL Framework: Towards a Formalism to Handle Spelling Variants, Compounds and Multi-words[END_REF]) for more details. For now, the morphosyntactic lexica are implemented as a set of text files but a migration to a database scheme is planned. Using a database will facilitate the integration of phonetic features and semantic lexica.

Lemma and paradigm view

The user can navigate the lexicon in two ways: lemmabased or inflection-paradigm-based navigation. As shown in figure 2, he/she can visualize all the lemmas in the left panel (a simple research option is proposed) and for each lemma, the system shows in the right panel all the inflection paradigms applied to that lemma. Visualization and maintenance are possible at the same time. Thus, as the user visualizes the inflection paradigm VREG/e applied to the lemma amaze, he/she can modify both lemma and inflection paradigm (upper buttons offer several functionalities).

As shown in figure 3, the paradigm view shows all the inflection paradigms on the left panel and, on the right panel, for each paradigm the resulting forms that are generated when applying this paradigm to a lemma prototype are visualized.

Both views are coordinated. As a result, if the user double clicks over a lemma in the lemma view, the system switches the view and shows the relevant paradigm for the lemma. The interaction done the other way round is more powerful: if the user double clicks over an inflection paradigm name, the editor switches to the lemma view and shows, by automatically filtering (see section 5.3), all the lemmas that have the paradigm applied.

Adding new words: Import

Adding new lemmas to the lexicon can be carried out by the import function. The linguist imports a text file with a list of lemmas corresponding to the same inflection paradigm, he/she selects the corresponding paradigm out of a list of paradigms that can be filtered by word category (noun, adjective, verb, adverb or specific function word). A pop-up window shows an alert if one or more of the words are already in the lexicon and shows the word and the paradigm assigned in the lexicon. If any conflict appears, the user can undo the import of the lemmas by removing them.

Finding words by criteria: Advanced filtering

As lexicons are usually big and continue to grow, searching and filtering are essential functions. The basic search functionalities are not enough powerful when the user wants to access all the forms that match several conditions (e.g. all the noun forms generated from a lemma that finishes by -ness‖ whose plural is generated from the inflection paradigm N-es). This is why the editor offers an advanced filtering tool. The user can define conditions over:

-the lemma type (simple, compound, both) -the lemma and form string (regex or literal) -the complete tags (regex allowed) -the variants type -the inflection paradigms: number of paradigms applied to a lemma, set of paradigms applied to a lemma (at least one, all, exactly all).

Lexicon formatting: Export

The aim of the editor is to enhance the maintenance and handling of the lexicon. To export the whole lexicon, there is a functionality to compile the lexicon in different formats (cf. table 2). The export format depends on the application that will use the lexicon. The exported lexicon does not necessarily contain all the forms. Indeed, the user can use the advanced filtering (see previous section) to decide which forms are to be exported. For example, table 2 shows three different formats: (1) form-lemma-tag format with a single couple lemma-tag per form, (2) formlemma-tag format with one or more lemma-tag couple and (3) lemma type (Simple or Compound)-form-tagflexion paradigm. The [S] stands for simple lemma as opposed to a compound word.

Until now we have not needed a XML format, or to follow standards such as ISO TC37/SC4. However, in case of need, new output formats are easy to integrate.

Quality check

A further functionality checks the consistency of the paradigms. Thus, when the user clicks the ladybug button (cf. figure 5), the editor shows all the lemmas that have a consistency problem when a paradigm is applied. This is especially useful after a massive import or an automatic integration of guessed items.

Another useful functionality is the search on the internet of all the forms generated for a lemma. The user can access, for each form, to the hits count and to the matches (showed as a KWIC). This is particularly useful for neologisms, compound words and low frequent terms.

Conclusions and further work

In this paper we have presented the NLP architecture OAL, which helps the linguist to develop linguistic resources in an industrial context. We saw that OAL provides different functionalities to assist the linguist during the whole process of lexicon enrichment: a relevant corpus may be crawled from the Web, a linguistic pipeline processes corpora and includes a guessing tool to help the linguist to add new words by suggesting possible inflection paradigms for each new lemma for simple nouns, adjectives and verbs. This userfriendly environment improves the development of linguistic resources and assures its quality.

The resources definitions and tools are proprietary, as they have been designed and developed for internal use at Syllabs. The development of OAL has followed an incremental strategy. At present, semantic lexica, a named entities guesser and a named entities phonetizer are being developed. These new functionalities imply the enrichment of the SylLex formalism and the migration of the lexicons to an unified database scheme (for morphosyntactic and semantic lexicons). Further work will concentrate on how to integrate aligned multilingual lexica to the lexicon editor.

  Different export format examples.

Figure 4 :

 4 Figure 4: Lexicon filter.

Figure 5 :

 5 Figure 5: Consistency of the inflection paradigms.

Annexe 1: Lexicon editor figures


E H El Abdalaoui

Jean-Paul Thouvenot

GENERALIZED RIESZ PRODUCTS ON THE BOHR COMPACTIFICATION OF R p˚q

Keywords: . 2012. hal-00703137v4 AMS Subject Classifications (2010) Primary: 42A05, 42A55; Secondary: 11L03, 42A61 Generalized Riesz products, almost periodic functions, Bohr compactification, mean value, Besicovitch space, Kac Central Limit Theorem, Kakutani criterion

come

Generalized Riesz Products on the Bohr compactification of

El Houcein El Abdalaoui

Introduction

We stress that the main purpose of this paper is to extend and complete the study of the notion of generalized Riesz product associated to the rank one flows on R formulated in the same manner as Peyriére in [START_REF] El Abdalaoui | Spectral disjointness in some class of rank one flows[END_REF]. The authors in [START_REF] El Abdalaoui | Spectral disjointness in some class of rank one flows[END_REF] mentioned that Peyriére extended the notion of Riesz product to the real line using a class of kernel functions. Furthermore, it is noted by Peyriére in his pioneer paper [START_REF] Peyrière | Étude de quelques propriétés des produits de Riesz[END_REF] that an alternative extension of the classical Riesz products can be done using the Bohr compactification of R.

Indeed, it is usual that the extension of some notions from the periodic setting to the almost periodic deals with the Bohr compactification bR of R (or more generally, the Bohr compactification of Local Abelian Groups). The Bohr compactification plays in the almost periodic case the same role played by the torus T def " tt P C, |t| " 1u in the periodic case as the domain of the fast scale variables. As opposed to the torus, the Bohr compactification is often a non-separable compact topological space and this lack of separability is a source of difficulties in trying to adapt the arguments from the periodic context to the almost periodic one. Peyriére [START_REF] Peyrière | Étude de quelques propriétés des produits de Riesz[END_REF] mentioned this difficulty and introduced the Riesz products on R associated to some class of kernels.

Notice that here we use the notion of separability used in Hirotada-Kakutani paper [START_REF] Hirotada | Bohr compactifications of a locally compact Abelian group[END_REF], That is, the topological group (or space) is called separable if it satisfies the second countability axiom of Hausdorff which means that it has a countable basis. In that paper, Hirotada and Kakutani established that the Bohr compactification of a given locally compact Abelian group has a countable basis if and only if the union of the spectrum of all almost periodic functions is countable.

Our analysis here is also motivated by the recent growing interest in the problem of the flat polynomials suggested by A.A. Prikhod'ko [START_REF] Prikhod'ko | On flat trigonometric sums and ergodic flow with simple Lebesgue spectrum[END_REF] in the context of R. It turns out that the main idea developed in [START_REF] Prikhod'ko | On flat trigonometric sums and ergodic flow with simple Lebesgue spectrum[END_REF] does not seem well adapted to the context of our extension of generalized Riesz products to the Bohr compactification of R. This is due to the fact that the sequence of trigonometric polynomials constructed by A.A. Prikhod'ko is only locally L 1 -flat and not L 1 -flat in the usual sense (see, for instance, Remark 6.11).

The paper is organized as follows. In section 2 we review some standard facts on the almost periodic functions including the ergodicity of the action of R by translations on its Bohr compactification. In section 3 we define the notion of the generalized Riesz products on the Bohr compactification of R.

In section 4 we summarize and extend the relevant material on the Kakutani criterion and the Bourgain criterion on the singularity of the generalized Riesz products introduced in section 3. In section 5 we state and prove the Central Limit Theorem due to M. Kac. In section 6 we apply the Central Limit Theorem of Kac to prove our main result concerning the singularity of a large class of generalized Riesz products on bR. Finally, in the appendix, we consider the problem of the flat polynomials on the Bohr compactification of R and we add a short note based on Hirodata-Kakutani paper [START_REF] Hirotada | Bohr compactifications of a locally compact Abelian group[END_REF] on the Bohr compactification compared with the Stone-Čech compactification.

The Bohr compactification of R

The Bohr compactification of R is based on the theory of almost periodic functions initiated by H. Bohr [START_REF] Bohr | Almost Periodic Functions[END_REF] in connection with the celebrated ζ-function of Riemann. In this section we are going to recall the basic ingredients of this theory. For the basic facts about almost periodic functions and generalizations of this concept the reader is referred to the classical presentation of Bohr [START_REF] Bohr | Almost Periodic Functions[END_REF] and Besicovitch [START_REF] Besicovitch | Almost Periodic Functions[END_REF].

We point out that the theory of almost periodic functions can be extended to more general setting with applications in many context including the nonlinear differential equations [START_REF] Corduneanu | Almost periodic functions, With the collaboration of N[END_REF]. Definition 2.1. Let f : R ÝÑ C be a bounded continuous function and ε ą 0; we say that τ P R is an ε-almost period for f if sup xPR |f px `τ q ´f pxq| def " ||f p. `τ q ´f p.q|| 8 ă ε.

The mapping f is said to be almost periodic if for any ε ą 0 the set of ε-almost periods of f is relatively dense, i.e., there is l " lpεq ą 0 such that any interval with length l contains at least one ε-almost period.

The space of all almost periodic functions is denoted by AP pRq. From the above definition we easily deduce that AP pRq is a subspace of the space of bounded continuous functions on R. An important characterization of almost periodic functions is due to Bochner and it can be stated as follows Theorem 2.2 (Bochner's characterization of AP pRq). A bounded function f is almost periodic function if, and only if, the family of translates tf p. tqu tPR is relatively compact in the space bounded continuous functions on R endowed with the sup-norm topology.

The proof of Theorem 2.2 can be found in [START_REF] Bohr | Almost Periodic Functions[END_REF], [START_REF] Besicovitch | Almost Periodic Functions[END_REF] or [START_REF] Dunford | Linear Operators. I and II[END_REF]. Furthermore, we have the following fundamental theorem (see for instance [START_REF] Bohr | Almost Periodic Functions[END_REF] or [START_REF] Besicovitch | Almost Periodic Functions[END_REF]).

Theorem 2.3 (Bohr). A bounded continuous function f is almost periodic function if, and only if, f is uniformly approximated by finite linear combinations of functions in the set tcosptxq, sinptxqu tPR .

The space of all continuous functions on bR is denoted by CpbRq. CpbRq is a commutative C ‹ -algebra under pointwise multiplication and addition. Below, we give an important topological characterization of the Bohr compactification of R due to Gelfand, Raikov and Chilov. They obtain this characterization as an application of their theory of commutative Banach algebras.

Theorem 2.4 (Gelfand-Raikov-Chilov [START_REF] Gelfand | Commutative normed rings[END_REF]). The group R, equipped with the usual addition operation, may be embedded as a dense subgroup of a compact Abelian group bR in such way as to make AP pRq the family of all restrictions functions f |R to R of functions f P CpbRq. The operator f Þ ÝÑ f |R is an isometric ‹-isomorphism of CpbRq onto AP pRq. Moreover, the addition operation `: R ˆR ÝÑ R extends uniquely to the continuous group operation of bR, `: bR ˆbR ÝÑ bR. The group bR is called the Bohr compactification of bR.

For simplicity of notation, for any f in AP pRq, we use the same letter f for its canonical extension to bR. As a consequence of Theorem 2.4 combined with the Riesz representation Theorem we have Theorem 2.5 ([18]). The dual of the space AP pRq is isometrically isomorphic to the space MpbRq of all Radon measures on the Bohr compactification of R. The isomorphism x ˚Þ ÝÑ µ x ˚is given by the formula

x ˚pf q " ż bR f ptqdµ x ˚ptq.
We recall in the following the definition of the characters. The characters play a important role in the Abelian group and, by Koopmann observation, in the spectral analysis of dynamical systems. Definition 2.6. Let G be an Abelian group and e its identity element, then a character of G is a complex valued function χ defined on G such that χpeq " 1 and χpstq " χpsqχptq for all s, t P G.

Let us recall the following basic fact on compact Abelian groups due to Peter and Weyl.

Theorem 2.7 (Peter-Weyl). Let G be a compact Abelian group, with BpGq its Borel field and h its Haar measure. Then the set of continuous characters is fundamental both in CpGq and in L 2 pG, BpGq, hq.

The proof of Peter-Weyl Theorem can be found in [START_REF] Rudin | Fourier analysis on groups[END_REF] and for the thorough treatment we refer the reader to [START_REF] Rudin | Fourier analysis on groups[END_REF].

It is obvious that the continuous characters of bR are the functions e iω : bR ÝÑ T. In addition the orthogonality of two distinct characters can be checked directly. Indeed, ż bR e iωt .e iω 1 t dhptq " lim

T ÝÑ`8 1 2T
ż T ´T e ipω´ω 1 qt dt " 0, whenever ω ‰ ω 1 .

For f P AP pRq we denote by $ R f ptqdt the asymptotic mean value of f , given by

$ R f ptqdt " lim T ÝÑ`8 1 2T ż T ´T f ptqdt.
As a consequence of the averaging properties of almost periodic functions we have the following Lemma 1. For any f P AP pRq we have

ż bR f dhptq " $ R f ptqdt,
where dhptq is the Haar measure in bR, normalized to be a probability measure, and dt is the usual Lebesgue measure in R. Moreover we have

ż bR f ptqdhptq " lim T ÝÑ`8 1 T |K| ż T.K f ptqdt,
where K is any bounded subset of R with |K| ‰ 0, |K| is the Lebesgue measure of K.

Following [START_REF] Katznelson | An introduction to harmonic analysis[END_REF], for any f P AP pRq, we introduce the notation

p f `tλu ˘" $ R f ptqe ´iλt dt.
That is, `p f `tλu ˘˘λPR are the Fourier coefficients of f relative to orthonormal family te iλt u λPR ; the inner product is defined by

ă f, g ą"ă f, g ą" ż bR f pκqgpκqdhpκq.
2.1. On Besicovitch space. Since the functions f P AP pRq correspond to restrictions, f " r f |R , of continuous functions f on bR, a natural question is whether it is possible to define a class of functions f which correspond to "restrictions" f " r f |R , of functions r f P L 1 pbRq. This motivates the following definition.

Definition 2.8. Given p P r1, `8q the space BAP p pRq, of Besicovitch's generalized almost periodic functions on R, consists of those functions f P L p loc pRq for which there exists a sequence f n P AP pRq satisfying lim nÝÑ`8 lim sup

T ÝÑ`8 1 2T
ż T

´T

ˇˇf n pxq ´f pxq ˇˇpdx " 0 p1q

We denote BAP p pRq simply by B p pRq.

The space of generalized almost periodic functions B p pRq was introduced by Besicovitch, who also gave them a structural characterization. We refer to [START_REF] Besicovitch | Almost Periodic Functions[END_REF] for more details about functions in B p pRq. We immediately have AP pRq Ă B p pRq Ă B 1 pRq for any p ě 1 and it is easy to see that any f P B 1 pRq has the mean value property, that is, for any bounded measurable B Ă R, with |B| ‰ 0, we have

M pf q " lim LÝÑ`8 1 L|B| ż B L f dx,
where B L " tx P R : x L P Bu. The space corresponds to L p pbRq in a way similar to the one in which the space AP pRq corresponds to CpbRq. Indeed, notice first that the definition of B p pRq immediately gives that the asymptotic mean value $ ˇˇf ˇˇp dx of a function in B p pRq is well defined; moreover, any approximating sequence f n P AP pRq satisfying p1q can be viewed as a Cauchy sequence in L p pbRq and, hence, there exists r f P L p pbRq such that Ă f n converge to f in L p pbRq. Since r f is easily seen to be independent of the approximating sequence, in this way we may associate with each f P B p pRq a well determined function r f P L p pbRq which we may view as an "extension" of f to bR. Notice that the map f Þ Ñ r f is a linear map and that the approximation procedure together with Lemma 1 show that $ R ˇˇf ˇˇp " ż bR ˇˇr f pκq ˇˇpdκ, @f P B p pRq. p2q

As a consequence, the kernel of the map f Þ ÝÑ r f is made by the functions f such that the asymptotic mean value of ˇˇf ˇˇp is 0. The corresponding quotient space is denoted by B p pRq{ ". For f P B p pRq we set

|f | p def " ¨$ R ˇˇf ˇˇpdx ‹ ‹ ' 1 p
, so that ˇˇˇˇp is a semi-norm on B p pRq. It is well known that B p pRq is complete with respect to the semi-norm ˇˇˇˇp (see for instance [START_REF] Morsli | On modular approximation property in the Besicovitch-Orlicz space of almost periodic functions[END_REF]). The space B 2 pRq is endowed with the scalar product

ă f, g ą" $ R f pxqgpxqdx " ż bR r f pκqr gpκqdκ, p3q
The second equality follows by p2q with p " 2, implying that the scalar product is preserved under the map f Þ Ñ r f . Finally, we define

B 8 pRq def " # f P č pě1 B p pRq : sup pě1 ˇˇf ˇˇp ă `8+ , Again, ˇˇˇˇ8
is a semi-norm on B 8 pRq and the corresponding quotient space is denoted by B 8 pRq{ ". We finish this paragraph by stating the following important fact on the properties of the map f Þ Ñ r f . The proof of it is left to the reader.

Proposition 2.9. The map f Þ Ñ r f is an isometric isomorphism between the Banach spaces B p pRq{ " and L p pbRq for any p P r1, `8s Furthermore, we point out that a suitable extension of Lebesgue and Fatou's convergence results is obtained in [START_REF] Morsli | On modular approximation property in the Besicovitch-Orlicz space of almost periodic functions[END_REF].

Gelfand-Raikov-Chilov Theorem 2.4 allows us to define the Bohr compactification of R. But, in the Harmonic Analysis, it is well known that the Pontrygain Theorem gives an alternative definition. We briefly recall it here and we refer the reader to the large literature on the subject [START_REF] Rudin | Fourier analysis on groups[END_REF] and for a deeper discussion on the relation between the almost periodic functions and the Bohr compactification of R to [START_REF] Khavin | Commutative harmonic analysis. I. General survey. Classical aspects[END_REF], [START_REF] Grigoryan | Shift-invariant uniform algebras on groups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne[END_REF].

Let G be locally compact Abelian group and let p G be its dual, i.e., p G is the set of the characters endowed with the topology inherited from G. Let p G d be p G with the discrete topology. Then

x x G d def " bG is the Bohr compactification of G. bG is a compact group such that G is a dense subset of bG.
We end this section by stating and proving the classical result on the ergodicity of the action of R by translation on bR. We recall that the action of R by translation is defined by τ x pκq " κ `x where the extended addition is given by Theorem 2.4. Clearly, the family pτ x q xPR is a flow acting on bR since τ x pτ x 1 pκqq " τ x`x 1 pκq and the Haar measure is invariant under translation.

Theorem 2.10. The action of R on bR is ergodic, that is, for any Borel set A Ă bR which is invariant under the translation action we have hpAq P t0, 1u, where hpAq denotes the normalized Haar measure of A. Moreover hpRq " 0.

Proof. Let A Ă bR be an invariant Borel set. We have

hpAq " hpA X τ x Aq " ż bR 1 1 A pβq 1 1 A pβ `xqdhpβq.
Now, translations are strongly continuous on L 2 pRq. Indeed, this is a standard consequence of the density of CpbRq in L 2 pbRq, which follows from Theorem 2.4, and the invariance of the Haar measure. Therefore, the righthand side is a continuous function of x, and so the identity still holds with x P bR. Hence we get, using Fubini theorem and the invariance of the Haar measure,

hpAq " ż bR ż bR 1 1 A pκq 1 1 A pκ `ξqdhpκqdhpξq " ż bR 1 1 A pκqdhpκq ż bR 1 1 A pξqdhpξq " hpAq 2 ,
from which it follows that hpAq P t0, 1u, as asserted. It remains to show that hpRq " 0. First we observe that R is a Borel subset of bR, since it is the union of a countable family of compact sets, e.g., the images of the intervals r´k, ks, k P N. Since R is invariant under the translation action we have hpRq P t0, 1u. But, for any κ P bRzR, κ`R is also an invariant Borel set and R Ş tκ `Ru " H. By the invariance of the Haar measure hpκ `Rq " hpRq. Hence, we conclude that hpRq " 0 and the proof is achieved.

Generalized Riesz Products on bR

Riesz products were discovered in 1918 by F. Riesz [START_REF] Riesz | Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung[END_REF] to answer affirmatively a special question in the theory of Fourier series, namely, whether there exists a continuous measure whose Fourier coefficients do not vanish at infinity. Roughly speaking, the Riesz products are a kind of measures on the circle constructed inductively. The pioneer Riesz product construction gives a concrete example. Since then, the Riesz construction proved to be the source of powerful ideas that can be used to produce concrete counterexample of measures with a number of desired properties (controllability of the convergence of the Fourier coefficients being the goal of the original construction).

Later, A. Zygmund extended Riesz construction and introduced what it is nowadays called classical Riesz products [53, p.208].

In 1975, that Riesz products appear as a spectral type of some dynamical systems was shown by F. Ledrappier [START_REF] Ledrappier | Des produits de Riesz comme mesures spectrales[END_REF]. Ten years later, M. Queffelec [START_REF] Queffelec | Mesures spectrales associées à certaines suites arithmétiques[END_REF], inspired by the work of Coquet-Kamae and Mandes-France [START_REF] Coquet | Mendés France, Sur la mesure spectrale de certaines suites arithmétiques[END_REF], showed that the specific generalized Riesz products are the right tool to describe the spectrum of the class of dynamical systems arising from the substitution (see [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF] and the references therein). In 1991, B. Host, J.-F. Méla, F. Parreau in [START_REF] Host | Nonsingular transformations and spectral analysis of measures[END_REF] realized a large class of Riesz products as the maximal spectral type of the unitary operator associated with a non-singular dynamical system and a cocycle. Finally, in the more general setting, J. Bourgain established the connections between some class of generalized Riesz products on the circle and the maximal spectral type of a class of maps called rank one maps [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF]. One year later, an alternative proof is given by Choksi-Nadkarni using the Host-Méla-Parreau argument [START_REF] Choksi | The maximal spectral type of rank one transformation[END_REF], [START_REF] Nadkarni | Spectral theory of dynamical systems[END_REF] and at the same time a simple proof is obtained by Klemes-Reinhold [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF] using the standard Fourier analysis argument.

In [START_REF] El Abdalaoui | Spectral disjointness in some class of rank one flows[END_REF], el Abdalaoui-Lemańczyk-Lesigne and Ulcigrai proved that the generalized Riesz products analogous to Peyriére-Riesz products can be realized as a spectral type of some class of rank one flows.

Here our aim is to extend the notion of generalized Riesz products to the Bohr compactification of R. It turns out that such generalization can be done directly. More precisely, in the study of the spectrum of some class of rank one flows, the following trigonometric polynomials on R appears

P k pθq " 1 ? p k ÿ p k ´1
j"0 e iθpjh k `q s j q , p4q where q s 0 " 0 and q s j " j´1 ÿ l"0 s k,l , j ě 1, with pp k q kPN is a sequence of positive integers greater than 1 and ps k,j q j"0,¨¨¨,p k is a sequence of positive real numbers with s k,0 " 0 for any k P N; the sequence ph k q is defined inductively by

h 0 " 1 and h k`1 " p k h k `pk ÿ l"0 s k,l . p5q
For simplicity, for any p, q P t0, ¨¨¨, p k ´1u we introduce the following sequence of real numbers q s n,p,q " maxpp,qq´1 ÿ j"minpp,qq s n,j , p6q and for any real number t, we put eptq " e it . Theorem 3.1 (Generalized Riesz Products on bR). Let pP n q nPN be a family of trigonometric polynomials given by p4q. Then the weak limit of the sequence of probability measures on bR

n ź k"0 ˇˇˇP k pθq ˇˇˇ2
$ R Q n dt " 1.
Therefore, σ n is a probability measure on bR. In addition, for any t P R, we have

Q n`1 ptq " Q n ptqP n ptq " Q n ptq `Qn ptq∆ n ptq Hence z σ n`1 `λ˘" z Q n`1 `tλuq " x Q n `tλuq `x Q n ˚x ∆ n pλq ě x Q n `tλuq " x σ n `λC
onsequently the limit r λ of the sequence px σ n `λ˘q exists. Now, since bR is compact and pσ n q is a sequence of probability measure on bR we can extract a subsequence pσ n k q which converge weakly to some probability measure on bR. We deduce that the limit of pσ n q exists in the weak topology and this finishes the proof.

The proof above is largely inspired by lemma 2.1 in [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF]; it gives more, namely, the polynomials P n can be chosen with positive coefficients and satisfying

$ ˇˇˇP n ptq ˇˇˇ2 dt " 1 and $ n ź j"1 ˇˇˇP j ptq ˇˇˇ2 dt " 1.
We mention also that we have

$ k ź j"1 ˇˇˇP n j ptq ˇˇˇ2 dt " 1, p7q
for any given sequence of positive integers n 1 ă n 2 ă ¨¨¨ă n k , k P N ˚.

We can now formulate our main result whose proof occupies all Section 6. Theorem 3.2 (Main result). Let pp m q mPN be a sequence of positive integers greater than 1 and pps m,j q pm´1 j"0 q mPN be a sequence of positive real numbers. Assume that there exists a sequence of positive integers m 1 ă m 2 ă ¨¨¨, such that for any positive integer j, the numbers ph m j , s m j ,0 , ¨¨¨, s m j ,pm j ´1q are rationally independent. Then the generalized Riesz product

µ " n ź k"0 ˇˇP k pθq ˇˇ2 dhpθq,
where

P k pθq " 1 ? p k ÿ p k ´1 j"0 e `iθ `jh k `q s k,0,j ˘˘.
is singular with respect to the Haar measure on bR.

4. On the Kakutani criterion and the Bourgain singularity criterion of the Riesz products on the Bohr compactification of R

The famous dichotomy theorem of Kakutani has a rather long history. In his 1948 celebrated paper [START_REF] Kakutani | On equivalence of infinite product measures[END_REF], Kakutani established a purity law for infinite product measures. Precisely, if P "

`8 â i"1 P i and Q " `8 â i"1
Q i are a infinite product measures, where P i , Q i are probability measures such that P i is absolutely continuous with respect to Q i , for each positive integer i, then

P ! Q or P K Q according as ź i ż ˆdP i dQ i ˙dQ i converges or diverges.
There are a several proofs of Kakutani criterion in literature (see [START_REF] Brown | Products of random variables and Kakutani's criterion for orthogonality of product measures[END_REF] and the references given there). For a proof based on the Hellinger integral we refer the reader to [17, p.60].

Kakutani's Theorem was specialized to the Gaussian measures on Hilbert space with identical correlation operators in [START_REF] Grenander | Stochastic processes and statistical inference[END_REF] and it was extended to Gaussian measures with non-identical correlation operators by Segal [START_REF] Segal | Distributions in Hilbert space and canonical systems of operators[END_REF], Hajek [START_REF] Hajek | On a property of normal distribution of any stochastic process[END_REF], Feldman [START_REF] Feldman | Equivalence and perpendicularity of Gaussian processes[END_REF] and Rozanov [START_REF] Rozanov | Certain remarks to the work "On the densities of Gaussian distributions an the Wiener-Hopf integral equations[END_REF]. Later, in 1979, Ritter in [START_REF] Ritter | On Kakutani's dichotomy theorem for infinite products of not necessarily independent functions[END_REF], [START_REF] Ritter | On dichotomy of Riesz products[END_REF] generalized Kakutani's Theorem to a certain non-products measures with application to the classical Riesz products.

Here, applying the Bourgain methods, we obtain a new extension of Kakutani's Theorem to the class of generalized Riesz products on the Bohr compactification of R. Indeed, we show that the independence along subsequence suffices to prove the singularity.

Nevertheless, our strategy is similar to the strategy of [START_REF] El Abdalaoui | Spectral disjointness in some class of rank one flows[END_REF] and it is based on the extension of Bourgain methods to the generalized Riesz products on the Bohr compactification of R combined with the Central Limit tools introduced in [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF].

Moreover, having in mind applications beyond the context of this paper, we shall state and prove a Guenais sufficient condition on the L 1 flatness of the polynomials which implies the existence of generalized Riesz products on bR with Haar component. We recall that the generalized Riesz products µ is given by µ "

`8 ź k"0 ˇˇP k pθq ˇˇ2 dhpθq
where

P k ptq " 1 ? p k ÿ p k ´1 j"0 e `iθpjh k `q s k,j q ˘and q s k,j " j´1 ÿ l"0 s k,l
Theorem 4.1 (bR version of Bourgain criterion). The following are equivalent (i) µ is singular with respect to Haar measure.

(ii) inf

$ & % ż bR L ź ℓ"1 ˇˇP n ℓ ˇˇdh : L P N, n 1 ă n 2 ă . . . ă n L , .
-" 0.

The proof of bR version of Bourgain criterion is based on the following lemma.

Lemma 4.2. The following are equivalent

ż bR N ź k"0 ˇˇP k ˇˇdh ´´´´Ñ N Ñ`8 (1)
(2) inf

$ & % ż bR L ź ℓ"1 |P n ℓ | dh : L P N, n 1 ă n 2 ă . . . ă n L , .
-" 0.

Proof. The proof is a simple application of Cauchy-Schwarz inequality. Consider n 1 ă n 2 ă . . . ă n L and N ě n L . Denote N " tn 1 ă n 2 ă . . . ă n L u and N c its complement in t1, ¨¨¨, N u. Let a ă b be two real numbers and define a probability measure on R by dλ a,b ptq " 1 1 ra,bs ptq b ´a dt, where dt is the Lebesgue measure. Then we have

ż N ź k"0 |P k | dλ a,b " ż ź kPN |P k | 1 2 ˆź kPN c |P k | 1 2 N ź k"0 |P k | 1 2 dλ a,b ď ¨ż ź kPN |P k | dλ a,b '1 2 ¨ż ź kPN c |P k | ˆN ź k"0 |P k | dλ a,b '1 2 ď ¨ż ź kPN |P k | dλ a,b '1 2 ¨ż ź kPN c |P k | 2 dλ a,b '1 4 ¨ż N ź k"0 |P k | 2 dλ a,b

'1

By letting b ´a goes to infinity, we get

$ N ź k"0 |P k | dt ď ¨$ ź kPN |P k | dt '1 2 ¨$ ź kPN c |P k | 2 dt '1 4 ¨$ N ź k"0 |P k | 2 dt '1 4 " ¨$ ź kPN |P k | dt '1 2 .
The last equality follows from p7q.

Proof of Theorem 4.1. Assume that (i) holds. To prove that µ is singular , it suffices to show that for any ǫ ą 0, there is a Borel set E with hpEq ă ǫ and µpE c q ă ǫ. Let 0 ă ǫ ă 1.

Fix N 0 such that for any N ą N 0 , we have

ż bR N ź k"0 |P k | dh ă ǫ 2 . The set E " ! ω P bR : ś N k"0 |P k pωq| ě ǫ ) satisfies: hpEq ď 1 ǫ › › › › › N ź k"0 P k › › › › › 1 ď ǫ 2 {ǫ " ǫ,
and since E c is open set, it follows from the Portmanteau Theorem that we have

µpE c q ď lim inf M Ñ`8 ż E c M ź k"0 |P k | 2 dh ď lim inf M Ñ`8 ż E c N ź k"0 |P k | 2 M ź k"N `1 |P k | 2 dh ď ǫ 2 lim M Ñ`8 ż bR M ź k"N `1 |P k | 2 dh " ǫ 2 ă ǫ.
For the converse, given 0 ă ǫ ă 1, there exists a continuous function ϕ on bR such that:

0 ď ϕ ď 1, µptϕ ‰ 0uq ď ǫ and hptϕ ‰ 1uq ď ǫ. Let f N " N ź k"1 |P k |.
By Cauchy-Schwarz inequality, we have

ż f N dh " ż tϕ‰1u f N dh `żtϕ"1u f N dh ď hptϕ ‰ 1uq 1{2 ˆżbR f 2 N dh ˙1{2 `˜ż tϕ"1u f 2 N dh ¸1{2 hptϕ " 1uq 1{2 ď ? ǫ `ˆż bR f 2 N ϕ dh ˙1{2 .
Since µ is the weak limit of f 2 N dh, we have lim

N Ñ8 ż bR f 2 N ϕ dh " ż bR ϕ dµ ď µptϕ ‰ 0uq ď ǫ.
Thus, lim sup

ż bR f N dh ď 2 ? ǫ.
Since ǫ is arbitrary, we get lim

N Ñ8 ż bR f N dh "
0, and this completes the proof.

From now on, let M be a sequence of positive integers for which ph m , ps m,j q pm´1 j"0 q are linearly independent over the rationals and let us fix some subsequence N " tn 1 ă n 2 ă . . . ă n k u of M , k P N and m P M with m ą n k . Put

Q ptq " k ź i"1 ˇˇˇP n i ptq ˇˇˇ.
We define the degree of any trigonometric polynomial f by degpf q " maxt|ξ| : p f ptξuq ‰ 0u, p8q and we denote d m def " degpP m q. From equations p5q and p6q, we have

d m " h m`1 ´hm ´sm,pm ă h m`1 , p9q h m ď h m`1 {p m ď h m`1 {2. p10q
Since n j `1 ď n j`1 , telescoping we get

q k def " degpQ k q " d n 1 `dn 2 `¨¨¨`d n k ď ph n 1 `1 ´hn 1 q `ph n 2 `1 ´hn 2 q `¨¨¨`ph n k `1 ´hn k q p11q ă h n k `1.
In the same spirit as above it is easy to see the following lemma. The proof of it in the case of the torus is given in [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF] and [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF]. Lemma 2. With the above notations we have

ż bR Q ˇˇP m ˇˇdh ď 1 2 ˆżbR Qdh `żbR Q ˇˇP m ˇˇ2 dhpωq ˙´1 8 ˆżbR Q ˇˇˇˇP m ˇˇ2 ´1ˇˇd h ˙2.
The following proposition is a simple extension of Proposition 2.4 in [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF].

ż bR Q ˇˇˇˇP m ˇˇ2 ´1ˇˇd h ˙2.
Now, in the following lemma, we state a sufficient condition for the existence of an absolutely continuous component with respect to the Haar measure for the given generalized Riesz product . In the case of Z action, the lemma is due to Mélanie Guenais [START_REF] Guenais | Étude spectrale de certains produits gauches en théorie ergodique[END_REF], and the proof is similar. Proof. We denote by } ¨}p the norm in L p phq. For all functions P and Q in L 2 phq, by Cauchy-Schwarz inequality we have

p12q }P } 1 }Q} 1 ´}P Q} 1 " ´ż p|P | ´}P } 1 q p|Q| ´}Q} 1 q dh ď }|P | ´}P } 1 } 2 }|Q| ´}Q} 1 } 2 ,
and by assumption,

p13q `8 ÿ k"1 b 1 ´}P k } 2 1 ă 8, hence `8 ÿ k"1
1 ´}P k } 2 1 ă 8 and the infinite product

ź k }P k } 1 is convergent: p14q `8 ź k"1 }P k } 1 ą 0.
Let n 0 ď n be a positive integers and take P "

P n and Q " n´1 ź k"n 0 P k , then }|P | ´}P } 1 } 2 " a 1 ´}P } 2 1 and }|Q| ´}Q} 1 } 2 ď 1 ;
hence by p12q we have

}P Q} 1 ě }P } 1 }Q} 1 ´b1 ´}P } 2 1 .
Using the fact that }P k } 1 ď 1, we obtain by induction

› › › › › n ź k"n 0 P k › › › › › 1 ě n ź k"n 0 }P k } 1 ´n ÿ k"n 0 b 1 ´}P k } 2 1 ě n ź k"n 0 }P k } 1 ´`8 ÿ k"n 0 b 1 ´}P k } 2 1 .
From p13q combined with p14q we deduce that for large enough n 0

lim nÑ`8 n ź k"n 0 }P k } 1 ´`8 ÿ k"n 0 b 1 ´}P k } 2 1 ą 0,
hence the sequence `śn

´1

k"1 |P k | 2 has only countably many zeros, we conclude that µ admits also an absolutely continuous component with respect to the Haar measure.

On the Kac Central Limit Theorem

The Kac Central Limit Theorem in the context of the Bohr compactification of R is stated and proved in [START_REF] Kac | Statistical independence in probability, analysis and number theory[END_REF]. For sake of completeness we prove it here using the standard probability arguments.

Definition 5.1. The real numbers ω 1 , ω 2 , ¨¨¨, ω r are called rationally independent if they are linearly independent over Z , i.e. for all n 1 , ¨¨¨, n r P Z, n 1 ω 1 `¨¨¨`n r ω r " 0 ùñ n 1 " ¨¨¨" n r " 0. Theorem 5.2 (M. Kac [START_REF] Kac | Statistical independence in probability, analysis and number theory[END_REF]). Let ω 1 , ω 2 , ¨¨¨, ω n , ¨¨¨be rationally independent. Then, the functions cospω 1 tq, cospω 2 tq, ¨¨¨, cospω n tq, ¨¨¨are stochastically independent with respect to the Haar measure of the Bohr compactification of R.

Proof. It is sufficient to show that for any positive integer k and for a given positive integers l 1 , l 2 , ¨¨¨, l k , we have

$ R ˆcospω 1 tq ˙l1 ˆcospω 2 tq ˙l2 ¨¨¨ˆcospω k tq ˙lk dt " $ R ˆcospω 1 tq ˙l1 dt $ R ˆcospω 2 tq ˙lk dt ¨¨¨$ R ˆcospω k tq ˙lk dt Write cospω j tq " 1 2 ˆeiω j t `e´ω j t ˙, j " 1, 2, ¨¨¨, t P R,
and recall that $ R e iαt dt " lim

T ÝÑ`8 1 2T ż T ´T e iαt dt " # 1, if α " 0 0, if not .
Hence

ˆcospω 1 tq ˙l1 ¨¨¨ˆcospω k tq ˙lk " k ź j"1 1 2 l j k ź j"1 ˆeiω j t `e´iω j t ˙lj " k ź j"1 1 2 l j k ź j"1 ˆÿl j r j "0 ˆlj r j ˙eip2r j ´lj qω j t ˙. Whence k ź j"1 ˆcospω j tq ˙lj " k ź j"1 1 2 l j ˆÿl 1 ,l 2 ,¨¨¨,l k r 1 ,r 2 ,¨¨¨,r k "0 ˆl1 r 1 ˙ˆl 2 r 2 ˙¨¨¨ˆl k r k ˙eip ř k j"1 p2r j ´lj qω j qt ˙.
Because of linear independence, ÿ k j"1 p2r j ´lj qω j can be zero only if 2r i " l i , for any i " 1, ¨¨¨, k, and thus it follows that

$ R k ź j"1 ˆcospω j tq ˙lj dt " $ ' & ' % k ź j"1 1 2 l j ˆlj l j 2
˙, if all l i are even 0, otherwise.

It is well-known that if

sup nPN ˆE`ˇˇX n ˇˇ1`ε ˘˙ă `8, p15q
for some ε positive, then tX n u are uniformly integrable. Let us mention that the convergence in distribution or probability does not in general imply that the moments converge (even if they exist). The useful condition to ensure the convergence of the moments is the uniform integrability. Indeed, we have Now let us state and prove the Kac Central limit Theorem.

Theorem 5.6 (Kac CLT [START_REF] Kac | Statistical independence in probability, analysis and number theory[END_REF]). Let pλ n q nPN be a sequence of rationally independent real numbers. Then the functions cospλ n tq `i sinpλ n tq, n " 1, ¨¨¨, are stochastically independent under the Haar measure of the Bohr compactification of R and converge in distribution to the complex Gaussian measure N C p0, 1q on C.

Proof. By Theorem 5.2 the functions cospλ n tq `i sinpλ n tq, n " 1, ¨¨¨, are stochastically independent under the Haar measure of bR and it is straightforward to verify that the hypotheses of Central Limit Theorem 5.3 are satisfied. We conclude that the sequence ˜1 ?

n n ÿ k"1 e iλ k t
¸ně1 converges in distribution to the complex Gaussian measure N C p0, 1q on C.

Proof of the main result (Theorem 3.2)

Using the analogous lemma of Féjer's Lemma [53, p.49] combined with the CLT methods introduced in [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF], we shall give a direct proof of the singularity of a large class of generalized Riesz products on bR. Therefore, our strategy is slightly different from the strategy of the proofs given by many authors in the case of the torus [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], [START_REF] Klemes | The spectral type of staircase transformations[END_REF], [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF], [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF]. Indeed, the crucial argument in their proofs is to estimate the following quantity

ż Q ˇˇˇˇP m ˇˇ2 ´1ˇˇd h. p16q
Precisely, they showed that the weak limit point of the sequence `ˇ| P m | 2 ´1ˇˇȋ s bounded below by a positive constant and it is well-known that this implies the singularity of the generalized Riesz products (see for instance [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF] or [START_REF] Klemes | The spectral type of staircase transformations[END_REF]).

Let us start our proof by proving the following lemma analogous to Féjer's Lemma [53, p. Proof. By our assumption the sequence `hm , ps m,pm´1 q ˘mPM is rationally independent. Hence, by Kac Theorem 5.2, for m ą h n k`1 , the function Q and P m are stochastically independent. This allows us to write

ż bR Q ˇˇP m ˇˇdh " ż bR Qdh ż bR |P m |dh,
which proves the lemma.

Proof of Theorem 3.2. Applying Lemma 3, we proceed to construct inductively the sequence n 1 ă n 2 ă n k ă ¨¨¨, such that, for any k ě 1, we have

ż bR k`1 ź j"1 ˇˇP n j ˇˇdh ď ? 51π 100 ż bR k ź j"1
ˇˇP n j ˇˇdh. p17q Indeed, by our assumption combined with Kac CLT 5.2, it follows that pP m q converges in distribution to the complex Gaussian measure N C p0, 1q on C. But, according to p15q, pP m q is uniformly integrable. Hence, from Theorem 5.5, we get lim mÝÑ`8

ż bR |P m |dh " ż C ˇˇz ˇˇdN C p0, 1qpzq " ? π 2 . p18q
We remind that the density of the standard complex normal distribution N C p0, 1q is given by

f pzq " 1 π e ´|z| 2 .
Now assume that we have already construct which yields by Bourgain criterion that µ is singular with respect to Haar measure and completes the proof. Remark 6.1. The fundamental argument in the proof above is based on Lemma 3 and therefore strongly depended on the assumption that along subsequence the positive real numbers `hm , ps m,pm´1 q ˘are linearly independent over the rationals. We argue that in the general case, one may use the methods of Bourgain [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], Klemes-Reinhold [START_REF] Klemes | The spectral type of staircase transformations[END_REF], Klemes [START_REF] Klemes | The spectral type of staircase transformations[END_REF] and el Abdalaoui [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF] to establish the singularity of a large class of generalized Riesz products on bR. In particular, the case when pp m q is bounded. In the forthcoming paper, we will show how to extend a classical results from the torus and real line setting to the generalized Riesz products on the Bohr compactification of R.

Appendix.I. On The Flatness problem on bR.

We are concerned here with the flat polynomials issue in bR. First, we recall briefly the relevant fact on the flatness problem in the torus T.

The problem of flatness go back to Littlewood in his 1968 famous paper [START_REF] Littlewood | On polynomials n ÿ ˘zm , n ÿ e αmi z m , z " e θi[END_REF]. In that paper, Littlewood introduce two class of complex polynomials G n and F n where n is a positive integer. The class G n is a class of those polynomials P pzq " ř n k"0 a k z k that are unimodular, that is, |a k | " 1, for k " t0, ¨¨¨, nu. F n is the subclass of G n with real coefficients, i.e., a k " ˘1, for k " t0, ¨¨¨, nu. The polynomials P in F n are nowadays called Littlewood polynomials. By Parseval's formula 1 2π ż 2π 0 ˇˇˇP pe it q ˇˇˇ2 dt " n `1, for all P P G n .

Therefore, for all P P G n , min |z|"1 ˇˇP pzq ˇˇă ?

n `1 ă max |z|"1

ˇˇP pzq ˇǏn

[37] Littlewood raised the problem of the existence of a sequence pP n q of unimodular polynomials such that

max tPR ˇˇP n pe it q ˇ? n `1 ´´´´Ñ nÑ8 1.
Such sequence of unimodular polynomials are called ultraflat. Precisely, the usual definition of ultraflatness is given as follows Definition 6.2. Let pn k q be a sequence of positive integers and pε n k q a sequence of positive real numbers tending to 0, we say that a sequence pP n k q of unimodular polynomials is

pε n k q-ultraflat if max tPR ˇˇˇˇˇP n k pe it q ˇ? n k `1 ´1ˇˇˇˇď ε n k .
The problem of existence of the ultraflat polynomials was solved affirmatively by Kahane in his 1980 paper [START_REF] Kahane | Sur les polynômes à coefficients unimodulaires[END_REF]. Precisely, Kahane proved that there exists a sequence of pε n q-ultraflat with

ε n " O ˆn´1{17 a logpnq ˙.
But as noted by Queffelec and Saffari [START_REF] Queffelec | On Bernstein's inequality and Kahane's ultraflat polynomials[END_REF] Kahane proof is some kind of miracle. This is due to the fact that Kahane work is inspired by Körner paper [START_REF] Körner | On a polynomial of Byrnes[END_REF] and Körner paper is based on Byrnes paper [START_REF] Byrnes | On polynomials with coefficients of modulus one[END_REF]. But, in 1996 J. Benedetto and his student Hui-Chuan Wu discovered that Theorem 2 in Byrnes paper [START_REF] Byrnes | On polynomials with coefficients of modulus one[END_REF] was erroneous and as a consequence invalidated Körner main result. Fortunately Kahane proof was independent of Theorem 2 in Byrnes paper.

The problem of the existence of the ultraflat Littlewood polynomials is unsettled to this date and as pointed by Erdélyi [START_REF] Erdélyi | How far is an ultraflat sequence of unimodular polynomials from being conjugate-reciprocal?[END_REF] it is a common belief that there is no ultraflat sequence of Littlewood polynomials. As a consequence no long Barker sequences exist (see [START_REF] Borwein | Barker sequences and flat polynomials[END_REF] and the references given there).

One more important class of polynomials is the class of polynomials with coefficients a k P t0, 1u called a class of idempotent polynomials for obvious reasons of being convolution idempotents and denoted by

I n def " tP P CrXs : P pXq " n´1 ÿ 0 a k X k , a k P t0, 1uu, n P N ˚.
A subclass of I n with constant term 1 is called a class of Newman polynomials and denoted by N n .

Another extremal open problem in the class of Littewood polynomials or in the class of Newman polynomials is the problem of L 1 -flatness. In the same way as the ultraflatness, the L 1 -flatness is defined as follows Definition 6.3. Let pn k q be a sequence of positive integers and pε n k q a sequence of positive real numbers tending to 0, we say that a sequence pP n k q of polynomials is pε n k q ´L1 -flat if

› › P n k pe it q › › 1 › › P n k pe it q › › 2 ě 1 ´εn k .
The problem of the existence of L 1 -flat sequence of unimodular polynomials was solved by Newman in [START_REF] Newman | An L 1 extremal problem for polynomials[END_REF]. Later, M. Guenais inspired by Newman work constructed [START_REF] Guenais | Étude spectrale de certains produits gauches en théorie ergodique[END_REF] a sequence of L 1 -flat Littlewood polynomials on ś `8 n"0 Z{p n Z where pp n q ně0 is an increasing sequence of prime numbers. As a consequence she established a existence of countable group action with simple spectrum and Haar component. Subsequently, el Abdalaoui and Lemańczyk [START_REF] El Abdalaoui | Approximately transitive dynamical systems and simple spectrum[END_REF] proved that the sequence of Littlewood polynomials constructed by Guenais is ultraflat. Before, J. Bourgain in his 1993 paper [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF] conjectured that the supremum of the L 1 norm by L 2 norm over all idempotent polynomials on the circle must be strictly less than one. Precisely, he make the following conjecture Conjecture 6.4 (Bourgain). [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF] sup ně1 " sup

P PNn ˆ› › P pe it q › › 1 › › P pe it q › › 2 ˙* ă 1.
Using the Bourgain ideas, M. Guenais [START_REF] Guenais | Étude spectrale de certains produits gauches en théorie ergodique[END_REF] connected the problem of the existence of L 1 -flat sequence of Littlewood polynomials or Newman polynomials and the Banach problem on whether there exist a dynamical system with simple Lebesgue spectrum. Ulam in his book [52, p.76] stated the Banach problem in the following form Question 6.5 (Banach Problem). Does there exist a square integrable function f pxq and a measure preserving transformation T pxq, ´8 ă x ă 8, such that the sequence of functions tf pT n pxqq; n " 1, 2, 3, ¨¨¨u forms a complete orthogonal set in Hilbert space? Let us formulate the Bourgain conjecture in context of Bohr compactification of R. For that, let PpRq be the subspace of trigonometric polynomials on R and pω j q jPN be an increasing sequence of real numbers. Put J ω,n " tP P PpRq : P ptq " n ÿ k"0 a k e iω k t , a 0 " 1, a k P t0, 1uu Then, we can state the Bourgain conjecture in context of Bohr compactification of R in the following form Conjecture 6.6 (Bourgain in bR). for any increasing sequence pω j q jPN of real numbers, we have

sup ně1 " sup P PJω,n ˆ› › P pe it q › › 1 › › P pe it q › › 2 ˙* ă 1, where › › P pe it q › › 2 2 " $ ˇˇř n´1 k"0 a k e iω k t ˇˇ2dt " 1 `řn´1 k"1 a 2 k .
For the sequence pω j q jPN of rationally independent numbers, we are able to prove the following proposition. Proposition 6.7. Let `ωpjq ˘qn´1 j"0 ˘nPN be a sequence of real numbers such that for any n P N ˚, the real numbers `ωpjq ˘qn´1 j"0 are rationally independent.

Then, we have

$ R ˇˇˇ1 ? q n qn´1 ÿ j"0 e iωpjqt ˇˇˇd t ´´´´Ñ nÑ8 c π 2 .
Since, for any n P N ˚, the real numbers `ωpjq ˘qn´1 j"0 are rationally independent, then we can apply the Kac CLT Theorem 5.2. But, the functions ˆ1 ? qn ř qn´1 j"0 e iωpjqt ˙are in L 2 pbRq we deduce that they are uniformly integrable. Hence, by Theorem 5.5, we conclude that

lim nÝÑ`8 $ R ˇˇˇ1 ? q n qn´1 ÿ j"0 e iωpjqt ˇˇˇd t " c π 2 .
This finish the proof of the proposition.

Remark 6.8. The discussion above allows us to formulate the following questions Question 6.9. Does there exist a sequence of L 1 -flat Littlewood polynomials or of the L 1 -flat Newman polynomials on bR? Subsequently Question 6.10. Does there exist a the ultraflat Littlewood or Newman polynomials on bR? Remark 6.11. We remind here that in [START_REF] Prikhod'ko | On flat trigonometric sums and ergodic flow with simple Lebesgue spectrum[END_REF], A. A. Prikhod'ko constructed a sequence of trigonometric polynomials pP n q nPN which is locally L 1 -flat, that is, for any 0 ă a ă b and ε ą 0 there exists a positive integer n 0 " n 0 pa, b, εq such that for any n ą n 0 , we have In addition, it is proved in [START_REF] El Abdalaoui | On the spectral type of some class of rank one flows[END_REF] that this sequence is not L 1 -flat on R.

Appendix.II. On the Bohr compactification and Stone-Čech compactification of locally Abelian group.

Let G be a Locally Abelian compact group and denoted by τ its topology. Let p G be a group of the characters on G, that is, the continuous homomorphism from G to the torus T def " tt P C : |t| " 1u. Put

K " T p G ,
the space of all functions from p G to T equipped with the product topology (i.e.; the pointwise convergence topology). Therefore, there is a canonical injective homomorphism from G to K given by e : G ÝÑ K

x Þ ÝÑ epxq : h P K Þ ÝÑ epxqphq " hpxq. e is called a dual homomorphism. By abuse of notation we denote by the same letter G the image of G under e. Therefore, G is equipped with the Bohr topology denoted by τ b and the topology inherited from G. Hence τ b Ă τ . By taking the closure of G with respect to the Bohr topology we get the Bohr compactification of G and we denoted it by bG. Thus, by construction, bG is compact. Let us state the following useful lemma. Lemma 4. Let pg n q be a sequence of elements in G. Then pg n q converge to g with respect to τ b if and only if, for any character h P p G we have hpg n q ´´´´Ñ nÑ8 hpgq.

Proof. By the definition of the product topology on K, a sequence of functions pf n q converge to some function f if and only if, for any h P p G, f n phq ´´´´Ñ nÑ8 f phq.

By taking f n " epg n q and f " epgq we get epg n qphq ´´´´Ñ nÑ8 epgqphq, which means that for any h P p G, hpg n q ´´´´Ñ nÑ8 hpgq, and the proof of the lemma is complete.

We deduce from the lemma the following crucial fact about the separability of the Bohr compactification of G. We recall that the topological space is separable if it is contains a countable, dense subset. Proposition 6.12. If G is a separable with respect to τ topology then G is separable with respect to Bohr topology.

Proof. It is straightforward since the Bohr topology is contained in the usual topology.

In the case of R, the lemma 4 say that the sequence of real numbers px n q converge in the sense of the Bohr topology to x if and only if, for any t P R,

e ixnt ´´´´Ñ nÑ8 e ixt .
But since the characters on R are Lipschitz we deduce easy that the Bohr topology is contained in the usual topology and Q is dense in the usual topology and Bohr topology.

Nevertheless, by the Hirodata-Kakutani Theorem, R equipped with the Bohr topology doesn't have a countable basis and it is often that the Bohr topology doesn't have a countable basis. Precisely, Hirodata-kakutani Theorem asserts Theorem 6.13 (Hirodata-kakutani). The Bohr compactification of a given locally compact Abelian group has a countable basis if and only if the union of the spectrum of all almost periodic functions is countable.

We recall that the Fourier coefficient of f on χ P p G is given by

p f `tχu ˘" ż G f pgqχpgqdhpgq,
where h is the Haar measure on G and χ P p G is in the spectrum of f if the Fourier coefficients of f on χ is not zero 6.1. The Stone-Čech compactification and Bohr compactification. In the following the comparative board of the compactification of Stone-Čech and the Bohr compactification. For short we write LC for Locally Compact and c.f. for continuous functions.

The Stone-Čech compactification

The Bohr compactification Let X a LC topological space Let G a LCA topological group Consider C " CpX, r0, 1sq the space of c. f. : X Ñ r0, 1s A " p G Embedding X in r0, 1s C embedding G in T A The embedding function is the valuation e x pf q " f pxq e g pf q " f pgq Take the closure of epXq Take the closure of epGq We denote the closure of epXq by βX and the closure of epGq by epGq " bG

4 . 4 .

 44 The sequence of probability measures ˇˇP m pzq ˇˇ2 dh converges weakly to the Haar measure.From the proposition 4.3 and Lemma 2 we deduce the following Proposition With the above notations we have

  bR|P k | dh ˙2 ă 8, then µ admits an absolutely continuous component.

Theorem 5 . 5 . 8 E

 558 If the sequence of random variables tX n u converges in distribution to some random variable X and for some p ą 0, supnPN `Ep|X n | p q ˘"M ă `8, then for each r ă p, lim nÝÑ`ˆˇX n ˇˇr ˙" E ˆˇX ˇˇr Ḟor the proof of Theorem 5.5 we refer the reader to[5, p.32-33] or[13, p.100].

49 ] 3 .

 493 Lemma With the above notations we have lim sup

n 1 ă n 2 ă n 3 ă ¨¨¨ă n k and apply p18q with ε " ? π 100 combined with Lemma 3 to get m ą n k such that ż bR Q|P m |dh ď 51

 51 Put n k`1 " m. Therefore the inequality p17q holds and by letting k ÝÑ `

  Proof. Let R n ptq " P 0 ptq ¨¨¨P n ptq, Q n ptq " |R n ptq| 2 and σ n " |R n pκq| 2 dhpκq.

	exists and is denoted by
	8 ź k"0	ˇˇˇP k pθq ˇˇˇ2 dhpθq,
	By the definition of P n we have ˇˇP n ptq ˇˇ2 " 1 `∆n ptq with ∆ n ptq "	1 p n	ÿ p‰q e ˆi`p p ´qqh n `q s n,p,q	˘tȧ
	nd it is obvious that
				dhpθq,

  is a sequence of rational numbers which goes to 0 as n goes to 8.But, this sequence is not L 1 -flat on the Bohr compactification of R. Indeed, one may show that we have

			$	ˇˇP n pxq ˇˇdx ´´´´Ñ nÑ`8	c	π 2	.
		b	1 ´a ż b a	ˇˇˇˇP n pxq ˇˇ2 ´1ˇˇd x ă ε.
	The polynomials P n , n P N are given by
			P n pxq "	1 ? p n	p"0 pn´1 ÿ	e iωnppqx ,
	where, ω n ppq " mn.pn ε 2 n with p n ´´´´Ñ	e	εn pn p , pm n , p n q nPN is a sequence of positive integers

nÑ8 `8 and m n ´´´´Ñ nÑ8 `8, ε n

Acknowledgment 1. The author wishes to express his gratitude to M. Lemańczyk, J-P. Thouvenot, E. Lesigne, A. Bouziad, J-M. Strelcyn and A. A. Prikhod'ko for fruitful discussions on the subject. It is a pleasure for him to acknowledge the warm hospitality of the Poncelet French-Russian Mathematical Laboratory in Moscow where a part of this work has been done.

We conclude that

and this finish the proof of the theorem.

In the following we recall the classical well known multidimensional Central Limit Theorem in probability theory [16, p.81] stated in the following forms Theorem 5.3 (Multidimensional CLT Theorem). Let pZ n,k q 1ďkďkn, ně1 be a triangle array of random variables vectors in C d and put

(3) For any ε ą 0, lim

EpZ n,k q " 0.

(5) For each j, l P t1, ¨¨¨, du,

Then Γ is a hermitian non-negative definite matrix and the sequence of random vectors ˜1 ?

converges in distribution to the complex Gaussian measure N C p0, Γq on C d .

We also need the following important and classical fact from Probability Theory connected to the notion of the uniform integrability.