Leslie Cartz-Piver

Clément Polin

Cecile Diebolt

Marianne Chouly

Lydie Anne Garçonnot

Jean Pierre Clément

Benjamin Calvet

Philippe Couratier

Therapeutic patient education in Alzheimer's disease and related disorders: clinical experience and perspectives

Keywords: caregivers, Alzheimer's disease and related disorders, chronic disease, patient education, depression and anxiety

Therapeutic patient education (TPE) is the process by which health professionals impart information to help patients self-manage their chronic disease: it is an essential part of treatment of long-term diseases and conditions. Memory loss and other cognitive disorders are usually considered as obstacles to TPE for patients with Alzheimer's disease or related disorders (ADRD). Over 100 patients with different forms of ADRD and caregivers have benefited from TPE programs since 2011 at the University Memory Clinic of Limoges. Participants report better understanding of the disease and improved relationships. TPE may prevent anxiety and depression in patient and in caregivers, and reduce burden of caregivers. General guidelines and perspectives for TPE in ADRD are outlined.

Introduction

Therapeutic patient education (TPE) is the process by which health professionals impart information to help patients self-manage their chronic disease. TPE is designed to train self-managing and coping skills to patients: it is an essential part of treatment of longterm diseases and conditions [START_REF]Educational handbook for health personnel[END_REF].

The ongoing French National Health Plan recommends TPE for people affected by neurodegenerative disorders. Whereas TPE programs are well defined for Parkinson's disease [START_REF] Ory Magne | A therapeutic educational program in Parkinson's disease: ETPARK[END_REF], for multiple sclerosis [START_REF] Wendebourg | Development and Feasibility of an Evidence-Based Patient Education Program for Managing Fatigue in Multiple Sclerosis: The "Fatigue Management in MS" Program (FatiMa)[END_REF] and for Huntington's disease [START_REF] A'campo | The Patient Education Program for Huntington's Disease (PEP-HD)[END_REF], there is yet little experience of TPE for dementia patients.

Memory loss and other cognitive disorders due to Alzheimer's disease or related disorders (ADRD) are usually considered as obstacles to TPE. Many programs address the family caregivers of people with ADRD, but not the people themselves [START_REF] De Rotrou | Do patients diagnosed with Alzheimer's disease benefit from a psycho-educational programme for family caregivers? A randomised controlled study[END_REF]. At our Clinical and Research Memory Center (CRMC), people with ADRD and their caregivers have benefited from TPE programs since 2011; this paper reports the clinical experience and suggests general guidelines for TPE in ADRD.

MATERIAL AND METHODS

2.1

The beginning of the project. In 2010, several people with posterior cortical atrophy (PCA) were diagnosed at the CRMC. Because PCA syndrome is a rare condition, a support group was organized for the people with PCA and their caregivers. The exchanges occurring during the support group meetings revealed that people with PCA were able to adopt coping attitudes and strategies and that sharing their experience was very positive for their self-esteem [START_REF] Videaud | Impact of drug-free care in posterior cortical atrophy: Preliminary experience with a psycho-educative program[END_REF]. This experience incited professionals to elaborate a TPE program for people with ADRD, which was authorized by the Local Health Agency and started in 2011.

Participants:

The ADRD diseases targeted were "orphan diseases" such as PCA, non-fluent and logopenic forms of primary progressive aphasia (PPA), behavioral frontotemporal lobar degeneration (FTLD) and also early onset Alzheimer's disease (EOAD).

These rare disorders are more inclined to cause isolation, anxiety and misunderstanding because they are generally unknown by the general public and by primary healthcare professionals. People with these rare disorders and caregivers find it difficult to understand and to accept their disease. their diagnosis, using the criteria defined by the Diagnostic and Statistical Manual of Mental Disorders (DSM-5). For each person with ADRD, the main family caregiver was included in the TPE program.

People with ADRD who did not acknowledge their disease because of pronounced anosognosia or denial were not included in the program. People with ADRD who were not able to give their written consent because of their cognitive deterioration were also excluded from TPE. In these cases of non-inclusion, a family support program was offered to their caregivers. In some cases, people with ADRD and their caregivers did not wish to participate because of the distance from their home to the CRMC or because they felt constrained by the TPE program schedule.

TPE groups:

Preselected people with ADRD and their family caregivers were informed about the TPE program during follow-up at the CRMC. Those interested were interviewed by one of the neuropsychologists who gave detailed information about the program and collected written consent. The neuropsychologist then established for each participant (person with ADRD and caregiver separately) an individual educational needs assessment describing the participant's objectives for TPE (figure 1).

Thus, groups of people suffering from the same ADRD (or the same syndrome), with similar degrees of cognitive deterioration, accompanied by one or two family caregivers, were formed; groups did not exceed 15 participants.

The TPE professional team:

The ADRD TPE program was coordinated by a specialized neurologist of the CRMC. During a TPE session, one of the neuropsychologists of the team became the referent for the group: this professional conducted the initial and final assessments and was present at every meeting. At least two professionals of the TPE team were present at every meeting conducted by the referent neuropsychologist: the neurologist, the nurse or other psychologists of the CRMC. All the professionals of the team had been trained for TPE. Other professionals such as social workers or speech therapists were invited for special meetings when needed.

The TPE sessions:

Separate sessions were designed for each ADRD: PCA, PPA, FTLD and EOAD. The sessions consisted of eight monthly meetings (Table 1) with the referent neuropsychologist and one or more other member of the TPE team. A light snack was offered at the beginning and the end of each meeting to help the participants feel welcome and relaxed.

The first meeting was always dedicated to introducing the program as well as all the professionals of the TPE team and all the participants. It was important to create group cohesion and reduce the stress by underlining common points among the participants.

The next meetings were dedicated to explaining the effect of the disease on the brain, and the symptoms and changes due to the disease. As the meetings went, participants were encouraged to discuss their own difficulties and ask their questions in an open manner.

Most meetings were designed for the people with ADRD and their caregivers together.

Only during the meeting about "Feelings about ADRD" were people with ADRD and caregivers separated to express their feelings freely, and then reunited at the end. Active learning methods were used and the contents were adjusted according to the participants' needs on the basis of continuous assessment.

Evaluation:

To evaluate the impact of TPE on the people with ADRD and the caregivers, three types of tools were used: quantitative evaluation by standardized scales filled out before and after TPE, qualitative evaluation and educative evaluation.

For quantitative evaluation, the standardized scales used were the Hospital Anxiety and Depression scale (HAD) [START_REF] Zigmond | The Hospital Anxiety and Depression Scale[END_REF], the Quality of Life in Alzheimer's Disease scale (QOL-AD) Socio-demographic characteristics of participants were described by frequencies and percentages for qualitative variables and by means and standard deviation for continuous variables. To compare quantitative variables between before and after TPE on continuous data (HAD scores, QOL-AD scores, ZBI-22 scores), paired samples Wilcoxon tests were used. All analyses were performed using SPSS for Windows 25.0 (IBM Corp., Armonk, NY, USA). Level of significance was P value < 0.05.

RESULTS

3.1: TPE activity:

From 2011 to 2017, twelve sessions of the ADRD TPE program took place. Three sessions were dedicated to PPA, one session was dedicated to PCA, four sessions were dedicated to FTLD and four sessions were dedicated to EOAD (Table 2).

3.1.1: Population:

A total of 105 people (52 people with ADRD and 53 family caregivers) participated in the ADRD TPE program from 2011 to 2017. Globally, the people with ADRD were aged from 45 to 80 years old with a mean age of 65.5 years. The PPA group was the oldest, on average 70.3 years old (59 to 80), whereas the EOAD was the youngest, on average 62.4 years old (52 to 66). The FTLD group was on average 66.1 years old (45 to 79); the PCA group was on average 64.5 years old (51 to 77). The people with ADRD were more often men than women (30 men versus 22 women). Only the EOAD group presented more women than men (11 versus 7).

Different socio-cultural backgrounds were represented amongst people with ADRD: 49% had elementary or secondary school education (level one = 6 to 9 years of schooling), 33% had completed secondary school (level two = 9 to 12 years) and 18% had university education (level three = over 12 years). The EOAD group was the less educated (61% level one; 22% level two; 17% level three), versus the FTLD group that was the most educated (38% level one; 31% level two; 31% level three).

Fifteen people with ADRD were working at the time of diagnosis of their ADRD (3 PPA, 2 PCA, 3 FTLD, and 7 EOAD); they all had stopped working at the time of TPE. Three people with ADRD had children living at home at the time of diagnosis (2 EOAD and 1 FTLD) and during TPE. All the people with ADRD were accompanied by their spouses and for one person, a second caregiver that was a child. Eighteen of the caregivers were working during the time of the TPE sessions.

3.1.2: Stages of ADRD:

The delay between the first symptoms and the diagnosis was three years on average, with a wide range varying from less than one year to 13 years.

The delay between the diagnosis and the TPE program was 1 to 4 years; in several cases, the TPE program took place the year the diagnosis was established. The average Mini-Mental State score was 18/30 for the EOAD groups (5/30 to 28/30), 23/30 for the FTLD groups (18/30 to 27/30). In the PPA groups, Mini-Mental State scores varied widely, from 2/30 to 28/30 depending on the severity of aphasia.

3.2: Individual educational assessment

The assessment was performed before TPE, individually for the people with ADRD based on an interview and questionnaires. The caregivers were also assessed individually with the same tools. During the interviews, the participants expressed spontaneously their needs and requests, whereas questionnaires were used to collect data on predetermined information items.

The assessment of individual educational needs was used as baseline to guide the TPE program and at the end of the program to determine whether the participant's skills had progressed. Each participant could express one or more requests (Figure 2).

Family caregivers generally expressed more requests than people with ADRD, who could have reduced capacities to express themselves due to their disorders. "Understand the disease and its consequences", as well as "Maintain good relationships", especially with their spouse, were the main requests of people with ADRD (45% and 42% respectively)

and of the family caregivers (60% and 76% succeeded in completing correctly all the scales before and after TPE (Figure 3).

Before TPE, HAD scales (N= 62) showed average levels of anxiety and low depression levels in both people with ADRD (8.2 ± 4.7 anxiety and 6.2 ± 3.2 depression) and caregivers (9.0 ± 5.1 anxiety and 6.6 ± 4.2 depression); after TPE (N = 26) there were no significant changes in HAD scales in people with ADRD (p=0.31 and p=0.88) or in caregivers (p=0.17 and p=0.12) even though the symptoms and the disorders had Before TPE, caregivers (N = 32) expressed a mild burden (ZBI-22 = 35); after TPE (N = 16), burden was similar except for the item "Do you feel that you will be unable to take care of your relative much longer?" which had a tendency to decrease (p=0.05).

Qualitative evaluation was performed by following attendance and participation during the TPE session and by a Satisfaction Questionnaire completed at the end of the session TPE by the participants.

Attendance was incomplete for eight people with ADRD and their nine caregivers (16%)

for different reasons. Two couples did not wish to continue after the first meeting, so follow-up was continued individually. Five couples discontinued TPE because the person with ADRD was admitted to a nursing-home during the session. One couple interrupted TPE because the person with ADRD developed breast cancer that needed urgent treatment. In another case, the caregiver interrupted TPE because of stroke, but the person with ADRD continued to participate.

Participation evolved during the TPE session. There was always an important emotional strain during the first meeting, when the people with ADRD and their caregivers discovered other people living with the same disease. It was important to let them all express themselves. The meeting concluded with an outline of the similarities between people with ADRD, between caregivers, and between all participants to help create cohesion in the group. As the session went on, the participants relaxed and began to look forward to seeing each other from one meeting to another. Friendships were formed and some couples exchanged phone numbers and continued to see each other once the TPE session was over.

The Satisfaction Questionnaire revealed that one of the strong points of the TPE program was meeting other people with ADRD and caregivers living with the same disease. The participants also reported better understanding of the disease and its consequences, and better knowledge about help and support systems.

Educative evaluation: An interview at the end of the TPE session took place individually for people with ADRD and for family caregivers with their referent neuropsychologist.

The evaluation for people with ADRD's mainly showed progress in "Understanding the disease", and in "Maintaining a good relationship". For caregivers, progress was noted in all requests, even those not expressed spontaneously, such as "Getting Help and Support" and "Communication".

Discussion:

People with ADRD are most often discredited because of cognitive decline, and the caregiver becomes the main discussion partner for professionals. Many programs and other facilities have been conceived and organized to help caregivers, the common belief being that when the caregiver gets support, the person with ADRD is indirectly getting care and "being better/not getting worse", especially when behavioral symptoms occur [START_REF] De Rotrou | Do patients diagnosed with Alzheimer's disease benefit from a psycho-educational programme for family caregivers? A randomised controlled study[END_REF]. Participation of people with ADRD in these interventions is generally found to be beneficial [START_REF] Gilhooly | A meta-review of stress, coping and interventions in dementia and dementia caregiving[END_REF]. TPE is different from psychosocial interventions because the person with ADRD is the center of attention, and the objective is not so much to stimulate his/her capacities rather than to give the opportunity to learn about his/her disease and to adapt to the situation. TPE aims to enable people with chronic disease to acquire and maintain abilities to optimally manage their lives with their disease. It is therefore a patientcentered continuous process: it includes organized awareness, information, self-care learning and psychosocial support regarding the different aspects such as the disease, prescribed treatment and care, organizational information, and behavior related to health and illness [START_REF]Educational handbook for health personnel[END_REF].

Following progress in biomarkers, ADRD diagnosis can be performed in very early stages of the disease, where people with ADRD usually have preserved capacities to participate in discussions and decisions. It is then very important to offer TPE, so that they can understand and accept the disease and its consequences, and learn to manage their lives with the disease as long as they can. This early stage is also favorable for professionals to inform people with ADRD and know about their decisions for the future, as well as to get their consent for research [START_REF] Gauthier | Diagnosis and management of Alzheimer's disease: past, present and future ethical issues[END_REF].

The participation of the main caregiver in the TPE program is necessary because the relationship between the person with ADRD and the caregiver is the core part of their life, and maintaining the quality of this relationship is one of their main concerns (Figure 2). In several cases, the TPE professionals realized that the disease had never been discussed in the family circle, and so TPE meetings were opportunities for the person with ADRD and their caregiver to exchange about their feelings. Finally, as the disease progressed, the caregiver progressively had to take on more activities and responsibilities:

TPE allows the caregiver to be armed for this eventuality, and to be aware of the person with ADRD's wishes.

It is interesting to note that people with different types of ADRD benefited from TPE.

The first sessions (2011-2013) were designed for groups of people with focal deficits due to PPA or PCA. Since 2014, the TPE team developed sessions adapted to people with EOAD, who usually have very good insight about their situation and many questions to ask at the beginning of their disease. TPE in advanced stages can still be profitable to people with EOAD and their family caregivers, thanks to preserved social skills that allow good participation, which is very important for self-esteem. For people with FTLD, the TPE team was worried about behavior problems, so in 2014 the first FTLD session was designed with separate meetings for caregivers and for patients. As soon as the first meeting started, it became obvious that people with FTLD wanted to be with their caregivers. Moreover, their behavior problems did not disrupt the TPE meetings: people with FTLD suffering from apathy benefited from the stimulation and those with disinhibition brought a welcome sense of humor to the meetings. Since 2015, the TPE meetings address the people with FTLD and their caregivers together.

Initial levels of anxiety and depression were low in the study population. After TPE levels of anxiety and depression did not increase even though the people with ADRD's disorders had progressed with time, so TPE probably had a favorable effect on preventing anxiety and depression in both people with ARDR and their caregivers. While quality of life has been reported to increase following TPE [START_REF] Villars | Impact of a therapeutic educational program on quality of life in Alzheimer's disease: results of a pilot study[END_REF], this was not the case in our experience. Indeed, the caregivers' quality of life decreased significantly, mainly because of financial worries (item 12). The tendency to decrease the burden in the item ZBI-22 "Do you feel that you will be unable to take care of your relative much longer?" may reflect better self confidence of the caregivers, which is an interesting outcome for TPE. Recommendations in TPE practice (1) consider that individual educative assessment before and after TPE is the best measure of the effect of TPE. In this experience, general satisfaction of the participants was very good. After TPE, participants reported better understanding of the disease and its consequences as well as new professional and/or friendly contacts for further help.

Sessions of TPE are composed of scheduled meetings with planned progressive themes.

From the requests expressed by people with ADRD and the caregivers at initial assessment (Table 2), the TPE team developed the targets for TPE in ADRD (Table 3).

These targets can be used for people with ADRD and for their caregivers. The delay between the meetings is useful to let people with ADRD and caregivers try new attitudes or actions. However this experience showed that as the disease progresses, people with ADRD and caregivers tend to drop out or are unable to participate. Our revised TPE program is shortened to six sessions (Table 4). Successful TPE for people with ADRD is a question of good timing: not too late, not too long.

Strength and weakness:

This report is an original clinical experience that addressed a significant number of cases of rare pathologies, recruited over the years by a single CRMC. Moreover, in absence of curative medication there is increasing interest in studies about non-pharmaceutical care of people with ADRD and their caregivers. Even though authorization by an ethical committee was not obtained (because not mandatory at the time), ethical considerations were met because written consent was collected from all participants, people with ADRD and caregivers.

One of the limits of this experience is the absence of a control group, because it is difficult to organize in rare pathologies, and because the people with ADRD who refused to participate were considered clinically different from those included due to more pronounced anosognosia. The main weakness of this clinical experience is linked to the few measures completed before and after the TPE program, due mainly to progressive worsening of symptoms that is the characteristic feature of ADRD: because of this, the TPE team decided to reduce the number of meetings from eight to six.

Conclusions

TPE is designed to help patients with chronic disease and their families to understand the disease, manage their situation and cooperate with healthcare providers, so they can maintain or improve their quality of life. The ongoing experience of TPE at the CRMC demonstrates that TPE is possible to achieve for people with a wide variety of ADRDs and their family caregivers, especially in early stages of the disease. TPE can prevent anxiety and depression in people with ADRD and in caregivers, and may reduce the burden of caregivers. The general guidelines outlined in this paper should help more professionals engage in this activity and offer TPE to people with ADRD as soon as they are diagnosed. Because there is currently no curative treatment for ADRD, TPE is an ethical necessity. As more memory centers engage in TPE, it would be very interesting to

SELF MANAGEMENT SKILLS

Understand and be able to explain -the disease, the impact on the brain -the signs and symptoms -the evolution -the effect of the treatment and care Maintain good relationships -with the spouse -within the family circle -be aware of the impact of the disease on the family and on work or social life Self care and management -take appropriate medication -monitor weight /adapt food and diet -keep adapted social activities -practice sport and/or adapted physical activities -adapt environment Prevent complications of the disease -be aware of the risk of errors of medication intake -be aware of the risk for driving -be aware of the risk of swallowing difficulties -be aware of the risk of getting lost Get support and help -know about financial support -know about respite services -know about legal protection -accept help and assistance COPING and ADAPTATION SKILLS

Adapt emotions and feelings

-reduce fear and self stigma -improve self-esteem and sense of self-reliability -accept the consequences of the disease -have adapted projects -accept to plan ahead Develop communication skills -express feelings and emotions -communicate with family, with friends -communicate with professionals

  , and the Zarit Burden Inventory version 22 items (ZBI-22) (9, 10). Qualitative evaluation was based on attendance and a Satisfaction Questionnaire SQ filled out by the participants at the end of the session. Educative evaluation was based on interviews comparing the individual needs assessment before and after TPE.

  progressed.Quality of life (QOL-AD) before TPE (N = 46) was similar for people with ADRD and caregivers (35 ± 4.3 and 32.7 ± 5.5). After TPE (N = 29), quality of life had decreased significantly for caregivers (32.7 ± 5.5 reduced to 29.7 ± 5.4; p = 0.04), especially concerning their financial situation (item 12: 2.8 ± 0.68 reduced to 2.3 ± 0.94; p = 0.04), whereas quality of life did not change for people with ADRD (35.4 ± 4.3 to 35.2 ± 6.1; p = 0.69).

Figure 1 :Figure 2 :

 12 Figure 1: Selection of TPE participants

Quantitative evaluation before and after TPETable 1 : General Framework of the ADRD TPE program MEETINGS THEME SKILLS METHODS PROFESSIONALS

 1

	45% 65% Caregivers AFTER 0% 10% 20% 30% 40% 50% 60% 70% 80% Caregivers BEFORE Patients AFTER Patients BEFORE Interview Information and written consent 42% 36% 27% 76% 60% 76% 8,1 8,8 9 8,2 7,4 6,6 6,9 6,2 29,7 32,3 35,2 35,4 Assessment of educational needs Meeting 1 Introduction Meeting 2 ADRD and the brain Meeting 3 Cognitive disorders due to ADRD UNDERSTAND -ADAPT 24% 21% 15% 41% 38% 38% 34,3 35 Anxiety Patients Caregivers Depression Quality of Life Burden BASELINE EVALUATION Standardized scales and Questionnaires CREATE GROUP COHESION UNDERSTAND-PREVENT Meeting 4 Emotional changes due to ADRD and communication skills MAINTAIN RELATIONSHIPS and COMMUNICATION SKILLS -ADAPT Meeting 5 Help and support systems MANAGE-GET HELP Meeting 6 Communication skills RELATIONSHIPS-COMMUNICATION Figure 3: Individual	Neuropsychologist = reference of the group Neuropsychologist TPE team Neuropsychologist Memory Doctor (Neurologist, Psychiatrist or Geriatrician) Neuropsychologist Psychologist or nurse Neuropsychologist Psychologist or nurse Neuropsychologist Social Worker Neuropsychologist Speech therapist or nurse
	Meeting 7	Feelings about ADRD	ADAPT-COMMMUNICATION	Neuropsychologist
				Psychologist or nurse
	Meeting 8	Continuing projects and activities	PREVENT -ADAPT	Neuropsychologist
		with ADRD		TPE team
	Individual	Assessment of educational progress FINAL EVALUATION	Neuropsychologist = reference of the group
	Interview		Standardized scales and Questionnaires

Table 2 : TPE sessions and participants from 2011 to 2017

 2

	ADRD	2011	2012	2013	2014	2015	2016	2017	Total 2011-2017
					4 patients
	PCA				4 caregivers				8
		5 patients	5 patients	4 patients
	PPA	5 caregivers	5 caregivers	4 caregivers					28
					4 patients	4 patients	8 patients
	FTLD				5 caregivers	4 caregivers	8 caregivers		33
							(2 GROUPS)
	EOAD				5 patients 5 caregivers	4 patients 4 caregivers		9 patients (2 GROUPS) 9 caregivers	36
	Total	10	10	8	27	16	16	18	105

Table 3 : Suggested Education TARGETS for ADRD EDUCATION TARGETS for People with ADRD and Caregivers

 3

Table 4 : Revised session of the ADRD TPE program MEETINGS THEME SKILLS METHODS TRAINED TPE PROFESSIONALS

 4

	Individual	Information and written consent	BASELINE EVALUATION	Neuropsychologist = reference of the group
	Interview	Assessment of educational needs	Standardized scales and Questionnaires
	Meeting 1	Introduction	CREATE GROUP COHESION	Neuropsychologist
			Round Table	TPE team
			Outline similarities
	Meeting 2	ADRD and the brain	UNDERSTAND-PREVENT	Neuropsychologist
			Questionnaire " NAME YOUR DISORDER"	Memory Doctor (Neurologist, Psychiatrist or
			Prevention Cards	Geriatrician)
	Meeting 3	Cognitive disorders due to ADRD UNDERSTAND -ADAPT	Neuropsychologist
			Symptom cards	Psychologist or nurse
	Meeting 4	Emotional changes due to ADRD	MAINTAIN RELATIONSHIPS and	Neuropsychologist
		and communication skills	COMMUNICATION SKILLS -ADAPT	Psychologist or nurse
			Symptom cards and Communication game(s)
	Meeting 5	Feelings about ADRD	ADAPT-COMMMUNICATION	Neuropsychologist
			Round tables (separate for people with ADRD and	Psychologist or nurse
			for caregivers)
	Meeting 6	Continuing projects and activities	ADAPT -MANAGE-GET HELP	Neuropsychologist
		with ADRD	Questionnaire about dysfunctional statements	TPE team
			Discuss help and support systems
	Individual	Assessment of educational progress FINAL EVALUATION	Neuropsychologist = reference of the group
	Interview		Standardized scales and Questionnaires

conduct a multiple-center study to increase the power of the experience and define the best indicators and the best methods for TPE for people with ADRD and their caregivers.

Disclosure of interest

The authors declare no conflict of interest


Pierre-Yves Bienvenu
email: pbienvenu@math.univ-lyon1.fr

¸ois Hennecart
email: francois.hennecart@univ-st-etienne.fr

ON THE DENSITY OR MEASURE OF SETS AND THEIR SUMSETS IN THE INTEGERS OR THE CIRCLE

Keywords:

Let d(A) be the asymptotic density (if it exists) of a sequence of integers A. For any real numbers 0 ≤ α ≤ β ≤ 1, we solve the question of the existence of a sequence A of positive integers such that d(A) = α and d(A + A) = β. More generally we study the set of k-tuples (d(iA)) 1≤i≤k for A ⊂ N. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of "pseudo sth powers". We consider similar problems for subsets of the circle R/Z, that is, we partially determine the set of k-tuples (µ(iA)) 1≤i≤k for A ⊂ R/Z.

Introduction

We respectively denote by N, Z, Q and R the set of all natural integers, the set of all integers, the set of all rational numbers and the set of all real numbers. Let T = R/Z be the torus (or the circle). For any finite set A, we denote its cardinality by |A|. For x ∈ R, we denote by x its integral part and by {x} = x -x its fractional part. For A ⊂ N and t > 1, we let A(t) = |A ∩ [1, t]|. We define if it exists the so-called asymptotic density of A by d A = lim t→∞ A(t) t .

Otherwise we define the lower and the upper asymptotic densities d(A) and d(A) using lim inf and lim sup instead of limits. More generally, if A ⊂ B ⊂ N, we define if it exists the density of A inside B as d B (A) = lim t→∞

A(t) B(t) .

The density of A inside N is therefore simply the density, and if B has a positive density, we have d

B (A) = d A /d B .
For a subset A of a semigroup G, let A + A = {a + b : a, b ∈ A}. For k ≥ 1, we denote by kA its k-fold sumset. From Kneser's Theorem [START_REF] Kneser | Abschätzung der asymptotischen Dichte von Summenmengen[END_REF], we know that for any subset A ⊂ N, the inequality d 2A < 2d A may only hold when d 2A is a rational number. Similarly, for any subset A of the circle T equipped with its Haar probability measure µ, a theorem of Raikov [START_REF] Raikov | On the addition of point-sets in the sense of Schnirelmann[END_REF] implies that µ(2A) ≥ min(1, 2µ(A)) where µ(A) = sup

F ⊂A F closed µ(F ).
In this paper, we firstly determine the possible values (α, β) of pairs (d A , d 2A ) and (µ(A), µ(2A)). We first completely settle the case β ≥ min(2α, 1). Theorem 1.1. Let (α, β) ∈ [0, 1] 2 . Suppose β ≥ min(2α, 1). Then the following statements both hold.

a) There exists A ⊂ N such that d A and d 2A exist and equal α and β respectively. b) There exists a measurable subset A ⊂ T such that 2A is measurable and µ(A) = α and µ(2A) = β. Further, for α > 0, we may take A to be open (in fact a finite union of open intervals).

The case β = 2α is obvious for the second item (with an interval A), and is a special case of a theorem by Faisant et al [START_REF] Faisant | Additive Complements for a given Asymptotic Density[END_REF] for the first item, whereas allowing different summands, Volkmann [START_REF] Volkmann | On uniform distribution and the density of sum sets[END_REF] proved that, given positive real numbers α 1 , α 2 and γ such that α 1 + α 2 ≤ γ < 1, there exist 1 A 1 , A 2 such that d A i = α i , i = 1, 2, and d A 1 + A 2 = γ; he actually proved the corresponding result for subsets of the circle too. A similar result was obtained by Nathanson [START_REF] Nathanson | Best possible results on the density of sumsets[END_REF], including a version for Schnirelmann's density.

More generally, we investigate the set D k of possible values of the tuple (d A , d 2A , . . . , d kA ) when A ranges over the set of sequences for which all of these densities exist. For any real number θ > 1, let [START_REF] Cabrelli | Sums of Cantor sets[END_REF] T k,θ = n ≥ 1 : 0 ≤ {θn} < 1 k + 1 .

Note that d T k,θ = 1/(k + 1) if θ is irrational by Weyl's criterion which we state in a moment (Theorem 1.3), while T k,θ = N if θ is an integer. In any case one has jT k,θ = {n ≥ 1 : 0 ≤ {θn} < j k+1 } for j ≤ k + 1, in particular. (k + 1)T k,θ = N. We now state the following key result which will be proved in Section 2 when k = 1. The general case k ≥ 2 will be tackled in Section 4.

Proposition 1.2. Let β ∈ [0, 1] and k ≥ 1 be an integer. There exists a set A ⊂ T k,θ such that iA has density 0 for any i < k, whereas kA has density β inside kT k,θ and (k + 1)A has density 1 inside N.

In particular, we have d kA = βk/(k + 1) if θ is irrational while d kA = β if θ is an integer. Thus any (k + 1)-tuple of the form (0, . . . , 0, β, 1) where β ≤ 1 belongs to D k+1 .

In parallel, we consider the similar problem in the circle T equipped with its Haar measure µ. Thus let E k be the set of all the possible values of (µ(A), . . . , µ(kA)) for A ⊂ T for which these measures exist. We may sometimes need to work with the subset E o k ⊂ E k of all the possible values of (µ(A), . . . , µ(kA)) when A ⊂ T is open and Riemann-measurable and similarly E c k , where we consider closed sets A. There is a close connection between E k and D k because of Weyl's criterion for equidistribution, of which we now state a direct consequence. For A ⊂ T and

λ ∈ R \ Q, let B λ,A = n ∈ N : {λn} ∈ A .
Theorem 1.3. For any irrational number λ and any Riemann-measurable function f : [0, 1] → R, we have

lim x→+∞ 1 x n≤x f ({λn}) = f.
1 In Volkmann's construction, the sets of integers are sets of relative integers and not necessarily positive integers though.

In particular, for any Riemann-measurable subset A ⊂ T, we have d B λ,A = µ(A).

The latter equality may be extended to open sets A.

The extension to open sets is [START_REF] Volkmann | On uniform distribution and the density of sum sets[END_REF]Lemma 4]. Further, Theorem 1.3 and a simple compactness argument shows that for any > 0 and irrational λ, there exists a constant C = C(λ, ) such that for any interval I of length at least the set B λ,I contains at least an integer less than C. Finally, the operation A → B λ,A behaves well with respect to set addition.

Lemma 1.4. Let k ≥ 2 and A i ⊂ T be open for i = 1, . . . , k and λ be irrational. Then d k i=1 B λ,Ai = µ( k i=1 A i ).
Proof. Let A ⊂ T be open, and for any > 0 let A be the set of all a ∈ A whose distance to the boundary of A is at least . Thus A = >0 A and µ(A) = lim →0 + µ(A ). Further,

k i=1 A i = >0 k i=1 A i . We observe that (2) B λ, k i=1 A i ⊂ k i=1 B λ,Ai ⊂ B λ, k i=1 Ai .
The rightmost inclusion is easy; for the leftmost one, let

x ∈ B λ, k i=1 A i , thus x = k i=1 a i where a i ∈ A i . Consequently, ]a i -/k, a i + /k[ ⊂ A i for i ∈ {1, . . . , k -1}. If n is large enough (larger than kC(λ, 2 /k)), there exists n 1 , . . . , n k-1 ≤ n/k such that {n i λ} ∈ ]a i -/k, a i + /k[ . Let n k = n -n 1 -• • • -n k-1 > 0. Then {n k λ} = {nλ} -{n 1 λ} -• • • -{n k-1 λ} mod 1, which implies {n k λ} mod 1 ∈ ]a k -, a k + [ ⊂ A k , in other words n k ∈ B λ,A k . Thus n ∈ k i=1 B λ,Ai .
Taking densities and applying Theorem 1.3 in equation ( 2), we find that

µ k i=1 A i = d(B λ, k i=1 Ai ) ≤ d k i=1 B λ,Ai ≤ d k i=1 B λ,Ai ≤ d(B λ, k i=1 Ai ) = µ k i=1 A i .
Letting → 0 + , we conclude.

Consequently, E o k ⊂ D k ; in particular, the second item of Theorem 1.1 implies the first one when α > 0, but we will provide another proof for it. Further, Raikov's theorem together with Theorem 1.1 means that

E 2 = E o 2 = E c 2 = {(α, β) ∈ [0, 1] 2 : β ≥ min(1, 2α)}.
To complete our description of D 2 , we need to understand which pairs (α, β) satisfying β < 2α belong to it, which we do in the next theorem. For an integer n, let v 2 (n) be its dyadic valuation; we extend it to rational numbers by letting

v 2 (p/q) = v 2 (p) -v 2 (q). Theorem 1.5. Let β ∈ Q ∩ ]0, 1[ such that v 2 (β) ≤ 0 and let α ∈ ]0, 1[ satisfy β < 2α.
We denote g 0 = min g ≥ 1 : gβ is an odd integer . Then there exists a sequence A ⊂ N such that d A = α and d 2A = β if and only if

β 2 < α ≤ β 2 + 1 2g 0 .
We briefly discuss iterated sumsets. It is not clear what constraints a tuple (α i ) 1≤i≤k must satisfy for a set A ⊂ T satisfying µ(iA) = α i to exist; we certainly need α i ≥ min(1, α j + α i-j ) for any j < i due to Raikov's theorem but it may not be sufficient. In particular, we will deduce the following constraint from a theorem of Gyarmati, Konyagin and Ruzsa [START_REF] Gyarmati | Double and triple sums modulo a prime[END_REF].

Theorem 1.6. There exists a constant c > 0 such that the following holds. Let A ⊂ T be closed, and suppose that µ(2A) < c. Then µ(3A) ≥ 3 2 µ(2A). In view of Lev's result [START_REF] Lev | Structure theorem for multiple addition and the Frobenius problem[END_REF]Theorem 3] asserting that for any finite set of integers A one has k(|(k + 1)A| -1) ≥ (k + 1)(|kA| -1), one may more generally imagine that µ((k + 1)A) ≥ k+1 k µ(kA) under certain restrictions on µ(kA). Note that another result from [START_REF] Gyarmati | Double and triple sums modulo a prime[END_REF] implies that the constant c may not be taken to be 1. Gyarmati et al. conjecture that its optimal value is 1/2. On the other hand, due to the Plünnecke-Ruzsa inequalities, we know that if d 2A ≤ Kd A , we must have d 3A K 3 d A . Similarly, in the circle, if µ(2A) < 3µ(A) and µ(A) is small enough, Moskvin et al. [START_REF] Moskvin | Inverse problems of additive number theory and local limit theorems for lattice random variables. (Russian) Number-theoretic studies in the Markov spectrum and in the structural theory of set addition[END_REF] showed that A must satisfy strict structural conditions that imply µ(3A) ≤ 3(µ(2A) -µ(A)).

We solve partially the problem with k = 3. For general k, our understanding of E k and D k is yet poorer. Note that in general, our sets A ⊂ N satisfy d (k + 1)A ≥ k+1 k d kA , which, in view of the aforementioned result of Lev, may be inevitable. In the following statement we compile all the results that we can prove in this general context.

Theorem 1.7. Let (α, β, γ) ∈ ]0, 1]
Theorem 1.8. Let α = (α 1 , . . . , α k+1 ) ∈ [0, 1] k+1 , where k ≥ 1. a) If α 1 = • • • = α k-1 = 0 and α k+1 ≥ k+1 k α k , or α k+1 ≥ α k and α k+1 is the inverse of an integer, then α ∈ D k+1 . b) If α 1 = • • • = α k = 0, then α ∈ E k+1 . c) If α i = iα
for each i and some α ≤ 1/(k + 1), then α ∈ E 0 k+1 ⊂ D k+1 . The last item is obvious by taking an interval of length α, and was also proved somewhat differently for D k in [START_REF] Faisant | Additive Complements for a given Asymptotic Density[END_REF].

In the next section, we prove the complete description of D 2 and E 2 given in Theorems 1.1 and 1.5.

Sumsets in the integers

2.1. A preliminary reduction. We first show that Theorem 1.8 a) follows from the special case α k+1 = 1 stated in Proposition 1.2. Let α ∈ [0, 1] k+1 satisfy the hypothesis of Theorem 1.8 a), and let β = α k and γ = α k+1 . We distinguish several cases. a) We first assume that γ is an irrational number. Let A be the set given in Proposition 1.2 with parameters θ = 1 γ and β = β γ and A be defined by A = { θa : a ∈ A}.

Since A ⊂ T k,θ we have for any a 1 , . . . , a k+1 ∈ A

θa 1 + θa 2 + • • • + θa k+1 = θ(a 1 + a 2 + • • • + a k+1 ) .
Since θ > 1, we get d jA = θ -1 d jA , j = 1, 2, . . . , k + 1. This yields Theorem 1.8 a) when γ is an irrational number. b) If γ is the inverse of a positive integer q, we use again Proposition 1.2 with parameters θ = 1 γ and β = β γ to generate a set A. We define the set (3)

A (q) = {qa : a ∈ A} which satisfies d (k -1)A (q) = 0 < d kA (q) = β q < d (k + 1)A (q) = 1 q .
c) We finally assume that γ = s q is a rational number with 2 ≤ s < q. Upon multiplying numerator and denominator by appropriate numbers, we may assume that s = (k + 1)r for some integer r satisfying 3 ≤ r < q k+1 . Let U = {0, 1, . . . , r -2, r}. Then |jU | = jr for any j. Letting A = U + A (q) where A (q) is defined by (3), we thus obtain

             d (k -1)A = |(k -1)U | × d (k -1)A (q) = 0, d kA = |kU | × d kA (q) = krβ q = k k + 1 βγ , d (k + 1)A = |(k + 1)U | × d (k + 1)A (q) = (k + 1)r q = γ .
This concludes the proof of Theorem 1.8 a), assuming Proposition 1.2. We will now prove the latter, focussing first on the case k = 1 (so concerning twofold sumsets, that is Theorem 1.1), since it is much more simple than, while retaining some important features of, the general case, which we handle later. Theorem 2.1. For any sequence s j of elements of the torus T and any interval A, we have for any integers n and m the bound

1 n |{1 ≤ j ≤ n : s j ∈ A}| -µ(A) 1 m + 1 n m k=1 1 k n j=1 e 2iπsj k ,
where the implied constant is absolute.

Applying this with s j = {θj} for some irrational number θ and using the standard exponential sum bound

m j=1 e 2iπjθ ≤ 1 2 θ ,
where θ = min k∈Z |θ -k|, we obtain

1 n B θ,A (n) -µ(A) 1 m + 1 n m k=1 1 k θk , where (4) B θ,A (n) = 1 ≤ j ≤ n : {θj} ∈ A .
The series m k=1

1 k θk diverges as m tends to infinity, but selecting m = m(n) as a sufficiently slowly increasing function of n, one may achieve

1 n m(n) k=1 1 k θk 1 m(n) → 0
as n tends to infinity, and thus there exists a non increasing function η : N → R + (depending on θ only) which tends to zero such that

(5) 1 n B θ,A (n) -µ(A) ≤ η(n).
Note that the bound ( 5) is uniform in A; in particular, it is still valuable if A is replaced by a sequence A n of intervals of sufficiently slowly decaying measure (e.g.

µ(A n ) ≥ 2η(n))
. Also we note that using the sequence s j = {θ(j + X)}, we may obtain the more general bound

(6) 1 n B θ,A (n + X) -B θ,A (X) -µ(A) ≤ η(n)
for any integers X and n.

We now start the proof of Proposition 1.2 in the case k = 1. We will adopt a probabilistic construction. The setting corresponds to that given in [8, Chapter III, Theorem 13]. We shall use the notation P(A) for the probability measure of an event A and E(Z) for the expectation of a random variable Z.

Let θ be an irrational number, η be a non increasing function tending to 0 such that (5) holds.

Our aim is to study the number of representations of n under the form n = k 1 +k 2 with k 1 < n/2 < k 2 and k 1 , k 2 ∈ T 1,θ where T 1,θ is defined by (1). When {θn} < 1/2 we shall deduce from ( 5) the lower bound B θ,I (n/2) > n 2 ({θn} -η(n/2)) with I = ]0, {θn}[ . We thus need to assume that {θn} is sufficiently larger than η(n/2). Similarly when {θn} > 1/2 we need to assume {θn} is sufficiently smaller than 1 -η(n/2). So we are led to restrict our attention to those integers n that belong to

X θ = n ∈ N : 2η(n/2) < {θn} < 1 -2η(n/2)
. By ( 5) and the ensuing remarks we have [START_REF] Gyarmati | Double and triple sums modulo a prime[END_REF] d X θ = 1.

We now define our desired random sequence A. Let (ξ k ) k≥1 be a sequence of mutually independent Boolean random variables such that

P(ξ k = 1) = β k , k ≥ 1,
where β k is the constant sequence equal to β if β > 0 and the decaying sequence k -1/5 if β = 0. Let A be the random sequence consisting of the integers k ∈ T 1,θ such that ξ k = 1. Then the expectation of its counting function

A(x) = 0<k≤x ξ k satisfies E(A(x)) = k≤x {θk}<1/2 β k = O(x 4/5 ) if β = 0, βT 1,θ (x) if β > 0.
By the strong law of large numbers for the mean of mutually independent random variables (cf. [8, chapter III, Theorem 11]) we infer that almost surely andA(x) x 4/5 if β = 0. It follows that in both cases the density of A inside T 1,θ satisfies d T 1,θ (A) = β almost surely, as required. Now we prove that 2A ⊃ X θ \ F , where F is almost surely a finite set. This would imply that d 2A = 1. Let n ∈ X θ . We define

A(x) T 1,θ (x) ∼ β as x → ∞ if β > 0,
K n = {0 < k < n/2 : k ∈ T 1,θ ∩ (n -T 1,θ )}, and R(n) = k∈Kn ξ k ξ n-k .
Then by the independence of the ξ k 's ( 8)

P(R(n) = 0) = k∈Kn P(ξ k ξ n-k = 0) ≤ (1 -β 2 n ) |Kn| ≤ exp(-|K n | β 2 n ).
< {θn} < 1/2, we have {θ(n -k)} = {θn} -{θk} < 1/2. Thus k ∈ K n . This means that K n ⊃ 0 < k < n/2 : {θk} < {θn} , whence |K n | ≥ n 2 ({θn} -η(n/2)) ≥ n 2 η(n/2) by (5). If {θn} ∈ 1 -I = [1/2, 1 - 2η(n)[ instead,
it suffices to replace the condition {θk} < {θn} by 1 2 -{θk} < 1 -{θn} to obtain the same result.

One can choose η(n) to be arbitrarily slowly decaying, say η(n) ≥ n -1/2 . This way

|K n | n 1/2 , so that β 2 n |K n | n 1/10
and from (8) we get

n∈X θ P(R(n) = 0) < ∞.
We conclude by the Borel-Cantelli lemma (cf. [15, Lemma 1.2]) that almost surely, all but finitely many integers in X θ are sums of 2 terms from the random sequence A. The result follows from [START_REF] Gyarmati | Double and triple sums modulo a prime[END_REF]. This finishes the proof of Proposition 1.2 in the case where k = 1, and thus of Theorem 1.1.

We now determine which pairs (α,

β) ∈ R 2 with 0 < α ≤ β < 2α belong to D 2 , that is, we prove Theorem 1.5. Let A ⊂ N such that β = d 2A < 2d A = 2α
. By Kneser's theorem [START_REF] Kneser | Abschätzung der asymptotischen Dichte von Summenmengen[END_REF] there exists a (minimal) positive integer g such that x + g ∈ 2A for all but finitely many x ∈ 2A, and

d 2A ≥ 2d A - 1 g .
Let G = Z/gZ be the group of the residue classes modulo g and

A = x ∈ G : x ∩ A = ∅ , B = x ∈ A : |x ∩ A| = ∞ , C = x ∈ A : 0 < |x ∩ A| < ∞ = A \ B. Let A = x∈B (x ∩ N) ∪ x ∈ {0, 1, . . . , g -1} : x ∈ C . Then d A ≤ d A = |B| g , d 2A = |A + B| g .
If |A + B| < |A| + |B| -1 then by Kneser's theorem (cf. [START_REF] Kneser | Abschätzung der asymptotischen Dichte von Summenmengen[END_REF]; see equivalently [15, Theorem 5.5])

A + B would have a non trivial period in G x ∈ G : x + A + B = A + B = {0},
in contradiction with the minimality of g. We thus have

|A + B| ≥ |A| + |B| -1.
On the other hand our hypothesis d 2A

< 2d A implies |A| + |B| -1 < 2|B|. We deduce B = A and finally |2A| = 2|A| -1. Let r = |A|. Then β = 2r-1 g with 1 ≤ r ≤ g+1 2 . We get β 2 < α ≤ r g = β 2 + 1 2g .
We proved the following.

A = V + A (g) 0 .
Then A clearly admits the required density α. Finally we have 2A = 2V + 2A

Measures of sumsets in the circle

3.1. Twofold sumsets. To start with, we show that in order to achieve a large ratio µ(2A)/µ(A), a large number of connected components will be necessary.

Proof. Let A = k j=1 I j . So 2A = i≤j (I i + I j ). Let µ(I i ) = m i , so µ(I i + I j ) = m i + m j and µ(2A) ≤ i≤j (m i + m j ) = (k + 1) k i=1 m i . The equality case is clear.
We now attempt to prove the first item of Theorem 1.1 in the case α > 0. Let (α, β) ∈ ]0, 1] 2 satisfy β ≥ min(2α, 1). If β = min(2α, 1), the interval A = ]0, α[ satisfies µ(A) = α, µ(2A) = β. So we now suppose 0 < α < 1/2 and β > 2α.

First, note that for any k, if A = [0, ]∪{2 }∪• • •∪{(k -1) }, then 2A = [0, k ] so we can achieve a duplication ratio µ(2A)/µ(A) = k. The idea is then to somewhat "thicken" the singletons, in order to reduce the duplication ratio of the set.

Let k = β/α , thus k ≤ β/α < k + 1 and k ≥ 2.

Then let A = ]0, x[ ∪ {2x, . . . , kx}+ ] -, 0[ , for some x ≤ α and ≤ x/2 to be determined later. Note that

2A = ]0, (k + 1)x[ ∪ {(k + 2)x, . . . , 2kx}+ ] -2 , 0[ . Thus µ(A) = x + (k -1) and µ(2A) = (k + 1)x + 2(k -1)
. The doubling ratio is therefore

f ( /x) = (k + 1)x + 2(k -1) x + (k -1) = (k + 1) 1 + 2 k-1 k+1 x 1 + (k -1) x .
We have f (0) = k + 1 and while f (1/2) = 4k/(k + 1) ≤ k. Therefore by continuity of f , there is a value of the ratio y = /x for which the doubling ratio is the desired β/α. Then there remains to pick 3.2. Threefold sumsets. First we prove Theorem 1.6. We will derive it from the following theorem of Gyarmati, Konyagin and Ruzsa [START_REF] Gyarmati | Double and triple sums modulo a prime[END_REF].

x such that α = x + (k -1) = x(1 + (k -1)y), namely x = α 1+(k-1)

Proposition 3.2.

There exists an absolute constant c > 0 such that the following holds. Let p ≥ 29 be a prime. Let A ⊂ Z/pZ and let (n, s) = (|2A| , |3A|). If n < cp, then s ≥ 3n-1 2 . We derive the analogous result for measures in the circle T by a standard method. We first prove Theorem 1.6 for simple sets, that is, the union of finitely many closed intervals. Let A ⊂ T be a simple set. Let c be the constant given by Proposition 3.2 and suppose that µ(2A) < c. Let p ≥ 29 be a prime, that we will let tend to infinity ultimately. Let A p = j ∈ Z/pZ : j/p ∈ A .

One may check that |A p | = pµ(A) + O(1) as p tends to infinity. Further note that (kA) p = kA p for any k ∈ N. Since 2A and 3A are simple, one has |(kA) p | = pµ(kA) + O(1) for k = 2, 3; thus we have |(2A) p | < cp for p sufficiently large, so we can apply Proposition 3.2 and conclude in the case of simple sets. Now if A is closed (that is, compact), writing I δ = ]-δ, δ[ , we have A = δ>0 (A+ I δ ), in fact kA = δ>0 (kA + I kδ ) for any integer k ≥ 1. So for any fixed > 0, we can chose δ such that µ(kA+I kδ ) ≤ µ(kA)+ . Further, by compacity, there exists a simple set A (the union of finitely many translates of I δ ) such that A ⊂ A ⊂ A+I δ . We have

µ(3A) ≥ µ(3A ) -≥ 3 2 µ(2A) -.
Letting tend to zero, we conclude the proof of Theorem 1.6.

We prove Theorem 1.7. If α ≥ 1/3, the triplets (α, β, γ) that belong to E k are the ones for which β ≥ min(1, 2α) and γ = 1.

We now consider triplets where α < 1/3; we prove the following proposition, which implies Theorem 1.7.

Proposition 3.3. The set of triplets (µ(A), µ(2A), µ(3A)) for sets A ⊂ [0, 1/3] ⊂ T having at most two connected components is

(α, β, γ) ∈ [0, 1] 3 : β ∈ [2α, 3α], γ ∈ [3β/2, 2β -α[ or β = 3α, γ ∈ [3β/2, 2β] .
Proof. We may take A of the form ]0, x[ ∪ ]y, z[ for some 0

≤ x ≤ y ≤ z ≤ 1/3. So 2A = ]0, 2x[ ∪ ]y, x+z[ ∪ ]2y, 2z[ and 3A = ]0, 3x[ ∪ ]y, 2x+z[ ∪ ]2y, 2z +x[ ∪ ]3y, 3z[ .
We are seeking for which triplets (α, β, γ) the system    α =

x + z -y β = 3α -max(0, 2x -y) -max(0, x + z -2y) γ = 6α -max(0, 3x -y) -max(0, 2x + z -2y) -max(2z + x -3y, 0) admits solutions. We now discuss the existence of solutions according to the number of connected components of 2A and 3A, that is, for each max above, whether it is positive or not. In the following discussion, the necessary conditions we provide may always easily be seen to be sufficient, although we do not always explicitly state it.

1) If 2A is an interval, then so is 3A so γ = 3α = 3β/2.

2) If 2A has two connected components, so there is exactly one overlap between the intervals of 2A, we distinguish. a) If 2x > y and x + z < 2y, so 2A = ]0, x + z[ ∪ ]2y, 2z[ , we have β = x -2y + 3z. We have necessarily 3x > y and 2x + z > 2y, so 3A = ]0, 2z + x[ ∪ ]3y, 3z[ where the last two intervals may overlap or not. i) If they do, so 2z +x > 3y, we have γ = 3z. So β = x-2y +γ and α = x -y + γ/3. Get α -β = y -2γ/3 so y = α -β + 2γ/3 while x = 2α -β + γ/3. We check that the inequalities are satisfied: 2x-y = 3α-β > 0 so β < 3α, 2y-x-z = -β+2γ/3 > 0 implies γ > 3β/2. Further, we need 2z +x-3y = -α+2β -γ > 0 which amounts to 3β/2 < γ < 2β -α < 5α. Conversely, whenever these conditions are satisfied, the system has solutions. ii) Ohterwise we have 2z + x < 3y, hence γ = 3(z -y) + 2z + x.

Thus a solution exists if and only if γ = 2β -α. b) Now if 2x < y and x + z > 2y, so 2A = ]0, 2x[ ∪ ]y, 2z[ , we have β = 2x -y + 2z. We have necessarily 2x + z > 2y and 2z + x > 3y, so 3A = ]0, 3x[ ∪ ]y, 3z[ , where the two intervals may or not overlap. i) If they do, so 2x < y < 3x, we have γ = 3z. Further we find y = β -2α, and x = β -α -γ/3. So y -2x = -β + 2γ/3 > 0 implies yet again γ > 3β/2. Further y -3x = -2β + α + γ < 0 implies γ < 2β -α. Also x + z -2y = 3α -β > 0 amounts to β < 3α. ii) Otherwise, so y > 3x, we find γ = 3z -y + 3x = 3α + 2y and again γ = 2β -α. 3) If 2A has three connected components (no overlap), then β = 3α. We have 2x < y and x+z < 2y. We distinguish according to the presence of overlaps or not in 3A. a) If there is no overlap, we have γ = 6α. It is realisable, just take x, then y > 3x, then y < z < min((3y -x)/2, 1/3), then all constraints are realised. We can achieve that for any value of α ≤ 1/6. b) If 3A is connected, γ = 3z. Now the conditions 2x < y and x + z < 2y imply z < 3(y -x), which is equivalent to 2z > 3(x+z -y), and finally γ > 3β/2. c) If there is exactly one overlap, that is, if 3A has three connected components, we distinguish. i) Suppose 3x > y. So 2x + z < 2y and 2z + x < 3y. Then γ = 6α -3x + y. This imposes γ ∈ ]5α, 6α[ = ]5β/3, 2β[ . ii) Now suppose 2x + z > 2y. So y > 3x and 2z + x < 3y. Then γ = 6α -2x -z + 2y = 5α -x + y > 5α. iii) If only the last gap is overcome, γ = 6α -2z -x + 3y = 5αz + 2y > 5α. d) If 3A has two connected components, we distinguish. i) If all but the last gap are overcome, γ = 6α-3x+y-2x-z+2y = 5α -4x + 2y > 5α. ii) If all but the middle gap are overcome, γ = 6α -3x + y -2z -

x + 3y = 5α -3x -z + 3y > 5α. iii) If all but the first gap are overcome, γ = 6α -2x -z + 2y -2z -

x + 3y > 5α.

Regarding sets with k connected components when k ≥ 3, the determination of the possible triplets (α, β, γ) becomes untractable by this method. Nevertheless, we can easily see that the structure of the set of the possible triplets remains similar, that is, a connected union of finitely many (in fact O k (1) many) polytopes, where a polytope is the intersection of finitely many half-spaces.

3.3. Further iterated sumsets. We now prove Theorem 1.8 b). Let β ∈ (0, 1] and k ≥ 1 an be integer. We argue as in the proof of Theorem 1.1 b) in the particular case α = 0. Let C k+2 be the Cantor set of initial segment [0, 1] ⊂ R and ratio of dissection 1/(k + 2). It is known [START_REF] Cabrelli | Sums of Cantor sets[END_REF]Corollary 2.3] that µ(kC k+2 ) has measure 0 whereas (k + 1)C k+2 = [0, k + 1]. The suitable re= of C k+2 by the factor β/(k + 1) provides the desired construction.

Note that this does not imply Theorem 1.8 a) in the particular case α k = 0 since the openness condition of Lemma 1.4 may not be removed. Indeed, if A ⊂ T has measure zero, one may see that B λ,A is empty for almost all λ ∈ R \ Q, since the map λ → nλ on the circle is measure-preserving for any integer n. So we need the specific argument based on Proposition 1.2 we gave in Section 2 when k = 1. The general case k ≥ 2 will follow from Proposition 4.6.

Iterated sumsets in the integers

We now prove Proposition 1.2 for k ≥ 2. The (probabilistic) argument we will use subsumes, but is significantly more complicated than, the one used in Section 2, which is why we preferred to present it separately. First of all we collect a number of useful but technical results. 4.1. Preliminary lemmas. First we need to somewhat generalise the bound (6) obtained via the Erdős-Turán theorem. Proposition 4.1. Let k, D, M, X be integers. Let f = k i=1 P i 1 Ii where (I i ) i≤k is a family of pairwise disjoint intervals in [0, 1[ and P i a polynomial of degree less than D whose coefficients are all at most M . Then

1 N X<n≤N +X f ({θn}) -f = O(M Dk η(N )).
A function f satisfying the above hypothesis will naturally be referred to as piecewise polynomial.

Proof. It suffices to prove it for monomials and for k = 1, the general case following by linear combinations (incurring an extra factor M k). Thus let a < b be in [0, 1[ , and let d ≤ D and f be defined by f (x) = x d 1 (a,b) . Using the bound (6), we note that

a d ((b -a) -O(η(N )) ≤ 1 N X<n≤N +X {θn} d 1 (a,b) ({θn}) ≤ b d ((b -a) + O(η(N ))
Further, observe that

a d (b -a) ≤ b a x d dx ≤ b d (b -a). Hence (a d -b d )(b -a) -O(η(N )) ≤ 1 N X<n≤N +X {θn} d 1 (a,b) ({θn}) - b a x d dx ≤ (b d -a d )(b -a) + O(η(N )).
Given that b d -a d ≤ d(b -a), we find that

X<n≤N +X {θn} d 1 (a,b) ({θn}) - b a x d dx ≤ d(b -a) 2 + O(η(N )). Then splitting the interval [a, b] into O( η(N ) -1
) consecutive intervals of size η(N ) , we obtain, for each of these intervals, an error term of size O(dη(N )), and so in total, an error term of size O(D η(N )).

A certain type of sums will appear in the sequel, for which we now give an asymptotic.

Lemma 4.2. Let 0 < α < 1, 0 ≤ β < ∞ and (9) J N (α, β) := 0<x<N 1 x α (N -x) β . Then J N (α, β) =      B(1 -α, 1 -β)N 1-α-β + O(N -min(α,β) ) if β < 1, N -α log N + O(N -α ) if β = 1, ζ(β)N -α + O(N -α-1+1/β ) if β > 1,
where B(•, •) denotes the Euler beta function defined by

B(x, y) = 1 0 t x-1 (1 -t) y-1 dt and ζ(•) is the Riemann zeta function.
This can be proved by considering Riemann sums; we omit the standard details. The beta function satisfies the following functional equation involving Euler's gamma function:

B(x, y) = Γ(x)Γ(y) Γ(x + y) .
By induction, we may achieve the following simple lemma.

Lemma 4.3. Let (α 1 , . . . , α s ) ∈]0, 1[ s . Then 1≤u1,...,us≤n i ui=n i u -αi i = O(n s-1-i αi ).
Further, let : N → R + tend to 0. Then there exists a sequence depending only on that tends to zero such that 1≤u1,...,us≤n i ui=n

(u 1 ) i u -αi i = (n)n s-1-i αi .
Proof. We prove the second part for s = 2, the rest following by a simple induction. Let K δ be such that for all k ≥ K δ , we have (k) ≤ δ. Further let M be an upper bound for . Then

k<n (k)k -α1 (n -k) -α2 ≤ M k<K δ k -α1 (n -k) -α2 + δ k<n k -α1 (n -k) -α2
The right-hand side is O(K 1-α1-α2 δ +δn 1-α1-α2 ) by Lemma 4.2. We have K δ → ∞ (unless (k) = 0 eventually) as δ → 0, but choosing δ as a sufficiently slowly decaying function of n, we can make the error term as small as o(n 1-α1-α2 ) as desired.

For any real number 0 ≤ x ≤ 1 and any integer 1

≤ j ≤ k -1, let a j (x) = max 0, x - j k + 1
, b j (x) = min x, j k + 1 and I j (x) be the open interval

I j (x) = ]a j (x), b 1 (x)[ . Let f 1 = 1 [0,1[ and f j+1 (x) = b1(x) aj (x) f j (x -y)dy = bj (x) a1(x) f j (y)dy, 1 ≤ j ≤ k -1.
Then for any 1 ≤ j ≤ k -1 i) a j and b j are piecewise affine. Further a j (x) + b j (x) = x.

ii) µ(I j (x)) = b 1 (x) -a j (x) = max 0, min x, 1 k+1 , j+1 k+1 -x . As a result, f j is supported on ]0, j k+1 [ . iii) f j is a non negative, nonzero piecewise polynomial function. In fact f j has only finitely many zeros on ]0, j k+1 [ .

We will need the following estimate.

f j (x -{θu}) 1 u α 1 (N -u) β = J N (α, β)(f j+1 (x) + O(η (N ))),
where η is a function N → R + which tends to zero and depends only on θ.

Proof. We decompose the interval of summation [1, N [ into subintervals of some length m = f (N ) tending to infinity rather slowly, m = o(N ) at any rate, even m N o (1) but not too slowly either; we fix m = η(N ) -1/2 for definiteness. We write

[1, N [ = 0≤k< N m ]km, (k + 1)m] ∪ N m m, N
where the last interval has at most m elements.

Let K = N m . Let a = -α and b = -β. We note that n∈ ] N m m,N [ n a (N -n) b ≤ m(Km) a .
Denoting by S the sum to estimate, this implies that 1) ).

S = 0≤k<K n∈ ]km,(k+1)m] {θn}∈Ij (x) f j (x -{θn})n a (N -n) b + O(N a+o(
Also we note that when n ∈ ]km, (k + 1)m], the expression n a (N -n) b may be regarded as approximately constant, more precisely

n a (N -n) b = m a+b k a (K -k) b (1 + O(1/k))(1 + O(1/(K -k))).
We may restrict the sum over k to reasonably large k, like between √ K and K-√ K; indeed, we have

0<u≤m √ K u a (N -u) b ≤ (N -N 1/2+o(1)
) b N a/2+1/2+o(1) = N b+a/2+1/2+o (1) which is negligible to N a+b+1 . We may argue analogously to discard the sum over

k ≥ K - √ K. This way (1 + O(1/k))(1 + O(1/(K -k)) = 1 + O(1/ √ K) for any k considered. Thus S, up to an error O(N a+b+1 / √ K)), equals (10) m a+b (1 + O(1/ √ K)) √ K≤k<K- √ K k a (K -k) b n∈ ]km,(k+1)m] {θn}∈Ij (x)
f j (x -{θn}).

We now apply Proposition 4.1 to the inner sum. By definition of f j+1 , we obtain n∈ ]km,(k+1)m] {θn}∈Ij (x)

f j (x -{θn}) = m(f j+1 (x) + O(η( √ m))).
Injecting that in [START_REF] Landreau | Etude probabiliste des sommes de s puissances s-ièmes[END_REF], we find that

S = m a+b+1 (1+O(1/ √ K))(f j+1 (x)+O(η( √ m))) √ K≤k<K- √ K k a (K -k) b +O(N c )
for some c < a + b + 1. Now we have from ( 9)

m a+b+1 √ K≤k<K- √ K k a (K -k) b = J N (-a, -b) + O(N c )
by the same arguments as above. Finally, upon gathering all error terms together (whereby the term in O(η( √ m)) provides the largest one), we obtain the desired conclusion.

We are now ready to show the following key estimate. Lemma 4.5. There exists a positive function η depending only on θ and k, and which tends to 0 at infinity, such that for any integer n we have

(11) S k (n) := 0<u1<•••<u k <n ∀i, ui∈T k,θ n=u1+•••+u k (u 1 • • • u k ) -1+1/k = λ k f k ({θn}) + O (η (n)) , where λ k = Γ( 1 k ) k k! . Proof. Let E k (n) := 0<u1,••• ,u k <n ∃i =j:ui=uj n=u1+•••+u k (u 1 • • • u k ) -1+1/k and S k (n) := 0<u1,...,u k <n ∀i, ui∈T k,θ n=u1+•••+u k (u 1 • • • u k ) -1+1/k , so that S k (n) = O(E k (n)) + k!S k (n). We observe that E k (n) = O(n -1/k
). Further, reformulating the diophantine constraints using the intervals I j , we have the decomposition

(12) S k (n) = u1<n {θu1}∈I k-1 ({θn}) u -1+1/k 1 u2<n-u1 {θu2}∈I k-2 ({θ(n-u1)}) u -1+1/k 2 • • • • • • u k-1 <n-u1-•••-u k-2 {θu k-1 }∈I1({θ(n-u1-•••-u k-2 )}) u k-1 (n -u 1 -• • • -u k-1 ) -1+1/k .
To simplify the notation, let us denote

n j = n -u 1 -• • • -u k-j , thus n k = n and n j = n j+1 -u k-j .
We shall prove by induction on j ≤ k that

(13) S k (n) = C j 0<u1<n {θu1}∈I k-1 ({θn}) u -1+1/k 1 0<u2<n1 {θu2}∈I k-2 ({θn1}) u -1+1/k 2 • • • • • • 0<u k-j <nj+1 {θu k-j }∈Ij ({θnj+1}) u -1+1/k k-j (n j+1 -u k-j ) -1+j/k f j ({θn j }) + j (n) where C j = j-1 i=1 B 1 k , i k
and j tends to 0. When j = k, there is no more summation at all and (13) boils down to C k f k ({θn}) + k (n), which is the desired result since

C k = k-1 j=1 B 1 k , j k = k-1 j=1 Γ( 1 k )Γ( j k ) Γ( j+1 k ) = Γ 1 k k .
Equation ( 12) is the j = 1 case. We now suppose that (13) holds for some j ≤ k -1. Let A j (n) be the main-term of the right-hand side of (13). Using Lemma 4.4 on the innermost sum, and reparametrising by writing n j+1 = v 1 and u i = v i+1 in the error term, we find

A j (n) = A j+1 (n) + O v1,...,v k-j ≤n vi=n η (v 1 )v -1+ j+1 k 1 k-j i=2 v -1+1/k i .
By Lemma 4.3 and since η tends to 0, the expression between brakets in the above equation is certainly o(1). This concludes the induction step and therefore the proof of the lemma.

4.2. The construction. We argue as in Section 2 by the probabilistic method (see [START_REF] Halberstam | Sequences. Second edition[END_REF]Chapter III]). Let c > 0 and ξ m , m ≥ 1, be a sequence of independent Boolean random variables such that

P(ξ m = 1) = c m 1-1/k .
Let S be the random increasing sequence of the m's such that ξ m = 1. This is essentially a sequence of pseudo k-th powers. These objects have been well studied since their introduction by Erdős and Renyi [START_REF] Erdős | Additive properties of random sequences of positive integers[END_REF]. In particular Goguel [START_REF] Goguel | Über Summen von zufälligen Folgen natürlischen Zahlen[END_REF] computed the (almost sure) density of kS and Deshouillers and Iosifescu [START_REF] Deshouillers | Sommes de s + 1 pseudo-puissances s-ièmes[END_REF] found that the density of (k + 1)S is almost surely 1. Now we let A = S ∩ T k,θ , where T k,θ was defined by equation [START_REF] Cabrelli | Sums of Cantor sets[END_REF]. From now on we will suppose θ is irrational; if θ is an integer, T k,θ = N so A = S and the previous references apply. The treatment of this simpler case may still be read out from our proofs by discarding all the (then vacuous) diophantine conditions. The next proposition implies Proposition 1.2. ) we infer that with probability 1, A(x) ∼ x 1/k when x → ∞, thus for any 1 ≤ j ≤ k -1 (jA)(x)

x j/k , as x tends to infinity.

It follows that d jA = 0 almost surely.

b) Let n be a positive integer and observe that 0 < {θn} < 1. We denote I(t, k) the open interval

I(t, k) = max 0, t k - 1 k(k + 1) , min t k , 1 k + 1 . and R k+1 (n) = 0<u1<•••<u k <u k+1 <n n=u1+•••+u k+1 {θui}∈I({θn},k), (1≤i≤k) ξ u1 • • • ξ u k ξ u k+1 . Then R k+1 (n) > 0 implies that n ∈ (k + 1)A. Moreover R k+1 (n) = 0 = 0<u1<•••<u k <u k+1 <n n=u1+•••+u k+1 {θui}∈I({θn},k), (1≤i≤k) ξ u1 • • • ξ u k ξ u k+1 = 0 .
We denote by U(n) the set of the ordered (k + 1)-uples u such that n = k+1 i=1 u i and {θu i } ∈ I({θn}, k), i = 1, . . . , k.

The events A(u) = ξ u1 . . . ξ u k ξ u k+1 = 1 , u ∈ U(n), are not necessarily pairwise independent: for distinct (k + 1)-tuples u, v, the events A(u) and A(v) are not independent if and only if u ∼ v, where the notation ∼ means u i = v j for some i, j. We firstly have

µ n = c k+1 0<u1<•••<u k <n {θui}∈I({θn},k) (u 1 • • • u k (n -u 1 -• • • -u k )) -1+1/k .
The summand in the inner-sum is at least n k+1 -k+1/k , hence

µ n ≥ c k+1 B θ,I (n) k -k 2 B θ,I (n) k-1 k! n k + 1 -k+1/k ≥ c k+1 B θ,I (n) k - k 2 B θ,I (n) k-1 n -k+1/k
where I = I({θn}, k) and B θ,I (n) is defined by [START_REF] Erdős | On a problem in the theory of uniform distribution[END_REF]. By equation ( 5),

B θ,I (n) n ≥ min {θn} k , 1 k(k + 1) , 1 -{θn} k -η(n).
Hence if 2kη(n) < {θn} < 1 -2kη(n), we have Notice that if s + r = k + 1 with s > 0, then s -2(k -r) > 0 implies t = 2 -s = 1 and s = 1. We can now inject our upper bounds for ∆ n (s, r) in equation ( 16), in which the main contribution is given by s = 1, from the above discussion. We get

∆ n k c 2k+1 n 1/k + O k,c (1)
.

By ( 14) and ( 15) with the Borel-Cantelli lemma, we infer that almost surely, all but finitely many integers n such that 2kη(n) < {θn} < 1 -2kη(n) are sums of k + 1 members of A and that d (k + 1)A = 1 since their complementary set in N, namely n ∈ N : 0 ≤ {θn} ≤ 2kη(n) ∪ n ∈ N : 1 -2kη(n) ≤ {θn} < 1 has density 0. c) Let n such that 0 < {θn} < k/(k + 1). We consider

(17) R k (n) := k! 0<u1<•••<u k <n ui∈T k,θ n=u1+•••+u k ξ u1 • • • ξ u k
that is the random variable counting the representations of n as a sum of k distinct members of A. The key result is Lemma 4.5.

As in the study of R k+1 (n) in the previous paragraph we need to show that the probabilistic dependence between the events {ξ u1 • • • ξ u k = 1} is not too large. We

2. 2 .

 2 Twofold sumsets. Before embarking on the proof of Proposition 1.2 in the case k = 1, we need a quantitative version of Weyl's criterion (Theorem 1.3), due to Erdős and Turán [4, Theorem III].

Proposition 2 . 2 . 2

 222 Let A such that d 2A < 2d A . Then there exist two positive integers g and r ≤ g+1 β ∈ ]0, 1[ ∩ Q have non positive dyadic valuation, and g be the smallest positive integer for which gβ is odd, thus β = 2r-1 g for some integer r ≥ 1. Let α satisfy β Then let V = {0, . . . , r -1}, so that |2V | = 2r -1 ≤ g. Let β 0 = αg/r, thus β 0 ∈ ]0, 1]. We apply Proposition 1.2 with k = 1 to get a set A 0 such that d A 0 = β 0 and d 2A 0 = 1. The associated rescaled set A (g) 0 = {ga : a ∈ A 0 } satisfies d A (g) 0 = β 0 /g = α/r and d 2A (g) 0 = 1/g. Finally let A be the set

5 . 2 . 3 .

 523 = 2r-1 g = β as desired. This completes the proof of Theorem 1.Example Assume that there exists a set A such that α = d A = 4/9 and β = d 2A = 5/9. Applying Proposition 2.2 we get g ≤ 3 and r ∈ {1, 2}, whence β = 1, 1/2 or 1/3, a contradiction. Example 2.4. Fixing α = 1/5 and β = 3/10 in Theorem 1.5 yields g 0 = 10. Further the bounds β < 2α ≤ β + 1/g 0 are satisfied hence there exists a set A such that d A = 1/5 and d 2A = 3/10. The only admissible choice for g, r in Proposition 2.2 is g = 10 and r = 2.

Lemma 3 . 1 .

 31 Let A be a disjoint union of k intervals. Then µ(2A) ≤ (k + 1)µ(A). If the intervals are open, the equality case happens when all the k+1 2 intervals of the sum are pairwise disjoint.

  y , and then the corresponding . In the case α = 0, a radically different construction will be necessary. Let C ⊂ [0, 1] be the classical ternary Cantor set. It is well known that C + C = [0, 2] (cf. [1, Corollary 2.3]). Scaling C by a factor β/2 and projecting it to the circle, we obtain the set A = {βc/2 : c ∈ C} of measure 0 such that 2A = [0, β], thus µ(2A) = β.

Lemma 4 . 4 .

 44 Let (α, β) ∈ ]0, 1[ 2 .Let θ > 1 be irrational and x ∈ ]0, 1[ . Then for any j, we have 0<u<N {θu}∈Ij (x)

Proposition 4 . 6 .

 46 Almost surely we have a) d jA = 0, for any j = 1, . .. , k -1, b) d (k + 1)A = 1, c) d kA = k k+1 -F k (c) where F k (c) is a continuous function and increasing from 0 to k/(k + 1) when c is decreasing from ∞ to 0.Proof. a) By the appropriate version of the strong law of large numbers (cf.[START_REF] Halberstam | Sequences. Second edition[END_REF] chapter III, Theorem 11]

2 n 2 (

 22 Let µ n = u∈U (n) P(A(u)), ∆ n = u =v∈U (n) u∼v P(A(u) ∩ A(v)).By Janson's inequality [15, Theorem 1.28] (14)P(R k+1 (n) = 0) ≤ exp -µ µ n + ∆ n ) .

( 15 )c 2 .

 152 µ n ≥ (1 -o(1))c k+1 n 1/k η(n) k .Now we examine ∆ n . By a discussion according to the number s ≤ k -1 of positions where two distinct (k + 1)-tuples in U(n) agree, and ignoring the diophantine conditionss+2(k+1-s) ∆ n (s, k + 1 -s), where ∆ n (s, r) := 0<u1,...us<ns i=1 ui<n (u 1 • • • u s ) -1+1/k    0<v1,...vr<n n=u1+•••+us+v1+•••+vr (v 1 • • • v r ) -1+1/k   Applying Lemma 4.3, we see that the inner sum is (n -u 1 -• • • -u s ) -1+r/k. For every fixed tuple (u 1 , . . . , u s-1 ) in the sum above, we now apply Lemma 4.2 to the sumus<n-s-1 i=1 ui u -1+1/k s (n -u 1 -• • • -u s ) -1+r/k . If 1 -r/k ≥ 1/2, we obtain ∆ n (s, r) log n 0<u1,...us<n n=u1+•••+us (u 1 • • • u s ) -1+1/k log n n 1-s/kwhere we used Lemma 4.3 for the second inequality. If 1 -r/k < 1/2 then by Lemmas 4.2 and 4.3 again∆ n (s, r) 0<u1,...us<n u1+•••+us<n (u 1 . . . u s ) -1+1/k (n -u 1 -u 2 -• • • -u s ) -1+(2r/k-1) (2r/k-1)+s/k 1 if s ≤ 2(k -r),0<u1,...ut<n u1+•••+ut<n (u 1 • • • u t ) -1+1/k n t/k if t := s -2(k -r) > 0.

  Suppose for instance {θn} ∈ I, the case {θn} ∈ 1 -I is similar. Then for any k such that {θk}

Now we need a lower bound for |K n |. By definition k < n/2 belongs to K n if and only if {θk} < 1/2 and {θ(n -k)} < 1/2. Let I = ]2η(n/2), 1/2[ . Since n ∈ X θ , we have {θn} ∈ I ∪ (1 -I).

Acknowledgments

The authors are thankful to Georges Grekos for useful conversations and to the anonymous referee for valuable suggestions.

This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

shall use Landreau's work on sums of k pseudo k-th powers (cf. [START_REF] Landreau | Etude probabiliste des sommes de s puissances s-ièmes[END_REF]Lemma 1 (i) and Lemma 5 (iii)]):

Let ζ n , n ≥ 1, be the sequence of Boolean random variables defined by

and

By (18) we have

Hence the expectation of X n satisfies

We get by Theorem 1.3 and the fact that f k is supported on ]0, k/(k + 1)[ the asymptotic

We denote by F k (c) the above integral.

We follow the arguments used in the proof of [8, chapter III, Theorem 4 (iii)] or alternatively [START_REF] Landreau | Etude probabiliste des sommes de s puissances s-ièmes[END_REF]Section 4] to estimate the variance V(X N ). We may ignore the diophantine conditions in (17), the only resulting effect being to increase the related variance. We finally get V(X N ) = O(N -1/k ) and consequently by [8, chapter III, lemma 34] that with probability 1, lim

Hence almost surely d kA = k k+1 -F k (c). Observing that f k is a non negative piecewise polynomial function that has finitely many zeros on ]0, k/(k + 1)[ , we see that F k (c) is a decreasing continuous function satisfying lim c→0 + F k (c) = k/(k + 1) and lim c→∞ F k (c) = 0; this ends the proof of Proposition 4.6.