François Dubois
email: francois.dubois@cnam.fr

Dimitri Stoliaroff
email: dimitri.stoliaroff@astrium.eads.net

Isabelle Terrasse
email: isabelle.terrasse@airbus.com

Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics

Keywords: Stokes, Zhukovsky, Fraeijs de Veubeke, vector fields, integral boundary operator. AMS classification: 70E99, 76B07

Fluid motion in tanks is usually described in space industry with the so-called Lomen hypothesis which assumes the vorticity is null in the moving frame. We establish in this contribution that this hypothesis is valid only for uniform rotational motions. We give a more general formulation of this coupling problem, with a compact formulation.

We consider the mechanical modeling of a rigid body with a motion of small amplitude, containing an incompressible fluid in the linearized regime. We first establish that the fluid motion remains irrotational in a Galilean referential if it is true at the initial time. When continuity of normal velocity and pressure are prescribed on the free surface, we establish that the global coupled problem conserves an energy functional composed by three terms. We introduce the Stokes -Zhukovsky vector fields, solving Neumann problems for the Laplace operator in the fluid in order to represent the rotational rigid motion with irrotational vector fields. Then we have a good framework to consider the coupled problem between the fluid and the rigid motion. The coupling between the free surface and the ad hoc component of the velocity potential introduces a "Neumann to Dirichlet" operator that allows to write the coupled system in a very compact form. The final expression of a Lagrangian for the coupled system is derived and the Euler-Lagrange equations of the coupled motion are presented.

Scope of the problem

Sloshing of liquid in tanks is an important phenomenon for space and terrestrial applications. We think for example of sloshing effects in road vehicles and ships carrying liquid cargo. The question is to know the magnitude of the wave and the total effort on the structure due to the movement of the fluid. For this kind of problematics, a lot of references exist and we refer the reader i.e. to the book of H. Morand and R. Ohayon [START_REF] Morand | Interactions fluides structures[END_REF], to the review proposed by R. Ibrahim, V. Pilipchuk and T. Ikeda [START_REF] Ibrahim | Recent Advances in Liquid Sloshing Dynamics[END_REF], to the book of R.A. Ibrahim [START_REF] Ibrahim | Liquid Sloshing dynamics: theory and applications[END_REF], the book of O.M. Faltinsen and A.N. Timokha [START_REF] Faltinsen | Sloshing[END_REF] or to the review article of G. Hou et al. [START_REF] Hou | Numerical methods for fluid-structure interaction -a review[END_REF].

Moreover, for industrial applications, we would have a movable rigid tank with liquid free surface with six possible rigid movements and without needing a complete study of the elastic body, as studied e.g. in H. Bauer et al. [START_REF] Bauer | Interaction of a Sloshing Liquid With Elastic Containers[END_REF], S. Piperno et al. [START_REF] Piperno | Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and twodimensional application[END_REF], C. Farhat et al. [START_REF] Farhat | Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity[END_REF], J.F. Gerbeau and M. Vidrascu [START_REF] Gerbeau | A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows[END_REF], K.J. Bathe and H. Zhang [START_REF] Bathe | Finite element developments for general fluid flows with structural interactions[END_REF], T.E. Tezduyar et al. [START_REF] Tezduyar | Space-time finite element techniques for computation of fluid-structure interactions[END_REF] and the previous references.

In our case relative to space applications, the fundamental hypothesis of this contribution is the existence of some propulsion. We do not consider in this study the very complicated and nonlinear movement due to the quasi-disparition of gravity field. We refer for such studies to the contributions of F. Dodge and L. Garza [START_REF] Dodge | Studies of propellant sloshing under low-gravity conditions[END_REF][START_REF] Dodge | Experimental and Theoretical Studies of Liquid Sloshing at Simulated Low Gravity[END_REF], S. Ostrach [START_REF] Ostrach | Low-Gravity Fluid Flows[END_REF], H. Snydera [START_REF] Snydera | Sloshing in microgravity[END_REF], C. Falcón et al. [START_REF] Falcón | Capillary wave turbulence on a spherical fluid surface in low gravity[END_REF] and P. Behruzi, et al. [START_REF] Behruzi | Ballistic Phase Management for Cryogenic Upper Stages[END_REF] among others. On the contrary, a gravity field is supposed to be present in our contribution and moreover an extra-gravity field is added due to the propulsion system. Then it is legitimus to linearize all the geometrical deformations and the equations of dynamics. In this kind of situation, the knowledge of the action of the fluid on the structure is mandatory. The question has been intensively studied during the sixties under the impulsion of NASA (see e.g. H. Bauer [START_REF] Bauer | Fluid Oscillations in the Containers of a Space Vehicle and Their Influence on Stability[END_REF], D. Lomen [START_REF] Lomen | Liquid propellant sloshing in mobile tanks of arbitrary shape[END_REF][START_REF] Lomen | Digital Analysis of Liquid Propellant Sloshing in Mobile Tanks with Rotational Symmetry[END_REF], H. Abramson [START_REF] Abramson | The dynamic behavior of liquids in moving containers, with applications to space vehicle technology[END_REF], L. Fontenot [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF]) and in European countries in the seventies (see e.g. J.P. Leriche [START_REF] Leriche | Ballottements des liquides dans un réservoir de révolution[END_REF]) or in the context of Ariane 5 studies (B. Chemoul et al. [START_REF] Chemoul | Ariane 5 flight environments[END_REF]).

We observe that due to its own intrinsic movement, the structure has also some influence on the fluid displacement. This question has been rigorously studied by the Russian school in the sixties (N. Moiseev and V. Rumiantsev [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF]). It is sufficient in a first approach to consider the solid as a rigid body and to neglect all the flexible deformations.

In fact, we are in front of a complete coupled problem. The fluid is linearized and has an action on the solid, considered as a rigid body. The solid is a "six degrees of freedom" system that can also be considered as linearized around a given configuration. This coupled problem does not seem to have been considered previously under this form in the literature. We observe that this quite old problem raises actually an intensive scientific activity. As examples, we mention the contributions of O. Faltinsen, O. Rognebakke, I. Lukovsky and A. Timokha [START_REF] Faltinsen | Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth[END_REF][START_REF] Gavrilyuk | Eigenoscillations of threeand two-element flexible systems[END_REF] who derived a variational method to analyze the sloshing with finite water depth. Note also that K. London [START_REF] London | A fully coupled multi-rigid-body fuel slosh dynamics model applied to the Triana stack[END_REF] analyzed the case of a multi-body model with applications to the Triana spacecraft, and J. Vierendeels at al.  [START_REF] Vierendeels | Analysis and Stabilization of Fluid-Structure Interaction Algorithm for Rigid-Body Motion[END_REF] proposed to use the Flow3D computer software (Fluent, Inc) to analyze numerically nonlinear effects involved in the coupling of a rigid body with sloshing fluid, L. Diebold at al. [START_REF] Diebold | Effects on sloshing pressure due to the coupling between seakeeping and tank liquid motion[END_REF] studied the effects on sloshing pressure due to the coupling between seakeeping and tank liquid motion. In the thesis of A. Ardakani, the general rigid-body motion with interior shallow-water sloshing is studied in great detail and we refer to the communication of A. Ardakani and T. Bridges [START_REF] Ardakani | Dynamic coupling between shallow-water sloshing and horizontal vehicle motion[END_REF]. A time-independent finite difference method to solve the problem of sloshing waves and resonance modes of fluid in a tridimensional tank is also considered by C. Wu and B. Chen [START_REF] Wu | Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method[END_REF].

We begin this article with classical considerations on sloshing in a fixed solid. We focus on the free surface and to usual physical ingredients: the continuity of normal velocity and the continuity of pressure. The coupling between the free surface and the velocity potential introduces a "Neumann to Dirichlet" integral operator that allows to write the coupled system in a very compact form. Then in Section 2, we recall fundamental aspects of the dynamics of a six degrees of freedom rigid body dynamics: description of the rigid body and its infinitesimal motion, the incompressible fluid and its linearization. We discuss the so-called "Lomen hypothesis" intensively used for industrial space applications and prove that, with a good generality, the fluid motion remains irrotational in a Galilean referential. We introduce some special vectorial functions that we call the "Stokes -Zhukovsky vector fields", independently rediscovered by multiple generations of great scientists during the two last centuries (see e.g. G. Stokes [START_REF] Stokes | On some cases of fluid motion[END_REF], N. Zhukovsky [START_REF] Zhukovsky | On the motion of a rigid body having cavities, filled with a homogeneous liquid drops[END_REF] and B. Fraeijs de Veubeke [START_REF] Fraeijs De Veubeke | The inertia tensor of an incompressible fluid bounded by walls in rigid body motion[END_REF]). These vector fields solve Neumann problems for the Laplace operator in the fluid and allow the representation of a rigid body displacement by an irrotational field. It is a good framework to consider the coupled problem. Then the dynamics equations of the rigid body in the presence of an internal sloshing fluid are established. In Section 3, we study the coupled problem. We do not incorporate any dissipation and in consequence we establish the conservation of energy for this simple case. We propose a compact set of variables to describe the entire coupled dynamics. Then the coupled system appears in a very simple form formally analogous to a scalar harmonic oscillator! Finally, the expression of a Lagrangian for the coupled system is proposed.

When the explanations of the mathematical results are not detailed, we refer the reader to the classical books of N. Moiseev and V. Rumiantsev [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF], H. Morand and R. Ohayon [START_REF] Morand | Interactions fluides structures[END_REF], R.A. Ibrahim [START_REF] Ibrahim | Liquid Sloshing dynamics: theory and applications[END_REF], O.M. Faltinsen and A.N. Timokha [START_REF] Faltinsen | Sloshing[END_REF] or to the preliminary edition [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF] of this contribution.

1) Sloshing in a fixed solid

In this section, the studied mechanical system is the fluid. The liquid is contained inside the solid S, it occupies a volume Ω(t) variable with time, with a constant density ρ L . The total mass m L of liquid is the integral of the density ρ L on the volume Ω(t). At the boundary ∂Ω of liquid, we have a contact surface Σ(t) between liquid and solid and Σ(t) = ∂Ω ∩ ∂S, as described in Figure 1, and a free surface Γ(t) where the liquid is in  François Dubois, Dimitri Stoliaroff and Isabelle Terrasse thermodynamical equilibrium with its vapor. The liquid is submitted to a gravity field g 0 . This vector is collinear to an "absolute" vertical direction associated with a vector e 3 , third coordinate of a Galilean referential (e 1 , e 2 , e 3 ):

g 0 = -g e 3 .
Note that g > 0 with this choice, as illustrated in Figure 1. The velocity field of the liquid u(t) is measured relatively to an absolute referential, following e.g. the work of L. Fontenot [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF]. The liquid is assumed incompressible:

(1) div u = 0 in Ω(t) . General view of the sloshing problem in a solid at rest. The free boundary Γ(t) is issued from the equilibrium free boundary at rest Γ 0 with the help of the elongation η.

• Liquid as a perfect linearized fluid The pressure field p(x) is defined in the liquid domain Ω

x -→ p(x) ∈ IR. The conservation of momentum for a perfect fluid is written with the Euler equations of hydrodynamics:

∂u ∂t + (curl u) × u + ∇ p ρ L + 1 2 | u | 2 = g 0 in Ω(t).
In this contribution, we make a linearization hypothesis. In particular, we neglect the nonlinear terms in fluid dynamics and replace the previous equation by:

(2)

∂u ∂t + 1 ρ L ∇p = g 0 in Ω(t) .
• Velocity potential We suppose moreover that the fluid is irrotational:

(3) curl u = 0 .

If the domain Ω(t) is simply connected (be careful with this hypothesis for toric geometries !), the simple hypothesis (3) implies that the velocity field can be generated by a potential ϕ:

(4) u(x) = ∇ϕ(x) , x ∈ Ω(t)



• In order to have precise information concerning this velocity potential, we recall the Bernoulli theorem. We inject the velocity field u = ∇ϕ in the dynamical equations. We introduce a point P : ∇ ∂ϕ ∂t + p ρ L -g 0 • (x -x P ) = 0 for x ∈ Ω. We add some time function to the scalar potential of velocity (and assume that the domain Ω is connected). Then: [START_REF] Bauer | Fluid Oscillations in the Containers of a Space Vehicle and Their Influence on Stability[END_REF] ∂ϕ ∂t

+ p ρ L -g 0 • (x -x P ) = 0 , x ∈ Ω.
• We take now into consideration the incompressibility hypothesis (1) together with the potential representation of the velocity field (4). We then obtain the Laplace equation: (6) ∆ϕ = 0 in Ω(t). A first boundary condition for this equation is a consequence of the continuity of the normal velocity u•n at the interface Σ between solid and liquid: [START_REF] Behruzi | Ballistic Phase Management for Cryogenic Upper Stages[END_REF] ∂ϕ ∂n = 0 , x ∈ Σ(t) .

•

Free surface Consider as a reference situation the solid at rest. Then the free surface at equilibrium has a given position Γ 0 as presented in Figure 1. We note η n 0 the displacement of the free boundary at position y ∈ Γ 0 , where n 0 denotes the outward normal direction to Γ 0 . In this case of a fixed solid, we have n 0 = e 3 . The point x new position takes into account the variation of the free surface: [START_REF] Chemoul | Ariane 5 flight environments[END_REF] x = y + η(y) n 0 , y ∈ Γ 0 , x ∈ Γ. We denote by x 0 the center of gravity of the frozen free surface Γ 0 : (9)

Γ 0 y -x 0 dγ = 0 .
Note that due to incompressibility condition, we have: [START_REF] Diebold | Effects on sloshing pressure due to the coupling between seakeeping and tank liquid motion[END_REF] Γ 0 η dγ ≡ 0 .

Moreover, with x given on Γ(t) according to (8), we have,

Γ 0 (x -x 0 ) dγ = 0 .
We observe that, thanks to [START_REF] Diebold | Effects on sloshing pressure due to the coupling between seakeeping and tank liquid motion[END_REF], Γ 0 (x -x 0 ) dγ = Γ 0 (y -x 0 + η(y) n 0 ) dγ = Γ 0 (y -x 0 ) dγ = 0 due to the definition [START_REF] Delnevo | Numerical methods: Fast multipole method for shielding effects[END_REF]. We introduce also the coordinates X 1 , X 2 , X 3 , of a point x in the referential (x 0 , e 1 , e 2 , e 3 ): x -x 0 = X 1 e 1 + X 2 e 2 + X 3 e 3 .

•

Proposition 1. Neumann boundary condition on the free surface If we keep only the first order linear terms, the boundary condition for the velocity potential on the free surface can be written as a kinematic condition:

(11) ∂ϕ ∂n = ∂η ∂t , x ∈ Γ 0 .
Then equations ( 7) and ( 11) can be written in a synthetic form:

(12) ∂ϕ ∂n =    0 on Σ ∂η ∂t on Γ 0 .  • Proof of Proposition 1.
We introduce the equation F (X 1 , X 2 , X 3 , t) = 0 of the free surface. We take the total derivative relative to time of this constraint and replace the velocity dX dt by the gradient ∇ϕ of the potential. We obtain ∇F • ∇ϕ+ ∂F ∂t = 0. The normal vector n can be written as n = ∇F / | ∇F | and we have ∂ϕ ∂n ≡ ∇ϕ

• n = ∇ϕ • ∇F |∇F| .
The previous equation can be written as [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF] ∂ϕ ∂n

+ 1 | ∇F | ∂F ∂t = 0 .
We parameterize the surface with an explicit function η, id est

(14) F (X 1 , X 2 , X 3 , t) ≡ X 3 -η(X 1 , X 2 , t).
Then linearizing the problem, we suppose that the free surface is close to its reference value Γ 0 at rest and we can neglect the gradient ∇η of the free surface equation ( 14) compared to the unity. Thus we have

| ∇F | = 1 + O(|η| 2
). Due to the particular form ( 14), we deduce that we have ∂F ∂t = -∂η ∂t on the free boundary and the relation ( 11) is a direct consequence of ( 13) and the fact that the norm of ∇F is of order unity.

•

Proposition 2. Pressure continuity across the free surface On the free surface Γ, the continuity of the stress tensor can be written for a perfect fluid as a dynamic condition: [START_REF] Falcón | Capillary wave turbulence on a spherical fluid surface in low gravity[END_REF] p = 0 on Γ .

It takes the following linearized form:

(16) ∂ϕ ∂t + g η = 0 , x ∈ Γ 0 .
• Proof of Proposition 2. The proof is classical and is explained in classic books as [START_REF] Faltinsen | Sloshing[END_REF][START_REF] Ibrahim | Liquid Sloshing dynamics: theory and applications[END_REF][START_REF] Morand | Interactions fluides structures[END_REF]. We give it here for completeness of the study. We choose the point P for the Bernoulli equation ( 5) on the frozen free surface Γ 0 equal to the center x 0 introduced in (9). Due to Bernoulli theorem [START_REF] Bauer | Fluid Oscillations in the Containers of a Space Vehicle and Their Influence on Stability[END_REF] and continuity of the pressure on Γ, we deduce the following relation on the free surface:

(17) ∂ϕ ∂t -g 0 • (x -x 0 ) = 0 , x ∈ Γ(t).
We have the following calculus:

-g 0 • (x-x 0 ) = g e 3 • X 1 e 1 + X 2 e 2
+ η e 3 = g η, and the condition p = 0 of pressure continuity on the free surface is expressed by ∂ϕ ∂t + g η = 0 which is exactly relation [START_REF] Faltinsen | Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth[END_REF]. The proof is established.

• Free surface potential and Neumann to Dirichlet operator

We introduce the "free surface potential" Ω x -→ ψ(x) ∈ IR satisfying the following Neumann boundary-value problem for the Laplace equation:

(18)    ∆ψ = 0 in Ω ∂ψ ∂n = 0 on Σ η on Γ 0 .


Coupling Linear Sloshing with Rigid Body Dynamics

We consider a free surface η such that the global incompressibility condition (10) holds. We introduce the functional space

F 1/2 (Γ 0 ) ≡ η : Γ 0 -→ IR, Γ 0 η dγ = 0 .
We consider the "free surface potential" ψ associated to a given η ∈ F 1/2 (Γ 0 ) in the following way. The function Ω x -→ ψ(x) ∈ IR is uniquely defined by the Neumann problem [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF] with the additional condition ( 19)

Γ 0 ψ dγ = 0 .
We consider the restriction ζ (the trace) of the function ψ on the surface Γ 0

(20) Γ 0 x -→ ζ(x) ≡ ψ(x) ∈ IR The mapping F 1/2 (Γ 0 ) η -→ ζ ∈ F 1/2 (Γ 0 ) is the "Neumann to Dirichlet" operator.
We denote it with the letter W:

(21) ζ ≡ W • η .
A precise mathematical definition of the space F 1/2 (Γ 0 ) in the context of Sobolev spaces can be found in [START_REF] Lions | Problèmes aux limites non homogènes et applications; volume 1[END_REF] or [START_REF] Nédélec | Acoustic and Electromagnetic Equations; Integral Representations for Harmonic Problems[END_REF].

• Proposition 3. Positive self adjoint operator The operator W : F 1/2 (Γ 0 ) η -→ ζ ∈ F 1/2 (Γ 0 )
with ζ defined by the relations ( 18), [START_REF] Fraeijs De Veubeke | The inertia tensor of an incompressible fluid bounded by walls in rigid body motion[END_REF], [START_REF] Gavrilyuk | Eigenoscillations of threeand two-element flexible systems[END_REF] and ( 21) is self-adjoint. If we denote by (•, •) the L 2 scalar product on the linearized free surface Γ 0 , id est

(22) (η, ζ) ≡ Γ 0 η ζ dγ , η, ζ ∈ F 1/2 (Γ 0 ) , we have: (η , W • η) = (W • η , η), for all η, η ∈ F 1/2 (Γ 0 ).
In particular, with the free surface potential ψ defined in [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF], we have

(23) (η , W • η) = Ω ∇ψ • ∇ψ) dx ≥ 0 .
• Proof of Proposition 3. We have with the previous notations:

(η , W • η) = Γ 0 η ψ dγ = ∂Ω ∂ψ ∂n ψ dγ because ∂ψ ∂n = 0 on Σ and ∂ψ ∂n = η on Γ 0 = Ω div ψ ∇ψ ) dx due to Green formula = Ω ∇ψ • ∇ψ) dx because ∆ψ = 0 = ∂Ω ψ ∂ψ ∂n dγ because ∆ψ = 0 = Γ 0 ψ η dγ = (W • η , η) .


François Dubois, Dimitri Stoliaroff and Isabelle Terrasse

We observe that if η = η , then ψ = ψ and the scalar product (W • η , η) is given by the relation [START_REF] Ibrahim | Liquid Sloshing dynamics: theory and applications[END_REF] and is positive. The proof is complete.

• In conclusion of the section, the velocity potential ϕ satisfies [START_REF] Dodge | Experimental and Theoretical Studies of Liquid Sloshing at Simulated Low Gravity[END_REF], then its value on the free boundary Γ may be described in terms of the operator W . We have

ϕ = W • ∂η ∂t on Γ 0 .
Then the evolution equation ( 16) can be formulated only in terms of the a priori unknown free surface, parameterized by the function η and the operator W :

(24) ρ L W • ∂ 2 η ∂t 2 + ρ L g η = 0 on Γ 0 .
This equation is usually presented as a family of harmonic oscillators, through a diagonalization of the operator W with usual spectral methods [START_REF] Faltinsen | Sloshing[END_REF][START_REF] Ibrahim | Liquid Sloshing dynamics: theory and applications[END_REF][START_REF] Morand | Interactions fluides structures[END_REF]. Our formulation with a Neumann to Dirichlet operator can be solved with boundary element methods. See e.g. the books of J.C. Nédélec [START_REF] Nédélec | Acoustic and Electromagnetic Equations; Integral Representations for Harmonic Problems[END_REF] or O.M. Faltinsen and A.N. Timokha [START_REF] Faltinsen | Sloshing[END_REF]. For an explicit implementation of boundary integral methods, we refer to the work of one of us [START_REF] Delnevo | Numerical methods: Fast multipole method for shielding effects[END_REF]. We observe also that equation ( 24) will be modified by the coupling with the rigid movement.

2) Sloshing in a body with a rigid movement

In their book [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF], N. Moiseev and V. Rumiantsev study the problem of a completely fluid-filled reservoir. They introduce special functions to represent the effect of the fluid on the solid motion. In the present study, the reservoir is partially filled and the fluid has a free surface. Nevertheless, the system is now the rigid body submitted to various forces.

In this section, we derive the evolution equations of momentum and kinetic momentum of the solid. We adapt also the previous section in order to describe the fluid movement inside the moving rigid body.

• Rigid body

We consider a rigid moving solid S, of density ρ S and total mass m S . We introduce the center of gravity ξ. This solid is submitted to three forces. The first one is the gravity described previously. The weight of the solid S is then equal to m S g 0 . Secondly a force R at a fixed point A on the boundary ∂S. We can suppose that this force is a given function of time. Last but not least, the surface forces f on the boundary ∂S due to the internal fluid. We introduce a local referential ε j associated to the rigid body and issued from Galilean referential e j .

• Infinitesimal motion of the rigid body

The linearization hypothesis acts now in a geometrical manner. The center of gravity is a function of time ξ = ξ(t) and an infinitesimal rotation of angle θ = θ(t) allows to write a simple algebraic relation between the vectors ε j and e j : ε j = e j + θ × e j . Then

dε j dt = dθ dt × ε j for 1 ≤ j ≤ 3.
The solid velocity field u S (x) satisfies:

u S (x) = dξ dt + dθ dt × (x -ξ(t)) , x ∈ S .


Coupling Linear Sloshing with Rigid Body Dynamics

The kinetic momentum σ S and the tensor of inertia I S are defined as usual:

σ S ≡ S ρ S (x -ξ) × u S (x) dx and I S • y ≡ S ρ S (x -ξ) × y × (x -ξ) dx for y ∈ IR 3 .
For a rigid body, we have the classical relation:

σ S = I S • dθ dt .
• Dynamics equations of the rigid body By integration of the classical Newton laws of motion, the conservation of momentum takes the form:

(25) m S d 2 ξ dt 2 = m S g 0 + R + ∂S f dγ .
The momentum M S of the surface forces relatively to the center of gravity is given according to:

M S = ∂S (x -ξ) × f dγ.
Then the conservation of kinetic momentum takes the form:

(26)

I S • d 2 θ dt 2 = (x A -ξ) × R + M S .
• The force f on the boundary of the solid surface Σ admits the expression

f = p n , x ∈ Σ 0 , x ∈ ∂S \ Σ .
Then: ∂S f dγ = Σ p n dγ = ∂Ω p n dγ = Ω ∇p dx. Due to the previous expression of the momentum of pressure forces, we have after an elementary calculus:

(27) M S = Ω (x -ξ) × ∇p dx .
• About the fluid irrotationality hypothesis In the monograph [START_REF] Lomen | Liquid propellant sloshing in mobile tanks of arbitrary shape[END_REF], D. Lomen suppose the irrotationality for the motion of the liquid relatively to the motion of the rigid body. Then the velocity field of the liquid satisfies the conditions:

u(x) = dξ dt + dθ dt × (x -ξ(t)) + v , curl v ≡ 0 .
By taking the curl of this relation: curl u = 2 dθ dt . Consider now the time derivative of the previous relation and the curl of relation [START_REF] Ardakani | Dynamic coupling between shallow-water sloshing and horizontal vehicle motion[END_REF]. Then we obtain: d 2 θ dt 2 = 0 and the hypothesis of irrotationality in the relative referential done in [START_REF] Lomen | Liquid propellant sloshing in mobile tanks of arbitrary shape[END_REF] is physically correct only if the rotation of solid referential is uniform in time.

•

Irrotationality in the Galilean referential We observe that under an assumption of linearized dynamics, if vorticity curl u of liquid measured in the Galilean referential at initial time is null, then it remains identically null for all times: curl u ≡ 0 , t ≥ 0 , x ∈ Ω(t) .



François Dubois, Dimitri Stoliaroff and Isabelle Terrasse

To prove the previous relation, just take the curl of the linearized dynamics equation [START_REF] Ardakani | Dynamic coupling between shallow-water sloshing and horizontal vehicle motion[END_REF]. Then ∂ ∂t curl u = 0 and the property is established if it is true at t = 0. In the following, it is assumed that the fluid is irrotational in the Galilean referential. In this case, one can find a velocity potential even if the angular velocity is an arbitrary function of time.

• Stokes -Zhukovsky vector fields for fluid potential decomposition A natural question is the incorporation of the movement of a rigid body inside the expression of the potential ϕ of velocities. In other words, we have put in evidence the very particular role of the rigid movement for the determination of the velocity potential. We recall that this dynamics is a six degrees of freedom system described by the two vectors ξ(t) and θ(t). The remaining difficulty concerns the solid body velocity field which is rotational. Following an old idea due independently (at our knowledge) to G. Stokes [START_REF] Stokes | On some cases of fluid motion[END_REF], N. Zhukovsky [START_REF] Zhukovsky | On the motion of a rigid body having cavities, filled with a homogeneous liquid drops[END_REF] and B. Fraeijs de Veubeke [START_REF] Fraeijs De Veubeke | The inertia tensor of an incompressible fluid bounded by walls in rigid body motion[END_REF], we introduce a function ϕ such that:

(28)    ∆ ϕ = 0 in Ω ∂ ϕ ∂n = dξ dt + dθ dt × (x -ξ) • n on ∂Ω .
Due to linearity of the problem (28), we can decompose the vector field ϕ under the form:

(29) ϕ ≡ α • dξ dt + β • dθ dt .
The vector fields Ω x -→ α(x) ∈ IR 3 for translation and Ω x -→ β(x) ∈ IR 3 for rotation only depend on the three-dimensional geometry of the liquid. Observe that α(x) is homogeneous to a length and β(x) to a surface. We call them the "Stokes-Zhukovsky vector fields" in this contribution, as in the reference [START_REF] Faltinsen | Sloshing[END_REF].



Coupling Linear Sloshing with Rigid Body Dynamics

•

Free surface Consider as a reference situation the fluid at rest relative to the solid. Then the free surface has a given position Γ 0 as presented in Figure 2. During sloshing, two processes have now to be taken into account. First, the rigid motion of the surface Γ 0 and secondly the free displacement η n 0 of the free boundary measured in the relative referential, where n 0 denotes the normal direction to Γ 0 at position y ∈ Γ 0 . The local coordinates X j are defined by the relation x -x 0 = 3 j=1 X j ε j , where x 0 is the barycenter of the free surface Γ 0 defined in [START_REF] Delnevo | Numerical methods: Fast multipole method for shielding effects[END_REF]. We introduce the center of gravity x L of the liquid: m L x L ≡ Ω ρ L x dx, which is a priori a function of time. We introduce also the center of gravity x F of the "frozen fluid" Ω 0 at rest. Note that Γ 0 is a part of the boundary of Ω 0 : m L x F ≡ Ω 0 ρ L x dx and we refer to Figure 2 for a representation of this point.

• Stokes-Zhukovsky vector fields for translation

We set α ≡ 3 j=1 α j ε j . Then the scalar function α j (x) satisfies clearly the following Neumann problem for the Laplace equation:

(30)    ∆α j = 0 in Ω 0 ∂α j ∂n = n j on ∂Ω 0 .
The problem [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF] has a unique solution up to a scalar constant if the domain Ω is connected. It has an analytical solution. We consider the center of gravity x 0 of the linearized free surface Γ 0 according to [START_REF] Delnevo | Numerical methods: Fast multipole method for shielding effects[END_REF]. Then: α j (x) = (x -x 0 ) • ε j for j = 1, 2, 3 if the condition Γ 0 α j dγ = 0. holds. Then in consequence, ∇α j = ε j , j = 1, 2, 3

• We introduce α by rotating the Stokes-Zhukovsky translation vector field α:

α ≡ (x -x 0 ) × ε 3 = α × ε 3 Then: Γ 0 α dγ = 0 and α = X 2 ε 1 -X 1 ε 2 . Moreover Γ 0 η (X 2 ε 1 -X 1 ε 2 ) dγ = Γ 0 η α dγ and (31) α • θ = (-X 1 θ 2 + X 2 θ 1
) .

• Stokes-Zhukovsky vector fields for rotation

Analogously to the definition (30) of Stokes-Zhukovsky vector fields for translation, we set β ≡ 3 j=1 β j ε j . The scalar function β j (x) satisfies the equations:

(32)    ∆β j = 0 in Ω 0 ∂β j ∂n = (x -ξ) × n j on ∂Ω 0 .
It is elementary (see e.g. P.A. Raviart and J.M. Thomas [START_REF] Raviart | Introduction à l'analyse numérique des équations aux dérivées partielles[END_REF]) to verify that the Neumann problem (32) is well set up to an additive constant. But, oppositely to the Stokes-Zhukovsky vector field for translation, we have no analytical expression for the Stokes-Zhukovsky functions β j for rotation. Nevertheless, the following relations show that beautiful algebra can be developed for the Stokes-Zhukovsky vector fields. They are proven in detail in [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF]:

(33) ρ L Ω ∇α j dx = m L ε j , ρ L Ω ∇β j dx = m L ε j × (x F -ξ) , j = 1, 2, 3 .
• Liquid inertial tensor With B. Fraeijs de Veubeke [START_REF] Fraeijs De Veubeke | The inertia tensor of an incompressible fluid bounded by walls in rigid body motion[END_REF], we introduce the so-called "liquid inertial tensor" I defined according to (34)

I j k ≡ ρ L Ω ∇β j • ∇β k dx .
We have the complementary results, proven also in [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF]:

(35)        ρ L Ω (x -ξ) × ∇α j dx = m L (x F -ξ) × ε j , j = 1, 2, 3 , ρ L Ω (x -ξ) × ∇β j dx = I • ε j , j = 1, 2, 3 .
Moreover, the liquid inertial tensor I defined in ( 34) is positive definite. We have θ ,

I • θ = ρ L Ω | ∇(β•θ) | 2 dx ≥ 0 for θ ∈ IR 3 . Moreover, if (θ , I • θ) = 0, then θ = 0 in IR 3 .

• Decomposition of the fluid velocity potential

The fluid velocity potential ϕ satisfies a continuity condition across the solid interface Σ(t) due to the non-penetration of the fluid inside the solid:

(36) ∂ϕ ∂n = dξ dt + dθ dt × (x -ξ) • n , x ∈ Σ.
Therefore we subtract to ϕ the potential ϕ introduced in ( 29) and the difference satisfies a homogeneous boundary condition on the solid interface. In an analogous way, Proposition 1 can be derived in the relative referential. Then the fluid velocity potential ϕ can be decomposed according to

(37) ϕ ≡ α • dξ dt + β • dθ dt + ∂ψ ∂t .
The free surface potential ψ introduced in (37) still satisfies the relations [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF]:

   ∆ψ = 0 in Ω ∂ψ ∂n = 0 on Σ η on Γ 0 .
Due to the definition [START_REF] Gerbeau | A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows[END_REF] of the Neumann to Dirichlet operator, this last expression admits the compact form

(38) ψ = W • η on Γ 0 .
• We introduce the position 0 of the center of gravity of the fluid relatively to the solid center of gravity:

(39) 0 ≡ x F -ξ . We observe that this vector is linked to the solid and we have in particular d 0 = dθ × 0 . We have the following relations, with η ∈ F 1/2 (Γ 0 ) and α, β defined in [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF] [START_REF] Nédélec | Acoustic and Electromagnetic Equations; Integral Representations for Harmonic Problems[END_REF] and ψ by the relation [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF]. The proofs are detailed in our report [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF].

(40)

       ρ L Ω ∇ ∂ 2 ψ ∂t 2 dx = ρ L Γ 0 ∂ 2 η ∂t 2 (X 1 ε 1 + X 2 ε 2 ) dγ ρ L Ω (x -ξ) × ∇ ∂ 2 ψ ∂t 2 dx = ρ L Γ 0 ∂ 2 η ∂t 2 β dγ .



Coupling Linear Sloshing with Rigid Body Dynamics • Proposition 4. Pressure continuity across the free surface The continuity of pressure on the free boundary Γ takes the following linearized form: [START_REF] Zhukovsky | On the motion of a rigid body having cavities, filled with a homogeneous liquid drops[END_REF] ∂ϕ ∂t

+ g α • θ + η = 0 , x ∈ Γ(t) .
In an equivalent way with the potential ψ introduced in ( 37):

(42)

∂ 2 ψ ∂t 2 + α • dξ 2 dt 2 + β • d 2 θ dt 2 + g α • θ + η = 0 on Γ 0 .
Compared to the relation ( 16), the new term g α • θ expresses the action of the external gravity field driving the fluid due to the rotational movement of the solid. In the stationary case, the local displacement η of the free boundary compensates exactly the rotational displacement of the solid.

•

Proof of Proposition 4. The proof is a small variation of the one proposed for Proposition 2. On the free surface Γ, the continuity of the stress tensor can be written p = 0 as previously. We choose the point P for the Bernoulli equation ( 5) on the frozen free surface Γ 0 equal to the center x 0 introduced in (9). Then x -

x 0 = X 1 ε 1 + X 2 ε 2 + η ε 3 + O(|η| 2
) . Due to Bernoulli theorem [START_REF] Bauer | Fluid Oscillations in the Containers of a Space Vehicle and Their Influence on Stability[END_REF] and continuity [START_REF] Falcón | Capillary wave turbulence on a spherical fluid surface in low gravity[END_REF] of the pressure on Γ, we deduce the following relation [START_REF] Faltinsen | Sloshing[END_REF] on the free surface. In order to show the angular displacement of the solid, we have the following calculus:

-g 0 • (x -x 0 ) = g e 3 • X 1 (e 1 + θ × e 1 ) + X 2 (e 2 + θ × e 2 ) + η ε 3 , = g (-X 1 θ 2 + X 2 θ 1 + η) + O(|η| 2 ) ,
and the condition [START_REF] Faltinsen | Sloshing[END_REF] of pressure continuity on the free surface is expressed by

∂ϕ ∂t + g (-X 1 θ 2 + X 2 θ 1 + η) + O(|η| 2 ) = 0 , x ∈ Γ(t) ,
which is exactly relation [START_REF] Zhukovsky | On the motion of a rigid body having cavities, filled with a homogeneous liquid drops[END_REF] due to the relation [START_REF] Morand | Interactions fluides structures[END_REF]:

α • θ = (-X 1 θ 2 + X 2 θ 1 ).
If we determine the velocity potential ϕ according to the left hand side of the relation [START_REF] Stokes | On some cases of fluid motion[END_REF], the continuity of the pressure field (15) across the free surface takes exactly the form (42) and the proof is completed.

• Proposition 5. Conservation of the solid momentum The conservation of the solid momentum conducts to the coupled relation:

(43) (m S + m L ) d 2 ξ dt 2 -m L 0 × d 2 θ dt 2 + ρ L Γ 0 α ∂ 2 η ∂t 2 dγ = (m S + m L ) g 0 + R .
• Proof of Proposition 5. Due to the linearized Euler equations (2), the pressure field action can be stated as

∇p = ρ L g 0 -∇ ∂ϕ ∂t , x ∈ Ω ,
the conservation of momentum of the solid can be written as:

(44) m S d 2 ξ dt 2 = (m S + m L ) g 0 + R - Ω ρ L ∇ ∂ϕ ∂t dx .
With the help of the Stokes-Zhukovsky vector fields, we can express the last term in the right hand side of ( 44) with the free surface potential ψ:

ρ L Ω ∇ ∂ϕ ∂t dx = ρ L Ω j ∇α j d 2 ξ j dt 2 + ∇β j d 2 θ j dt 2 + ∇ ∂ 2 ψ ∂t 2 dx c.f. (37) = m L d 2 ξ dt 2 + m L d 2 θ dt 2 × x F -ξ + ρ L Ω ∂ 2 η ∂t 2 (X 1 1 + X 2 2
) dγ due to [START_REF] Ostrach | Low-Gravity Fluid Flows[END_REF], [START_REF] Raviart | Introduction à l'analyse numérique des équations aux dérivées partielles[END_REF] and [START_REF] Wu | Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method[END_REF]. Then the conservation of impulsion of the solid (44) takes the form:

   (m S + m L ) d 2 ξ dt 2 + m L d 2 θ dt 2 × (x F -ξ) + ρ L Γ 0 ∂ 2 η ∂t 2 (X 1 ε 1 + X 2 ε 2 ) dγ = = (m S + m L ) g 0 + R .
Then due to [START_REF] Vierendeels | Analysis and Stabilization of Fluid-Structure Interaction Algorithm for Rigid-Body Motion[END_REF], the equations ( 25) for the conservation of impulsion of the solid takes the following expression (43) for the coupled problem. The proof is completed.

• Proposition 6. Conservation of the solid angular momentum The conservation of the solid angular momentum conducts to the coupled relation

(45)    m L 0 × d 2 ξ dt 2 + (I S + I ) • d 2 θ dt 2 + ρ L Γ 0 β ∂ 2 η ∂t 2 dγ + ρ L g Γ 0 α η dγ = = m L 0 × g 0 + (x A -ξ) × R .
The term m L 0 × d 2 ξ dt 2 of dynamic evolution of the kinetic momentum in the relation (45) is due to the non-coincidence of the center of gravity ξ of the solid and the frozen center of gravity x F of the frozen liquid (see Figure 2). With the help of (40), the kinetic momentum of the fluid relative to the center of gravity of the solid is represented by the term ρ L Γ 0 β • ∂ 2 η ∂t 2 dγ. Last but not least, the restoring torque of the gravity field due to the weight of the liquid displaced by the movement of the free boundary is described by ρ L g Γ 0 α η dγ. We can interpret the elongation of the free surface by a continuous distribution of small harmonic oscillators that have an impact on the global conservation of the solid angular momentum.

•

Proof of Proposition 6. We explain in the following how the center of gravity x L of the liquid depends on the position η relative to the free boundary. The conservation of kinetic momentum [START_REF] Lions | Problèmes aux limites non homogènes et applications; volume 1[END_REF] takes now the form:

I S • d 2 θ dt 2 = (x A -ξ) × R + m L (x L -ξ) × g 0 - Ω ρ L (x -ξ) × ∇ ∂ϕ ∂t dx.
We consider also the center of gravity x F of the "frozen fluid" and we denote by X 0 3 the vertical coordinate of the frozen free surface Γ 0 . Relatively to the rigid referential, we have the following calculus:  Coupling Linear Sloshing with Rigid Body Dynamics

m L x L = Γ 0 dX 1 dX 2 X 0 3 X 3 min ρ L x dX 3 + X 0 3 +η X 0 3 ρ L x dX 3 = m L x F + Γ 0 dX 1 dX 2 η 0 ρ L   X 1 X 2 X 3   dX 3 = m L x F + Γ 0 ρ L   X 1 η X 2 η 1 2 η 2   dγ .
At first order:

m L x L × g 0 = m L x F × g 0 + ρ L g Γ 0 η (-X 2 ε 1 + X 1 ε 2 ) dγ.
• In consequence the conservation of kinetic momentum can be written under the form:

(46)

     I S • d 2 θ dt 2 = (x A -ξ) × R + m L (x F -ξ) × g 0 + ρ L g Γ 0 η (-X 2 ε 1 + X 1 ε 2 ) dγ - Ω ρ L (x -ξ) × ∇ ∂ϕ ∂t dx.
In an analogous way, the last term of the right hand side of (46) can be developed with the help of the decomposition (37): [START_REF] Ostrach | Low-Gravity Fluid Flows[END_REF], [START_REF] Raviart | Introduction à l'analyse numérique des équations aux dérivées partielles[END_REF] and [START_REF] Wu | Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method[END_REF]. In consequence, the motion (46) of the solid around its center of gravity can be written as:

ρ L Ω x -ξ × ∇ ∂ϕ ∂t dx = = ρ L Ω x -ξ × j ∇α j d 2 ξ j dt 2 + ∇β j d 2 θ j dt 2 + ∇ ∂ 2 ψ ∂t 2 dx = m L x F -ξ × d 2 ξ dt 2 + I • d 2 θ dt 2 + ρ L Ω ∂ 2 η ∂t 2 β dγ due to
       (I S + I ) • d 2 θ dt 2 + m L (x F -ξ) × d 2 ξ dt 2 + ρ L Γ 0 ∂ 2 η ∂t 2 β dγ + + ρ L g Γ 0 η (X 2 ε 1 -X 1 ε 2 ) dγ = m L (x F -ξ) × g 0 + (x A -ξ) × R .
and due to [START_REF] Vierendeels | Analysis and Stabilization of Fluid-Structure Interaction Algorithm for Rigid-Body Motion[END_REF], this is exactly the relation (45) and the proof is completed.

• Towards a synthetic formulation With the help of the relation [START_REF] Tezduyar | Space-time finite element techniques for computation of fluid-structure interactions[END_REF] on the boundary Γ 0 , the continuity (42) of the pressure field across the free surface is simply written as:

(47) ρ L W • ∂ 2 η ∂t 2 + ρ L α • d 2 ξ dt 2 + ρ L β • d 2 θ dt 2 + ρ L g ( α • θ + η) = 0 on Γ 0 .
The term ρ L g α • θ is due to the weight of the fluid working in the rigid movement associated to the solid rotation. The coupled problem (43) (45) (47) is now formulated in an attractive mathematical point of view. The unknown is composed of the triple (ξ(t), θ(t), η(t)), with ξ(t) ∈ IR 3 , θ(t) ∈ IR 3 , η(t) ∈ F 1/2 (Γ 0 ) and the three equations (43) (45) (47) are considered in IR 3 , IR 3 and on Γ 0 respectively. The mathematical difficulty is due to the term W • ∂ 2 η ∂t 2 because W is an integral operator.

3) Coupled system structure

• We are now in position to aggregate the previous results. The solid movement is a six degrees of freedom motion described by the velocity dξ dt of its center of gravity and its instantaneous rotation dθ dt . The motion of the solid around its center of gravity has been obtained in relation (46). The two equations ( 44) and (46) admit as a source term the gradient of the velocity potential. The partial differential equation that governs this potential is simply the incompressibility of the liquid, expressed by the Laplace equation. The boundary conditions are the non-penetration [START_REF] Snydera | Sloshing in microgravity[END_REF] of the fluid inside the solid, the normal movement [START_REF] Fontenot | The Dynamics of Liquids in Fixed and Moving Containers[END_REF] of the fluid relatively to the free surface and the continuity [START_REF] Falcón | Capillary wave turbulence on a spherical fluid surface in low gravity[END_REF] of the pressure field across the free surface expressed by (42).

•

Energy conservation

We can now consider the three terms of the total energy: the uncoupled kinetic energy

T ≡ 1 2 dξ dt • m S dξ dt + 1 2 dθ dt I S • dθ dt + 1 2 Ω ρ L |∇ϕ| 2 dx ,
the energy of interaction with gravity

U ≡ 1 2 ρ L g Γ 0 |η| 2 dγ + ρ L g Γ 0 η (X 2 θ 1 -X 1 θ 2 ) dγ and the gravity potential V ≡ -m S g 0 • ξ -m L g 0 • x F .
With kinetic energy T , energy of interaction with gravity U and gravity potential V defined previously respectively, we have the following detailed expressions:

(48)

       T = 1 2 (m S + m L ) | dξ dt | 2 + 1 2 dθ dt , (I S + I ) dθ dt + ρ L 2 Γ 0 ∂η ∂t , W • ∂η ∂t dγ + m L 0 , dξ dt , dθ dt + ρ L Γ 0 α dξ dt + β dθ dt ∂η ∂t dγ (49) U = 1 2 ρ L g Γ 0 η 2 dγ + ρ L g Γ 0 ( α • θ) η dγ (50) V = -(m S + m L ) g 0 • ξ -m L g 0 • 0 .
We recognize the kinetic energy of the solid with the translation and rotation decoupled terms

1 2 (m S + m L ) | dξ dt | 2 , 1 2
d dt T + U + V = R • u A .
The proof is detailed in [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF].

• Operator matrices We consider now the global vector q(t) according to:

(52) q ≡ η , ξ, θ t .

Remark that when t ≥ 0, q(t) belongs to the functional space F 1/2 (Γ 0 ) × IR 3 × IR 3 , an infinite dimensional vector space denoted by Q 0 (Ω, S) in the following:

(53) Q 0 (Ω, S) ≡ F 1/2 (Γ 0 ) × IR 3 × IR 3 .
With this notation, the interaction between the liquid Ω and the solid S, through the free boundary Γ 0 , is defined through global operator matrices M and K. The mass matrix M is defined according to:

(54) M =       ρ L W • ρ L α • ρ L β • ρ L Γ 0 dγ α • m S + m L -m L 0 × • ρ L Γ 0 dγ β • m L 0 × • I S + I       .
Remark that this matrix is composed by operators. In particular the operator W at the position (1, 1) is defined in [START_REF] Gerbeau | A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows[END_REF]. Moreover, if q ∈ Q 0 (Ω, S), M • q ∈ Q 0 (Ω, S) and M is an operator Q 0 (Ω, S) -→ Q 0 (Ω, S). In an analogous way, we define the global rigidity matrix K:

(55) K =     ρ L g 0 ρ L g α • 0 0 0 ρ L g Γ 0 dγ α • 0 0    
and we obtain as previously an operator Q 0 (Ω, S) q -→ K • q ∈ Q 0 (Ω, S). We introduce also a global right hand side vector F (t) :

(56)

F (t) =   0 (m S + m L ) g 0 + R m L 0 × g 0 + (x A -ξ) × R  
and the relation F (t) ∈ Q 0 (Ω, S) is natural. We remark with these relatively complicated definitions (53), (54), (55), (56) that the global dynamical system composed by the relations (43), (45), (47) admits finally a very simple form:

(57) M • d 2 q dt 2 + K • q = F (t)
. This equation is the extension of the previous free fluid oscillators equation [START_REF] Ibrahim | Recent Advances in Liquid Sloshing Dynamics[END_REF] to the coupling with the solid motion.

•

Properties of the mass matrix The matrix M defined in (54) is symmetric and "positive definite". We have the following expression for the quadratic form:

q , M • q = m S | ξ | 2 + (θ , I S • θ) + ρ L Ω | ∇α • ξ + ∇β • θ + ∇ψ | 2 dγ .

•

Lagrangian function for the coupled system With the reduction of the coupled sloshing problem to the unknown q ≡ (η, ξ, θ) ∈ Q 0 (Ω, S) we first specify the energies according to this global field. The conservation of energy (51) has been established again from the compact form (57) of the evolution equation. If the external force R(t) is equal to zero, it is natural to introduce the Lagrangian L according to the usual definition:

(60) L = T -(U + V ) .
Then this Lagrangian is a functional of the state q defined in (52) and of its first time derivative. We have the final Proposition:

• Proposition 9. Euler-lagrange equations With the above notations when right hand side F (t) is reduced to the gravity term, the equations of motion (57) take the form

d dt ∂L ∂ dq dt = ∂L ∂q .
The proof of this proposition is elementary. We omit it and refer to [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF].

• With this general framework, the Lagrangian formulation is simple to use. It is sufficient for the applications to evaluate carefully the Lagrangian L given by the relations (48), (49), (50) and (60).

Conclusion

In this contribution, we started from our industrial practice of sloshing for rigid bodies submitted to an acceleration. We first set the importance of the irrotational hypothesis of the flow in the external Galilean reference frame. Then we derived carefully the mechanics of the solid motion (conservation of momentum and conservation of kinetic momentum) and of the fluid motion (Laplace equation for the velocity potential), with a particular emphasis for the coupling with the continuity of the normal velocity field and the continuity of pressure across the fluid surface. A first difficulty is the representation of the solid rotational velocity vector field with potential functions. This can be achieved with the Stokes-Zhukovsky vector fields that are particular harmonic functions associated to the geometry of the fluid. Efficient numerical methods like integral methods (see e.g. [START_REF] Nédélec | Acoustic and Electromagnetic Equations; Integral Representations for Harmonic Problems[END_REF]) could be used to go one step further. A much well known mathematical difficulty is the reduction of the fluid problem to a Neumann to Dirichlet operator for the Laplace equation. The use of integral methods is also natural for this kind of coupling (see e.g. [START_REF] Moiseev | Dinamika tela s polostiami, soderzhashchimi zhidkost[END_REF] and [START_REF] Morand | Interactions fluides structures[END_REF]). In particular, the integral algorithm used e.g. in our contribution [START_REF] Balin | Boundary element and finite element coupling for aeroacoustics simulations[END_REF] is appropriate for such numerical computation. The degrees of freedom of both Stokes-Zhukovsky and modal functions are located on the interface between solid and liquid and the total computational cost of such approach is reasonable. Last but not least, we have derived a general expression for the Lagrangian of this coupled system. The next step is to look to simplified systems and confront our rigorous mathematical analysis with the state of the art in the engineering community. In particular, we are interested in developing appropriate methodologies to define equivalent simplified mechanical systems as the ones presented in [START_REF] Abramson | The dynamic behavior of liquids in moving containers, with applications to space vehicle technology[END_REF]. We plan also to apply our formulation with a Neumann to Dirichlet operator with boundary element methods.

Figure 1 .

 1 Figure 1.General view of the sloshing problem in a solid at rest. The free boundary Γ(t) is issued from the equilibrium free boundary at rest Γ 0 with the help of the elongation η.

Figure 2 .

 2 Figure 2. General view of the sloshing problem in a six degrees of freedom rigid body. The free boundary Γ(t) is issued from the equilibrium free boundary at rest Γ 0 with the help of the elongation η. The other notations are explained in the corpus of the text.

  dθ dt , (I S + I ) dθ dt , the coupling between translation and rotation m L 0 , dξ dt , dθ dt , the kinetic energy of the free surface ρ L 2 Γ 0 ∂η ∂t , W • ∂η ∂t dγ and the coupling ρ L Γ 0 α dξ dt + β dθ dt ∂η ∂t dγ between the solid movement and the free boundary. • Proposition 7. Energy conservation Due to the lack of knowledge concerning the external force R, the conservation of energy takes the following form, with the previous notations: (51)

Contribution published in European Journal of Mechanics-B -Fluids, doi:10.1016/j.euromechflu.

2015.06.002, volume 54, pages 17-26, november-december 2015.

Acknowledgments

The authors thank their colleagues of Airbus Defence and Space Christian Le Noac'h for enthusiastic interaction, Gerald Pignié for helpful comments all along this work, and François Coron for suggesting us to study this problem. They thank also Roger Ohayon of Conservatoire National des Arts et Métiers in Paris for a detailed bibliography transmitted in January 2008. A special thanks to Antoine Mareschal of Institut Polytechnique des Sciences Avancées for his internship in Les Mureaux in 2013. Last but not least, the authors thank the referees for very constructive remarks. Some of them have been incorporated into the present edition of the article.

In other words, we have the expression T = 1 2 dq dt , M • dq dt for the kinetic energy developed in (48). The proof of this proposition is detailed in [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF].

• We consider now the same questions for the rigidity operator K. We recall that the tangential coordinates X j on the linearized free surface Γ 0 such that x = 3 j=1 X j ε j satisfy Γ 0 X j dγ = 0 for j = 1, 2 . We introduce a length a characteristic of this surface Γ 0 . Precisely, we suppose that (58)

We introduce also the

dγ of the free surface, in coherence with the scalar product proposed in the relation [START_REF] Hou | Numerical methods for fluid-structure interaction -a review[END_REF].

•

Proposition 8. Properties of the rigidity matrix The matrix K is symmetric: q , K • q = K • q , q for arbitrary global vectors q and q in the space Q 0 (Ω, S). The matrix K is positive: q , K • q ≥ 0 if the rotation θ of the solid is sufficiently small relatively to the mean quadratic value of the free surface, id est

This relation is quite precise concerning the validity of linearity hypotheses.

•

Proof of Proposition 8. The symmetry of the matrix K is elementary to establish. We refer to [START_REF] Dubois | Coupling Linear Sloshing with Six Degrees of Freedom Rigid Body Dynamics[END_REF]. We have also:

and the analogous inequality

for the other component. We deduce from the previous assessment the minoration:

and this expression is positive when

which is exactly the hypothesis (59). The proposition is established.


Rita Giuliano

Michel Weber

LOCAL LIMIT THEOREMS IN SOME RANDOM MODELS FROM NUMBER THEORY

Keywords: 2010 Mathematics Subject Classification: Primary: 60F15, 60G50 ; Secondary: 60F05 independent random variables, lattice distributed, Bernoulli part, local limit theorem, almost sure local limit theorem, effective remainder, random walk in random scenery

We study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we consider arise from standard random models used in arithmetical number theory. We next use the characteristic function method to prove new local limit theorems for weighted sums of Bernoulli variables. Further, we give an application of the almost sure local limit theorem to a representation problem in additive number theory due to Burr, using an appropriate random model. We also give a simple example showing that the local limit theorem, in its standard form, fails to be sharp enough for estimating the probability P{Sn ∈ E} for infinite sets of integers E, already in the simple case where Sn is a sum of n independent standard Bernoulli random variables and E an arithmetic progression.

Introduction.

This work is devoted to the study of the local limit theorem and of its recent developments, in the context of some standard random models used in arithmetical number theory. It is also somehow completing the recent paper [START_REF] Giuliano | Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery[END_REF]. We will be mainly interested in studying the local limit theorem for weighted sums of Bernoulli variables. As it will be clarified soon, this turns up to be a fundamental question in the local limit theorem theory. We first recall some basic results and the used methods. The local limit theorem was established already three centuries ago in the binomial case by De Moivre and Laplace around 1730. Based on Stirling approximation formula of n!, it is a very precise result for moderate deviations.

Lemma 1.1. Let 0 < p < 1, q = 1 -p. Let X be such that P{X = 1} = p = 1 -P{X = 0}. Let X 1 , X 2 , . . . be independent copies of X and let S n = X 1 + . . . + X n . Let 0 < γ < 1 and let β ≤ γ √ pq n 1/3 . Then for all k such that letting x = k-np √ npq , |x| ≤ βn 1/6 , we have

P{S n = k} = e -x 2 2 √ 2πnpq e E , with |E| ≤ |x| 3 √ npq + |x| 4 npq + |x| 3 2(npq) 3 2
+ 1 4n min(p,q)(1-γ) . This slightly more precise formulation than the one given in Chow and Teicher [START_REF] Chow | Probability Theory: Independence, Interchangability, Martingales, Third Edition[END_REF], p. 46, is easily extrapolated from their proof. More generally, let X = {X n , n ≥ 1} be a sequence of independent, square integrable random variables taking values in a common lattice L(v 0 , D) = {v 0 +Dk, k ∈ Z}, where v 0 and D > 0 are real numbers. Let also M n = n j=1 E X j , Σ n = n j=1 Var(X j ). We say that X satisfies a local limit theorem if (1.1) ∆ n := sup

N =v0n+Dk Σ n P{S n = N } - D √ 2π e -(N -Mn ) 2 2Σn
= o(1).

This fine limit theorem has connections with Number Theory, see for instance Postnikov [START_REF] Postnikov | Introduction to analytic number theory[END_REF]. If X is an i.i.d. sequence, then (1.1) holds if and only if the "span"D is maximal (D = sup d > 0; ∃a ∈ Z : P{X ∈ a + dZ} = 1 ). This is Gnedenko's well-known result, which is also optimal (Matskyavichyus [17]). Under stronger integrability conditions, the remainder term can be improved (see [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF] Theorem 4.5.3), [START_REF] Petrov | Sums of Independent Random Variables[END_REF] Theorem 6 p.197). The general form of the local limit theorem ( [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF], Th. 4.2.1) for i.i.d. random variables, states Theorem 1.2. In order that for some choice of constants a n and b n lim

n→∞ sup N ∈L(v0n,D) b n λ P{S n = N } -g N -a n b n = 0,
where g is the density of some stable distribution G with exponent 0 < α ≤ 2, it is necessary and sufficient that

(i) S n -a n b n D ⇒ G as n → ∞ (ii) D is maximal.
There are essentially two approaches used: the method of characteristic functions and the Bernoulli part extraction method. In the later case, this method is called the extraction method of the Bernoulli part of a random variable and was developed by McDonald [START_REF] Macdonald | On local limit theorems for integer valued random variables[END_REF], for proving local limit theorems in presence of the central limit theorem. Kolmogorov [START_REF] Kolmogorov | Sur les propriétés de fonctions de concentrations de M. P. Lévy[END_REF] (see also Kolmogorov's interesting comment p. 29) initiated twenty years before a similar approach in the study of Lévy's concentration function. We also mention Arratia, Barbour and Tavaré [START_REF] Arratia | Logarithmic Combinatorial Structures[END_REF][START_REF] Arratia | The Poisson-Dirichlet distribution and the scale-invariant Poisson process[END_REF] probabilistic approach in the study of the asymptotic behaviour of logarithmic combinatorial structures, and the recent work of Röllin and Ross [START_REF] Röllin | Local limit theorems via Landau-Kolmogorov inequalities[END_REF] based on Landau-Kolmogorov inequalities.

An important problem inside the general study of the local limit theorem concerns the case when the considered sums are weighted sums of Bernoulli variables, the "simple" case when the weights are increasing covering already non-trivial examples of random models used in number theory. The purpose of the next Section is to underline this in providing a few examples of such models, which we believe, are challenging problems for probabilists.

Additionally, for weighted sums of independent or i.i.d. random variables, the Bernoulli part extraction method reduces the problem to the case of weighted sums of Bernoulli variables, thereby making this case crucial too for the application of this method.

The goal of this work is to investigate the local limit theorem for weighted sums of Bernoulli variables. In Section 3, we use the characteristic function method to prove new local limit theorems. Next in Section 4, we give an application of the almost sure local limit theorem to Burr's representation problem in additive number theory, using an appropriate random model. Finally, we also give an example showing that the standard form (1.1) of the local limit theorem, fails to be sharp enough for estimating the probability P{S n ∈ E} for infinite sets of integers E; and this already in the simple case where S n is a sum of n independent standard Bernoulli random variables and E an arithmetic progression.

2. Some Random Models in Number Theory.

2.1.

A Probabilistic Model for the Dickman Function. This function originates from the study by Dickman of the asymptotic distribution of the largest prime factor P + (n) of a natural integer n. He has shown that the limit

(2.1) lim n→∞ 1 n # k; 1 ≤ k ≤ n : P + (k) ≤ n 1/u = ρ(u)
exists, and ρ(u), called the Dickman Function, is defined as the continuous solution of the differential-difference equation

uρ ′ (u) + ρ(u -1) = 0, (u > 1)
with the initial condition ρ(u) = 1 for 0 ≤ u ≤ 1. We have

∞ 0 ρ(v)dv = e γ
, where γ is Euler's constant. This is a function of first importance in analytic number theory, which has been thoroughly investigated by Hensley, Hildebrand, Tenenbaum notably, see [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF] for more details.

There is a probabilistic way of describing the Dickman Function. We refer to Hwang and Tsai [START_REF] Hwang | Quickselect and the Dickman function[END_REF]. Let X = {X j , j ≥ 1} be a sequence of independent random variables such that (2.2)

P{X j = j} = j -1 P{X j = 0} = 1 -j -1 . (j ≥ 1) Proposition 2.1. Let D n = n j=1 X j .
Then

lim n→∞ P n -1 D n < x = e -γ x 0 ρ(v)dv (x > 0).
Arratia, Barbour and Tavaré [START_REF] Arratia | The Poisson-Dirichlet distribution and the scale-invariant Poisson process[END_REF], Corollary 2.8 proved a (restricted) local limit theorem for D n

(2.3) lim n→∞ nP{D n = k n } = e -γ ρ(x), when lim n→∞ k n /n = x > 0.
The almost sure local limit theorem was recently established in Giuliano, Szewczak and Weber in [START_REF] Giuliano | Almost Sure Local Limit Theorem for the Dickman distribution[END_REF]. The proof is essentially based on a long and delicate study of the related correlations functions. A proof of the local limit theorem in the form (2.3) using only characteristic functions is also given, correcting the one indicated [START_REF] Hwang | Quickselect and the Dickman function[END_REF]. No local limit theorem for X (in the sense of (1.1)) is known.

Remarks 2.2. (i) Hensley [START_REF] Hensley | The convolution powers of the Dickman function[END_REF] has shown that the limiting law is infinitely divisible. In the same paper, he also constructed another very interesting probabilistic model, adapted to the "psixiology" i.e. to functions Ψ, Φ linked to P + , P -. (ii) Obviously D n also reads as D n = n j=1 jβ j where {β j , j = 1, . . . n} are independent Bernoulli random variables such that (2.4)

P{β j = 1} = j -1 P{β j = 0} = 1 -j -1 . (j = 1, . . . , n) (iii) Let Z 1 , .
. . Z n be independent Poisson distributed random variables with intensity E Z j = 1/j, and let T n = n j=1 jZ j . Then we have the exact formula P{T n = n} = e -n j=1 1/j , based on Cauchy formula for cycles of permutations ( [START_REF] Arratia | Logarithmic Combinatorial Structures[END_REF], formula (1.2)). (iv) Vervaart has shown that independent Bernoulli random variables can be embedded into a Poisson process (see [START_REF] Vervaat | Success epochs in Bernoulli trials with applications in number theory[END_REF], Chapter 4).

A Diophantine Equation.

Let N = {ν 0 , . . . , ν P } be a finite set of integers. Consider the diophantine equation (2.5) x 1 + . . . + x n = y 1 + . . . + y n , in which the unknown x i , y j , 1 ≤ i, j ≤ n, are subject to belong to N. Let N n (N) denote the number of 2n-uples (x 1 , . . . , x n , y 1 , . . . , y n ) ∈ N 2n which satisfy (2.5).

Examine the basic case N = {0, . . . , P -1} and note N n (P ) = N n (N). Recall the approach used in [START_REF] Postnikov | Introduction to analytic number theory[END_REF] §2.4. Let X be a random variable defined by

P{X = k} = P -|k| P 2 if 0 ≤ |k| < P, 0 if |k| ≥ P.
We easily verify that E X = 0, σ 2 = E X 2 = P 2 -1 6 and E |X| 3 ≤ CP 3 . Moreover, E e 2iπtX = (1/P )F P -1 (2πt) where F m is the Fejér kernel,

F m (u) = 1 m + 1 sin m+1 2 u sin u 2 2 .
Note that if u(k) is the number of solutions of the equation x-y = k, 0 ≤ x ≤ P -1, 0 ≤ y ≤ P -1, then u(k) = P -|k| if |k| < P , and u(k) = 0 if |k| ≥ P . So that in turn P{X = k} = u(k) P 2 , k ∈ Z. Let X 1 , . . . , X n be independent copies of X and note S n = X 1 + . . . + X n . As (x 1 -y 1 ) + . . . + (x n -y n ) = 0 if and only if x 1 -y 1 = k 1 , . . . , x n -y n = k n , for some integers k j verifying k 1 + . . . + k n = 0, we have

P{S n = 0} = k 1 +...+kn =0 |k i |<P P X 1 = k 1 . . . P X n = k n = N n (P ) P 2n .
We have, as a direct consequence of the approximate local limit theorem with effective remainder given in [START_REF] Giuliano | Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery[END_REF], Corollary 1.8,

(2.6) N n (P ) √ n P 2n-1 = 3/π + O 1 P 2 + P √ n ,
uniformly over n, P such that for n ≥ CP 2 .

Remark 2.3. (i) As P{S n = 0} = 1 0 | sin P πt P sin πt | 2n dt, it is easy to bound from below N n (P ) by CP 2n-1 /
√ n and to get the upper bound C ε P 2n-1+ε / √ n, for any ε > 0, uniformly in P and n. See for instance [START_REF] Weber | Small divisors of Bernoulli sums[END_REF], inequality (2.3).

In fact, one "can"take ε = 0.

Theorem 2.4 ([7], Th. 2.1).

There exist absolute constants C ′ , C ′′ such that for any positive integers P and n, (2.7)

C ′ P 2n-1 √ n ≤ N n (P ) ≤ C ′′ P 2n-1 √ n .
The proof depends on finer bounds of the previous Fejér integrals, requiring more elaborated calculations.

Remark 2.5. We don't exactly know how the normalized ratios Nn(P )

P 2n
behave when n and P vary simultaneously; a question which is tightly related to the variation properties of powers of the Fejér kernels {F n Pj (u), j ≥ 1} for growing sequences {P j , j ≥ 1}. 2.3. Freiman-Pitman's Probabilistic Model of the Partition Function. This is probably the most informative example. Let q m (n), m ≤ n, denote the number of partitions of n into distinct parts, each of which is at least m, namely the number of ways to express n as (2.8)

n = i 1 + . . . + i r , m ≤ i 1 < . . . < i r ≤ n.
Let X m , . . . , X n be independent random variables defined by (2.9)

P{X j = 0} = 1 1 + e -σj , P{X j = j} = e -σj 1 + e -σj .
The random variable Y = X m + . . . + X n can serve to modelize the partition function q m (n). There is a one-to-one correspondence between the number of partitions of n of the required type and the number of vectors (x m , . . . , x n ) with x j = 0 or 1 such that mx m + . . . + nx n = n. Notice that (1 + e -σj ) P{Y = n}.

In [START_REF] Freiman | Partitions into distinct large parts[END_REF] p. 387 and 389, the authors noticed that an appropriate local limit theorem would allow to write

P{Y = n} ∼ e -(E Y -n) 2 /(2Var(Y )) /
q m (n) ∼ e σn n j=m (1 + e -σj ) 1 B √ 2π .
In place, Freiman and Pitman directly estimated the integral in (2.10) in a long delicate work [START_REF] Freiman | Partitions into distinct large parts[END_REF].

Remark 2.6. By Euler's pentagonal theorem, q 0 (n) appears as a coefficient in the expansion of k≤n (1 + e ikθ ).

The basic problem illustrated by the previous examples states as follows.

Problem 2.7. Let {k j , j ≥ 1} be an increasing sequence of positive integers and {p j , j ≥ 1} be a sequence of reals in ]0, 1[. Describe the CLT and LLT for the sequence

S n = k 1 β 1 + . . . + k n β n , n ≥ 1,
where β j are independent Bernoulli random variables defined by (2.11)

P{β j = 1} = p j P{β j = 0} = 1 -p j . (j ≥ 1)
In the Freiman-Pitman model, the system of independent random variables varies with the choice of the integer. And so there is, properly speaking, no central limit theorem involved and thereby no local limit theorem either, except when placing the problem in the setting of triangular arrays. Corresponding forms of the central limit theorem exist. As to suitable versions of the local limit theorem for triangular arrays with remainder term, we don't know whether such a result exists in the litterature. Thus it makes sense to also consider a "local" version of the previous problem.

Problem 2.8 (Finite version). To obtain effective sharp estimates of

P{S n = N }.
We refer to [START_REF] Giuliano | Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery[END_REF] where this question is investigated.

Returning to the Freiman-Pitman model, we observe that the relevant question rather concerns the search of sharp estimates of P{S n = 0} (namely of 1/2 -1/2 E e 2iπtSn dt), the random variables being centered, than working out a local limit theorem, which is quite another problem. Nevertheless, this model, as well as others previously reviewed, sheds light on limitations to the domain of validity of the local limit theorem, in a quite informative way. Some further useful remarks are necessary. We note throughout {ς, ς j , j ≥ 1} a sequence independent standard Bernoulli random variables (namely associated with p j ≡ 1/2) and

T n = ς 1 + . . . + ς n n ≥ 1.
< p j < 1/2, 1 ≤ j ≤ n, S n = k 1 ε 1 ς 1 + . . . + k n ε n ς n . Problem 2.
8 reduces to first estimate (conditionnally to ε j ) a sum of the same kind

T ′ n = k ′ 1 ς 1 + . . . + k ′ n ς n
with k ′ j increasing, but where the Bernoulli random variables are standard. Remark 2.10. If 1/2 < a < 1, let τ 0 be verifying 0 < τ 0 < 2 min(α, 1 -α). Define a pair of random variables (V, ε) as follows.

P{(V, ε) = (1, 1)} = 0 P{(V, ε) = (1, 0)} = α -τ0 2 . P{(V, ε) = (0, 1)} = τ 0 P{(V, ε) = (0, 0)} = 1 -α -τ0 2 .
Let ς be independent from (V, ε).

Then V + ες L = β. Remark 2.11. Fix the integer n. Let T ′ m = k m ς m + . . . + k n ς n , 1 ≤ m ≤ n,
and consider the parallelogram H m = h = k i1 + . . . + k ir with i 1 < . . . < i r and 1 ≤ r ≤ m . Then we have the following formula

P{T ′ 1 = b} = 1 2 m h∈Hm∪{0} P{T ′ n-m = b -h}. (2.12)

Weighted Local Limit Theorems.

We use the characteristic function method to study the local limit theorem for the sums

B ν = β 1 + . . . + β ν , ν = 1, 2, . . .
where β j are independent random variables defined by

P{β j = 0} = ϑ j , P{β j = k j } = 1 -ϑ j , (3.1)
with 0 < ϑ j < 1 for each j, and k j are increasing positive weights. Let

Var(B ν ) = ν j=1 (1 -ϑ j )ϑ j k 2 j , ϑ = ν inf j=1 ϑ j (1 -ϑ j ).
Theorem 3.1. Let ̺n ≤ k < k + ν ≤ n where 0 < ̺ < 1, n is some positive integer, and let k j = k + j -1, j = 1, . . . , ν. Let 1/24 < ε < 1/6. For every m ∈ Z,

P{B ν = n} - e - ( k+ν-1 j=k (1-ϑ j )j-n) 2 2Var(Bν ) 2πVar(B ν ) ≤ C
Var(B ν )

ν 1/6-4ε ϑ 1/3 ρ + e -2π 2 ϑρ 2 δ 2 ν 1/3 . Put (3.2) P ν = ν j=1 (1 -ϑ j ), M ν = ν j=1 (1 -ϑ j )k j , B ν = ν j=1 (1 -ϑ j )ϑ j k 2 j .
Theorem 3.2.

sup n∈Z P{B ν = n} - 1 √ 2πB ν e -(n-Mν ) 2 2Bν ≤ C P ν . Further if n 0 := ν j=1 (1 -ϑ j )k j is integer, then P{β 1 + . . . + β ν = n 0 } - 1 √ 2πB ν ≤ C P ν .
Before passing to the proofs, we begin with making a brief analysis. Let ϕ j (t) = E e 2iπtβj , ϕ Bν (t) = E e 2iπtBν . By the Fourier inversion formula,

P{B ν = m} = |t|≤τ e -2iπmt ϕ Bν (t)dt + τ ≤|t|≤ 1 2 e -2iπmt ϕ Bν (t)dt := I τ (ν, m) + I τ (ν, m),
where τ > 0 will be chosen to be small, depending of m. The first integral term produces the main term and is easily tractable. The estimation of the second integral term is in fact the hard part of the problem, where all the difficulty is concentrated. It is necessary to show that

τ ≤|t|≤ 1 2 e -2iπmt ϕ Bν (t)dt ≪ 1
Var(B ν ) .

There seems to be no other way than controlling τ ≤|t|≤ 1 2

|ϕ Bν (t)|dt. From Lemma 3.3-(i) will follow that |ϕ Bν (t)| ≤ exp -2 ν j=1 ϑ j (1 -ϑ j ) sin 2 πtk j . The whole matter consequently directly depends on the behaviour of the sine sum ν j=1 ϑ j (1 -ϑ j ) sin 2 πtk j away from 0, an obviously difficult question. Thus, answers can be expected only for specific cases.

3.1. Estimates of I τ (ν, m). Recall Lemma 3 in [START_REF] Freiman | Partitions into distinct large parts[END_REF]. Although stated with the choice of probability values given by (2.9), this lemma is general. For completion, we have included a slightly shorter proof. Lemma 3.3. Let m be a positive real and p be a real such that 0 < p < 1. Let β be a random variable defined by P{β = 0} = p, P{β = m} = 1 -p = q. Let ϕ(t) = E e 2iπtβ . Then we have the following estimates, (i) For all real t, |ϕ(t)| ≤ exp -2pq sin 2 πtm

(ii) If q| sin πtm| ≤ 1/3, ϕ(t) = exp 2iπqmt -2π 2 pqm 2 t 2 + B(t) ,
and |B(t)| ≤ Cqm 3 t 3 , the constant C being absolute.

Proof. One verifies that |ϕ(t)| 2 = 1 -4pq sin 2 πmt. As moreover 1 -ϑ ≤ e -ϑ if ϑ ≥ 0, (i) follows.

Write now ϕ(t) = 1 + q e 2iπmt -1 = 1 + u and notice that |u| = 2q| sin πmt|. We use the fact that if |θ| ≤ 2/3, then

1 + θ = exp{θ -θ 2 + B}, |B| ≤ C|θ| 3 .
And C is an absolute constant. From the bound |e z -(1

+ z 1! + . . . + z n n! )| ≤ |z| n+1 (
u -q 2iπmt -2(πmt) 2 | ≤ Cqm 2 |t| 2 u 2 + (2qπmt) 2 | ≤ Cq 2 m 3 |t| 3 |u| 3 ≤ Cq 3 m 3 |t| 3 .
As we assumed q| sin πtm| ≤ 1/3, we consequently find that

ϕ(t) = 1 + u = exp{u -u 2 + B} = exp 2iπqmt -2π 2 pqm 2 t 2 + B(t) , with |B(t)| ≤ Cqm 3 t 3 .
The next Lemma provides an estimate for the main integral term. Let 0 < δ ≤ 1 3π and put

(3.3) τ = δ ( ν j=1 (1 -ϑ j )k 3 j ) 1/3 . Lemma 3.4. For every n ∈ Z, τ -τ e -2iπnt ϕ Bν (t)dt - e - ( ν j=1 (1-ϑ j )k j -n) 2 2Var(Bν ) 2πVar(B ν ) ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν )
Var(B ν ) .

Further if n

0 := ν j=1 (1 -ϑ j )k j is integer, then τ -τ e -2iπn0t ϕ Bν (t)dt = 1 2πVar(B ν ) 1 + B , with |B| ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν )
1 + Var(B ν ) .

Proof of Lemma 3.4. As δ ≤ 1 3π , we observe that for j = 1, . . . , ν,

sup |t|≤τ (1 -ϑ j )| sin πtk j | ≤ (1 -ϑ j )πτ k j = δ(1 -ϑ j )πk j ( ν j=1 (1 -ϑ j )k 3 j ) 1/3 ≤ δ(1 -ϑ j ) 1/3 π ≤ 1 3 . Lemma 3.3 thus implies, ϕ Bν (t) = ν j=1 ϕ j (t) = exp 2iπt ν j=1 (1 -ϑ j )k j -2π 2 t 2 Var(B ν ) + B 1 (t) ,
and

|B 1 (t)| ≤ C|t| 3 ν j=1 (1 -ϑ j )k 3 j . By (3.3), sup |t|≤τ |B 1 (t)| ≤ Cτ 3 ν j=1 (1 -ϑ j )k 3 j = Cδ 3 .
Noting then ς = ν j=1 (1 -ϑ j )k j -n and writing that ϕ

Bν (t) = exp 2iπt(ς + n) -2π 2 t 2 Var(B ν ) + B 1 (t) ,
we thus deduce the following bound

τ -τ e -2iπnt ϕ Bν (t) -e 2iπtς-2π 2 t 2 Var(Bν ) dt ≤ τ -τ e -2iπnt ϕ Bν (t) -e 2iπtς-2π 2 t 2 Var(Bν ) dt = τ -τ e 2iπtς-2π 2 t 2 Var(Bν ) e B(t) -1 dt ≤ τ -τ e -2π 2 t 2 Var(Bν ) e B(t) -1 dt ≤ τ -τ |B(t)|dt ≤ Cτ δ 3 .

Now we also have that

(x) = e x 2 /2 ∞ x e -t 2 /2 dt, π √ x 2 + 2π + (π -1)x ≤ R(x) ≤ π (π -2) 2 x 2 + 2π + 2x
for all x ≥ 0, we further have

|H| ≤ |u|≥2πτ √ Var(Bν ) e -u 2 2 du 2π Var(B ν ) = e -2π 2 τ 2 Var(Bν ) 2π Var(B ν )) R 2πτ Var(B ν ) ≤ C e -2π 2 τ 2 Var(Bν ) Var(B ν )(1 + Var(B ν )) . Consequently τ -τ e -2iπnt ϕ Bν (t)dt - e -ς 2 2Var(Bν ) 2πVar(B ν ) ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν ) Var(B ν )(1 + Var(B ν ) )
.

Now if there is an integer n such that ν j=1 (1-ϑ j )k j = n, then ς = 0 and ϕ Bν (t) = exp 2iπtn- 2π 2 t 2 Var(B ν ) + B 1 (t) ; whence τ -τ e -2iπnt ϕ Bν (t)dt = τ -τ e -2π 2 t 2 Var(Bν ) 1 + e B1(t) -1 dt = 1 + B τ -τ e -2π 2 t 2 Var(Bν ) dt, (3.4) with |B| ≤ Cτ δ 3 . As moreover τ -τ e -2π 2 t 2 Var(Bν ) dt - 1 2πVar(B ν ) ≤ C e -2π 2 τ 2 Var(Bν ) Var(B ν )(1 + Var(B ν ) ) , we have τ -τ e -2iπnt ϕ Bν (t)dt = 1 2πVar(B ν ) 1 + B 1 + B 1 ,
where

|B 1 | ≤ Ce -2π 2 τ 2 Var(Bν ) /(1 + Var(B ν ) ). We conclude to τ -τ e -2iπnt ϕ Bν (t)dt = 1 2πVar(B ν ) 1 + B 2 ,
with

|B 2 | ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν )
1 + Var(B ν ) .

3.2.

Estimates of I τ (ν, m). We assume here that k j = k + j -1, j = 1, . . . , ν. Then, Lemma 3.5.

|t|>τ |ϕ Bν (t)|dt ≤ e -ϑν 3 τ 2 2 .
Proof. This is an immediate consequence of the following lemma

j = k + j -1, j = 1, . . . , ν, for every n ∈ Z, τ 0 e -2iπnt ϕ Bν (t)dt - e - ( k+ν-1 j=k (1-ϑ j )j-n) 2 2Var(Bν ) 2πVar(B ν ) ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν )
Var(B ν )

≤ C ν 1/6-4ε ρ -1 + e -2π 2 ρ 2 δ 2 ν 1/3
Var(B ν ) .

By combining with Lemma 3.5 and using Fourier inversion formula,

P{B ν = n} - e - ( k+ν-1 j=k (1-ϑ j )j-n) 2 2Var(Bν ) 2πVar(B ν ) ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν )
Var(B ν ) + e -ϑν 3 τ 2

8

.

We have the following estimates

     (i) δ nν 1/3 ≤ τ = δ ( ν j=1 (1-ϑj )k 3 j ) 1/3 ≤ δ ρn(ϑν) 1/3 , (ii) ϑ(ρn) 2 ν ≤ Var(B ν ) = ν j=1 (1 -ϑ j )ϑ j k 2 j ≤ n 2 ν, (iii) τ 2 Var(B ν ) ≥ ϑδ 2 n 2 ν 2/3 (ρn) 2 ν = ϑρ 2 δ 2 ν 1/3 . Choose δ = ν -ε with 1/24 < ε < 1/6. Then τ δ 3 ≤ δ 4 ρn(ϑν) 1/3 = 1 ρnϑ 1/3 ν 1/3+4ε = ν 1/6-4ε ϑ 1/3 ρnν 1/2 ≤ ν 1/6-4ε ϑ 1/3 ρ Var(B ν )
.

We pass to the control of the error terms. For the major integral term we have,

τ δ 3 + e -2π 2 τ 2 Var(Bν ) Var(B ν ) ≤ 1
Var(B ν )

ν 1/6-4ε ϑ 1/3 ρ + e -2π 2 ϑρ 2 δ 2 ν 1/3 .
Consequently,

P{B ν = n} - e - ( k+ν-1 j=k (1-ϑ j )j-n) 2 2Var(Bν ) 2πVar(B ν ) ≤ C Var(B ν ) ν 1/6-4ε ϑ 1/3 ρ + e -2π 2 ϑρ 2 δ 2 ν 1/3 .
3.4. Other Estimates of I τ (ν, m). The following lemma is relevant. Introduce for q ≥ 1 integer,

= ϕ(q) := ν j=1 θ j cos 2πtk j 2q ν j=1 θ j . Lemma 3.7. For any 0 < c ≤ 1, 1 0 |ϕ Bν (t)|dt ≤ ϕ(q) c 2q + e -(1-c) ν j=1 ϑj (1-ϑj ) . (3.5) ϕ θ1,...,θν ,k1,...,kν (q)
Proof of Lemma 3.7. Let θ j = ϑ j (1-ϑ j ), j = 1 . . . , ν and note E = t; |t| ≤ 1 2 : ν j=1 θ j cos 2πtk j > c ν j=1 θ j . At first, by using Tchebycheff's inequality,

λ{E} ≤ c ν j=1 θ j ) -2q 1 2 -1 2 ν j=1 θ j cos 2πtk j 2q dt = c -2q ϕ(q) 2q , Since |ϕ Bν (t)| ≤ 1, we have 1 0 |ϕ Bν (t)|dt ≤ λ{E} + E c |ϕ Bν (t)|dt ≤ c -2q ϕ(q) 2q + E c
|ϕ Bν (t)|dt, By Lemma 3.3, using that 2 sin 2 a = 1 -cos 2a, we have for all real t,

|ϕ Bν (t)| = ν j=1 |ϕ j (t)| ≤ exp - ν j=1 2θ j sin 2 πtk j = exp - ν j=1 θ j exp ν j=1 θ j cos 2πtk j .
So that

E c |ϕ Bν (t)|dt ≤ E c e -ν j=1 2θj sin 2 πtkj dt ≤ e -ν j=1 θj E c e | ν j=1 θj cos 2πtkj | dt ≤ e -(1-c) ν j=1 θj . By combining 1 0 |ϕ Bν (t)|dt ≤ ϕ(q) c 2q + e -(1-c) ν j=1 θj .
Remark 3.8. Assume ϑ j = ϑ for all j. Then ϕ(q) = ν -1 ν j=1 cos 2πtk j 2q and further

ν j=1 cos 2πtk j 2q 2q = N 2q (N),
where N 2q (N) is the number of solutions of (2.5) with corresponding set of values N = {k 1 , . . . , k ν }. So that

1 0 |ϕ Bν (t)|dt ≤ N 2q (N) (cν) 2q + e -(1-c)ϑ(1-ϑ)ν .
-In the case when N = {1, . . . , ν}, this together with Theorem 2.4 gives

1 0 |ϕ Bν (t)|dt ≤ ν 2q-1 √ q(cν) 2q + e -(1-c)ϑ(1-ϑ)ν = 1 νc 2q √ q + e -(1-c)ϑ(1-ϑ)ν . Taking c = 1 -(2q) -1 gives 1 0 |ϕ Bν (t)|dt ≤ C ν √ q + e -ϑ(1-ϑ)ν/2q . Take q large, q ∼ ϑ(1 -ϑ)ν/3 log ν. It follows that 1 0 |ϕ Bν (t)|dt ≤ C log ν ν 3/2
whereas, in the other hand Var(B ν ) = ϑ(1 -ϑ) ν j=1 j 2 ∼ Cϑ(1 -ϑ)ν 3 ... Remark 3.9. We also have

ν j=1 ϑ j (1 -ϑ j ) sin 2 πtk j ≥ ν ν j=1 ϑ j (1 -ϑ j )
P{B ν = n} - e -(n-E Bν ) 2 2Var(Bν ) 2πVar(B ν ) ≤ C τ δ 3 + e -2π 2 τ 2 Var(Bν ) Var(B ν ) + 1 ν j=1 ϑ j (1 -ϑ j ) ≤ C 1 ( ν j=1 (1 -ϑ j )k 3 j ) 1/3 + e -2π 2 δL 2 ( ν j=1 (1 -ϑ j )k 2 j ) 1/2 + 1 ν j=1 ϑ j (1 -ϑ j ) . ≤ C ν j=1 ϑ j (1 -ϑ j )
.

Now similarly,

P{B ν = n 0 } - 1 2πVar(B ν ) ≤ B 2 2πVar(B ν ) + 1 ν j=1 ϑ j (1 -ϑ j ) ≤ C ( ν j=1 (1 -ϑ j )k 2 j ) 1/2 1 ( ν j=1 (1 -ϑ j )k 3 j ) 1/3 + e -2π 2 δL 2 + 1 ν j=1 ϑ j (1 -ϑ j ) ≤ C ν j=1 ϑ j (1 -ϑ j )
.

This achieves the proof.

4. An ASLLT related to Burr's problem.

Let λ 0 < λ 1 < . . . be a sequence of positive integers, call it A, and let

P (A) = i ε i λ i , ε i = 0 or 1, a i ∈ A and i ε i < ∞ .
Burr asked in [START_REF] Burr | Combinatorial theory and its applications III[END_REF] which sets S of integers are equal to P (A) for some A? He mentioned that if the complement of S grows sufficiently rapidly, then there exists such a sequence A. Hegyvari showed in [START_REF] Hensley | The convolution powers of the Dickman function[END_REF]

that if B = {b i , i ≥ 1} is such that 7 ≤ b 1 < b 2 < • • •
+ b j = b k + b ℓ implies i = k, j = ℓ or i = ℓ, j = k,
then there is no sequence A for which P (A) = N\B. We refer to [START_REF] Burr | Combinatorial theory and its applications III[END_REF][START_REF] Folkman | On the representation of integers as sums of distinct terms from a fixed sequence Canad[END_REF][START_REF] Hensley | The convolution powers of the Dickman function[END_REF] for similar questions. Here we examine a variant of the initial problem. Consider the set E composed with all finite sums (4.3)

λ j1 + . . . + λ jn , 0 ≤ j 1 ≤ . . . ≤ j n , n ≥ 1.
Let 0 < η < 1 and let E η ⊆ E be the set composed with all finite sums λ j1 + . . . + λ jn such that at most ⌊ηn⌋ summands may coincide. Now let {x n , n ≥ 1} be a sequence of integers increasing nearly linearly, so that it is a relatively "full" sequence. More precisely, we assume there are reals a > 1, δ > 0 such that (4.4)

x n -na ∼ δ √ n, n → ∞.

We are interested in estimating from below the proportion of terms from this sequence which may be represented by a sum λ j1 + . . . + λ jn , namely which belong to E.

Theorem 4.1. Let D = g.c.d. λ i -λ j , i > j ≥ 0 . Let also

(4.5) ρ = sup r : ∞ j=0 λ j r j < ∞ .
Assume that 0 < ρ ≤ 1. Then for some 0 < η < 1 depending on both ρ and a,

lim inf t→∞ 1 log t n≤t 1 √ n 1 {xn∈Eη} ≥ D √ 2πσ e -δ 2 2σ 2 .
Here we have noted σ 2 = (1 -r) ∞ j=0 λ 2 j r j -(1 -r) ∞ j=0 λ j r j 2 , and 0 < r < ρ is solution of the equation

(1 -r) ∞ j=0 λ j r j = a.
Further, there exists with probability one a random subsequence λ ′ j = λ ′ j (ω), j = 1, . . ., tending to infinity with n, such that for all n large enough, among λ ′ 1 , . . . , λ ′ n at most ⌊nη⌋ may coincide, and

lim t→∞ 1 log t n≤t 1 √ n 1 {xn=λ ′ 1 +...+λ ′ n } = D √ 2πσ e -δ 2 2σ 2 .
4.1. Preliminaries. We first recall some auxiliary results on which the proof is based. Let X be a square integrable random variable with lattice distribution function F and put

(4.6) µ = E X, σ 2 = E X 2 -(E X) 2 .
Let D be the maximal span of X. Let also {X k , k ≥ 1} be independent copies of X, and consider their partial sums S n = X 1 + . . . + X n , n ≥ 1. We assume throughout that σ > 0. Almost sure versions with rate of Gnedenko's theorem (see after (1.1)) were recently proved in [START_REF] Giuliano | Almost sure local limit theorems with rate[END_REF]. Let g(x) = D √ 2πσ e -x 2 /(2σ

n = κ + O η (log n) -1/2+η , then 1 log N n≤N 1 √ n 1 {Sn=κn} a.s. = g(κ) + O η (log N ) -1/2+η .
4.2. Proof. We consider the following random model. Let 0 < r < ρ and let X be a random variable defined by 2 . This case being excluded by construction, we have σ > 0. Let also {X k , k ≥ 1} be independent copies of X, and consider their partial sums S n = X 1 + . . . + X n , n ≥ 1 Now observe that

P{X = λ j } = (1 -r)r j , j = 0, 1, . . . . The function µ(r) = (1 -r) ∞ j=0 λ j r j is continuous on [0, ρ[. Further µ(0) = 0 and lim r↑ρ µ(r) = ∞. We can thus select a real r ∈]0, ρ[ so that E X = µ(r) = a. Next E X 2 = (1-r) ∞ j=0 λ 2 j r j < ∞. And because r < ρ, E X 2+α < ∞ for some positive α. It is further clear that σ 2 = E X 2 -(E X) 2 cannot vanish unless X is a constant almost surely, since σ 2 = E (X -E X)
P X i1 = X i2 = . . . = X i k = ∞ j=0 P{X = λ j } k = (1 -r) k ∞ j=0 r kj = (1 -r) k (1 -r k ) ≤ 2(1 -r) k ,
if k is large, which we do assume. Thus

P ∃1 ≤ i 1 < . . . < i k ≤ n : X i1 = X i2 = . . . = X i k ≤ 2C k n (1 -r) k .
We take k = ⌊nη⌋. Since n! ∼ √ 2πnn n e -n , we have for n large

C k n ≤ 2 n n -k n-k n k k n 2π(n -k)k 1/2 ≤ 1 1 -η n(1-η) 1 η nη = 1 1 -η 1 η -1 1 η nη . Let C n = ∃1 ≤ i 1 < . . . < i ⌊nη⌋ : X i1 = X i2 = . . . = X i ⌊nη⌋ , n = 1, 2, . . .
These sets are non-increasing. And so

P ∞ n=m C n ≤ P C m ≤ 1 -r η(1 -η) 1 η -1 mη . Since (1 -η) 1 η -1 = exp{-1-η η log 1 1-η } → 1 as η ↑ 1, and 0 < r < 1, it follows that one can select η so that 1 -r η(1 -η) 1 η -1 < 1. This choice implies that P lim sup n→∞ C n = 0.
Thus, with probability one, for all n large enough, there is no k-uple, 1 ≤ i 1 < . . . < i k , with k ≥ ⌊nη⌋, such that X i1 = X i2 = . . . = X i k . In particular, with probability one, for all n large enough, at most ⌊nη⌋ from the random variables X i , i ≤ n may coincide.

Besides, using Gnedenko's theorem we have, uniformly in N , √ n P{S n = N } = D σ √ 2π e -(N -na) as claimed. The second part of the Theorem is a direct consequence of (4.12).

A Concluding Remark.

A probably well-known fact is that the local limit theorem is not a sufficiently sharp tool for estimating the probability P{S n ∈ E}, where E is an infinite set of integers and S n = X 1 +. . .+X n a sum of independent copies of a random variable X. As we could not find in the litterature an explicit example, we mention here a very simple one given in [START_REF] Weber | Correlation properties of divisors of Bernoulli sums[END_REF] and showing that this already arises for bounded random variables and for elementary sets E, namely arithmetic progressions.

Let d be some positive integer and take E = dN. Let also B n = β 1 + . . . + β n , where β i are independent standard Bernoulli random variables. By using the sharpest form of the local limit theorem for standard Bernoulli random variables, derived from [START_REF] Petrov | Sums of Independent Random Variables[END_REF]Theorem 13 = O log 5/2 n n 3/2 , which is much better than (5.1). We refer to [START_REF] Weber | Correlation properties of divisors of Bernoulli sums[END_REF] for more details.

1 + 1 0e 1 0 n j=m 1 +

 1111 e -σj e 2iπαj e -2iπαn dα = e σn x j ∈{0,1} m≤j≤n -σ(mxm+...+nxn) e -2iπ(mxm+...+nxn-n)α dα = e σn x j ∈{0,1} m≤j≤n e -σn χ{mx m + . . .+ nx n = n} = q m (n).Hence the formula (in which σ only appears in the right-hand side) (2.10) q m (n) = e σn e -σj e 2iπαj e -2iπαn dα.This also implies (letting ϕ(t) = E e 2iπtY be the characteristic function of Y ) q m (n) = e )e -2iπαn dα = e σn n j=m

  2πVar(Y ). Choosing σ as being the unique solution of the equation E Y = n j=m j 1+e σj = n would then give P{Y = n} ∼ 1/B √ 2π, and by reporting

Remark 2 . 9 (

 29 Reduction to standard Bernoulli random variables). Let β be a Bernoulli random variable with P{β = 1} = α = 1 -P{β = 0}. Assume 0 < α < 1/2. Let ε, ς be such that β, ε, ς are independent and P{ε = 1} = 2α = 1 -P{ε = 0}. Trivially ες L = β. We can thus write when 0

  n+1)! e |z| , valid for z ∈ C and n ∈ N * ([18], 3.8.25), we get by applying it with z = 2iπmt,

2π 2 t 2

 2 Var(Bν ) dt = R e 2iπtς-2π 2 t 2 Var(Bν )and recalling Boyd's estimate[18, p. 179] of Mill's ratio R

Lemma 3 . 6 ( 1 j=m sin 2 πjt ≥ k 4 min 1 , (tk) 2 .-ϑ 2 ν j=1 sin 2 τ 2 2 . 3 . 3 .

 3611222233 [START_REF] Freiman | Partitions into distinct large parts[END_REF],Lemma 8). For |t| ≤ 1/2 and any positive integers m and k such that k ≥ 2 we have m+k-By Lemma 3.3-(i), |ϕ Bν (t)| ≤ exp -2 ν j=1 ϑ j (1 -ϑ j ) sin 2 πtk j ≤ exp -2ϑ ν j=1 sin 2 πtk j .Thus |t|>τ |ϕ Bν (t)|dt ≤ |t|>τ e πjt dt ≤ |t|>τ e -ϑν 2 min(1,ν|t|) 2 dt ≤ e -ϑν 3 Proof of Theorem 3.1. By Lemma 3.4 applied with k

1 )P

 1 By operating quite differently, we obtained in[START_REF] Weber | Small divisors of Bernoulli sums[END_REF] the following uniform estimate. Let Θ(d, m) be the Theta elliptic function defined byΘ(d, m) = ℓ∈Z B n ∈ dN -Θ(d, n) d = O log 5/2 n n 3/2 .By using Poisson summation formula, this implies thatsup 2≤d≤n P B n ∈ dN -2 πn z≡0(d)e -(2z-n) 2 2n

  By picking ω in a measurable set of full measure, we find by what proceeds and (4.11), that there exists a subsequenceλ ′ 0 ≤ λ ′ 1 ≤ . . ., λ ′Further, for all n large enough, at most ⌊nη⌋ from the summands λ ′ 1 , . . . , λ ′ n may coincide. Thusλ ′ n → ∞ with n. But x n = λ ′ 1 + . . . + λ ′n and the fact that among λ ′ 1 , . . . , λ ′ n , at most ⌊nη⌋ can coincide, implies that x n ∈ E η . We consequently deduce

	σ	D √ 2π e -δ 2 2σ 2 as n → ∞. Since the assumptions of Theorem 4.2 are also fulfilled, it follows that
	(4.11)	lim t→∞	1 log t n≤t	1 √ n	1 {Sn=xn}	a.s. =	D 2πσ √	e -δ 2 2σ 2 .
						n = λ ′ n (ω), such that
	(4.12)	lim t→∞	1 log t n≤t	1 √ n	1 {xn=λ ′ 1 +...+λ ′ n } =	D 2πσ √	e -δ 2 2σ 2 .
			lim inf t→∞	1 log t n≤t	1 √ n	1 {xn∈Eη} ≥	D 2πσ √	e -x 2 2σ 2 ,

2 /2nσ 2 + o(1). By assumption lim n→∞ (x n -na)/ √ n = δ, so that (4.8) is satisfied. Therefore √ n P{S n = x n } ∼