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In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setting., Comment: 41 pages. Revised version has minor corrections and additions. To appear in Math Z

We consider a free quantized Klein-Gordon field in a spacetime (M, g) containing a stationary black hole, more precisely a spacetime with a stationary bifurcate Killing horizon in the sense of Kay and Wald. We prove the existence of the Hartle-Hawking-Israel ground state, which is a pure state on the whole spacetime whose restriction to the exterior of the black hole is a thermal state at Hawking temperature.We show that the HHI state is a Hadamard state and is the unique Hadamard extension of the above thermal state to the whole spacetime. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderon projector for elliptic boundary value problems.

Aussi déposé sur ArXiv Math, reference arXiv:2010.09793; The present paper establishes equivalence between uniform rectifiability of the boundary of a domain and the property that the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). The results are novel in a variety of ways, in particular (1) this is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, appears in the literature, the particular form of the "direct" result established here is new even in the half space; (2) the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions; (3) to the best of the authors' knowledge, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition); (4) our theorems apply to all domains, with possibly lower dimensional boundaries: this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d < n − 1, in R n. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

We give a rigorous definition of the Unruh state in the setting of massless Dirac fields on slowly rotating Kerr spacetimes. In the black hole exterior region, we show that it is asymptotically thermal at Hawking temperature on the past event horizon. Furthermore, we demonstrate that in the union of the exterior and interior regions, the Unruh state is pure and Hadamard. The main ingredients are the H\"afner-Nicolas scattering theory, new microlocal estimates for characteristic Cauchy problems and criteria on the level of square-integrable solutions., Comment: 62 pages; v2: various remarks and references added, to appear in Ann. Sci. E.N.S

Construction de mesures elliptiques absolument continues sur un ensemble de Cantor du plan. 18p/10dessins; It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in R 2 whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.

It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of now well-understood additional topological constraints. The topological restrictions, while mild, are necessary, as the counterexamples of Bishop and Jones show, and no analogues of these results have been available for higher co-dimensional sets. In the present paper, we show that for any $d

Take an open domain $\Omega \subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $\Omega$ can be a ball on $\mathbb R^3$, minus one of its diameters $D$, or $\Omega \subset \mathbb R^3$ could be a so-called saw-tooth domain, with a boundary consisting of pieces of 1-dimensional curves intercepted by 2-dimensional spheres. Under appropriate geometric assumptions, such as the existence of doubling measures on $\Omega$ and $\partial \Omega$ with appropriate size conditions, we construct a class of degenerate elliptic operators $L$ adapted to the geometry, and establish key estimates of elliptic theory associated to those operators. This includes boundary Poincar\'e and Harnack inequalities, maximum principle, and H\"older continuity of solutions at the boundary. We introduce Hilbert spaces naturally associated to the geometry, construct appropriate trace and extension operators, and use them to define weak solutions to $Lu=0$. Then we prove De Giorgi-Nash-Moser estimates inside $\Omega$ and on the boundary, solve the Dirichlet problem and thus construct an elliptic measure $\omega_L$ associated to $L$. At last, we introduce Green functions, and use them to prove a comparison principle. Since our theory emphasizes measures, rather than the geometry per se, the results are new even in the classical setting of a half-plane $\mathbb R^2_+$ when the boundary $\partial \mathbb R^2_+= \mathbb R$ is equipped with a doubling measure $\mu$ singular with respect to the Lebesgue measure on $\mathbb R$. Finally, the present paper provides a generalization of the celebrated Caffarelli-Sylvestre extension operator from its classical setting of $\mathbb R^{n+1}_+$ to general open sets, and hence, an extension of the concept of fractional Laplacian to Ahlfors regular boundaries and beyond., Comment: 116 pages. In version 2, we completed our theory with Green functions and a comparison principle

International audience; In a previous work, we have proposed a method for the analysis of the bounce back boundary condition with the Taylor expansion method in the linear case. In this work two new schemes of modified bounce back are proposed. The first one is based on the expansion of the iteration of the internal scheme of the lattice Boltzmann method. The analysis puts in evidence some defects and a generalized version is proposed with a set of essentially four possible parameters to adjust. We propose to reduce this number to two with the elimination of spurious density first order terms. Thus a new scheme for bounce back is found exact up to second order and allows an accurate simulation of the Poiseuille flow for a specific combination of the relaxation and boundary coefficients. We have validated the general expansion of the value in the first cell in terms of given values on the boundary for a stationary ''accordion'' test case.

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central series of the fundamental group. This last result has yet to be extended to higher order homologies.

It has been observed that on a weighted graph, an extension of Wilson's algorithm provides an independent pair pT , Lq, T being a spanning tree and L a Poissonian loop ensemble. This association can be interpreted in the framework of symmetric and skew symmetric Fock spaces. Given a weighted graph, we show how to define two types of natural interactions which correspond to local interactions between two Fock spaces. The first type of interaction involves loop ensembles and spanning trees. The second type of interaction involves loop holonomies and random connections.

We investigate the relation to random configurations and combinatorial maps of the Eulerian networks defined by Poissonian emsembles of Markov loops.

We study the local regularity of sliding almost minimal sets of dimension 2 in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap films bounded by a curve, and their definition is similar to Almgren's. We aim for a local description, in particular near L and modulo $C^{1+$\epsilon$}$ diffeomorphisms, of such sets $E$, but in the present paper we only obtain a full description when $E$ is close enough to a half plane, a plane or a union of two half planes bounded by the same line, or a transverse minimal cone of type $Y$ or $T$. The main tools are adapted near monotonicity formulae for the density, including for balls that are not centered on L, and the same sort of construction of competitors as for the generalization of J. Taylor's regularity result far from the boundary., Comment: 337 pages, une trentaine de figures

International audience; These lecture notes give an exposition of microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes. We concentrate on free fields and the corresponding quasi-free states and mainly on Klein-Gordon fields.Updated version with some errors corrected.

We study the autocorrelation coefficients of the Rudin-Shapiro polynomials, proving in particular that their maximum on the interval $[1, 2^n)$ is bounded from below by $C_1 2^{\alpha n}$ and is bounded from above by $C_2 2^{\alpha' n}$ where $\alpha = 0.7302852...$ and $\alpha' = 0.7302867...$.

International audience; Clinical symptoms of dengue virus (DENV) infection, the most prevalent arthropod-borne viral disease, range from classical mild dengue fever to severe, life-threatening dengue shock syndrome. However, most DENV infections cause few or no symptoms. Asymptomatic DENV-infected patients provide a unique opportunity to decipher the host immune responses leading to virus elimination without negative impact on an individual's health. We used an integrated approach of transcriptional profiling and immunological analysis to compare a Cambodian population of strictly asymptomatic viremic individuals with clinical dengue patients. Whereas inflammatory pathways and innate immune response pathways were similar between asymptomatic individuals and clinical dengue patients, expression of proteins related to antigen presentation and subsequent T cell and B cell activation pathways was differentially regulated, independent of viral load and previous DENV infection history. Feedback mechanisms controlled the immune response in asymptomatic viremic individuals, as demonstrated by increased activation of T cell apoptosis-related pathways and FcγRIIB (Fcγ receptor IIB) signaling associated with decreased anti-DENV-specific antibody concentrations. Together, our data illustrate that symptom-free DENV infection in children is associated with increased activation of the adaptive immune compartment and proper control mechanisms, leading to elimination of viral infection without excessive immune activation, with implications for novel vaccine development strategies.

We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calder\'{o}n projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator., Comment: 36 p

R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by R\'enyi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the R\'enyi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of R\'enyi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of $\sigma$-algebras and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results., Comment: To appear in IEEE Transactions on Information Theory

International audience; We investigate the future asymptotic behavior of Gowdy spacetimes on T3T3, when the metric satisfies weak regularity conditions, so that the metric coefficients (in suitable coordinates) are only in the Sobolev space H1H1 or have even weaker regularity. The authors recently introduced this class of spacetimes in the broader context of T2T2-symmetric spacetimes and established the existence of a global foliation by spacelike hypersurfaces when the time function is chosen to be the area of the surfaces of symmetry. In the present paper, we identify the global causal geometry of these spacetimes and, in particular, establish that weakly regular Gowdy spacetimes are future timelike geodesically complete. This result extends a theorem by Ringström for metrics with sufficiently high regularity. We emphasize that our proof of the energy decay is based on an energy functional inspired by the Gowdy-to-Ernst transformation. In order to establish the geodesic completeness property, we prove a higher regularity property concerning the metric coefficients along timelike curves and we provide a novel analysis of the geodesic equation for Gowdy spacetimes, which does not require high-order regularity estimates. Even when sufficient regularity is assumed, our proof provides an alternative and shorter proof of the energy decay and of the geodesic completeness property for Gowdy spacetimes.

This is a first paper by the authors dedicated to the distribution of eigenvalues for random perturbations of large bidiagonal Toeplitz matrices., 34 pages, 4 figures

The main goal of this paper is to prove inequalities on the reconstruction error for kernel principal component analysis. With respect to previous work on this topic, our contribution is twofold: (1) we give bounds that explicitly take into account the empirical centering step in this algorithm, and (2) we show that a "localized" approach allows to obtain more accurate bounds. In particular, we show faster rates of convergence towards the minimum reconstruction error; more precisely, we prove that the convergence rate can typically be faster than n ?1/2. We also obtain a new relative bound on the error. A secondary goal, for which we present similar contributions, is to obtain convergence bounds for the partial sums of the biggest or smallest eigenvalues of the kernel Gram matrix towards eigenvalues of the corresponding kernel operator. These quantities are naturally linked to the KPCA procedure; furthermore these results can have applications to the study of various other kernel algorithms. The results are presented in a functional analytic framework, which is suited to deal rigorously with reproducing kernel Hilbert spaces of infinite dimension.

International audience; In this paper we consider a control system of type x+ = x + u, wnere the u take values in a set of 2m + 1 integer numbers, symmetric with respect to 0. and analyse the reachable set after K steps. Our aim is to find a set of m input values such that the reachable set after K steps contains an interval of integers [-N,.,., N] with N as large as possible. For m = 2 and 3, we completely solve the problem and characterize the metric associated to this quantized control system. Copyright © 2004 IFAC.

Motivation: The spatial conformation of the chromosome has a deep influence on gene regulation and expression. Hi-C technology allows the evaluation of the spatial proximity between any pair of loci along the genome. It results in a data matrix where blocks corresponding to (self-)interacting regions appear. The delimitation of such blocks is critical to better understand the spatial organization of the chromatin. From a computational point of view, it results in a 2D segmentation problem. Results: We focus on the detection of cis-interacting regions, which appear to be prominent in observed data. We define a block-wise segmentation model for the detection of such regions. We prove that the maximization of the likelihood with respect to the block boundaries can be rephrased in terms of a 1D segmentation problem, for which the standard dynamic programming applies. The performance of the proposed methods is assessed by a simulation study on both synthetic and resampled data. A comparative study on public data shows good concordance with biologically confirmed regions. Availability and implementation: The HiCseg R package is available from the Comprehensive R Archive Network and from the Web page of the corresponding author. Contact: celine.levy-leduc@agroparistech.fr

Nous obtenons un résultat général de finitude pour le H1 de certains schémas en groupes linéaires sur les anneaux de (S-)entiers des corps globaux.

International audience

We discuss scaling limits of random planar maps chosen uniformly over the set of all $2p$-angulations with $n$ faces. This leads to a limiting space called the Brownian map, which is viewed as a random compact metric space. Although we are not able to prove the uniqueness of the distribution of the Brownian map, many of its properties can be investigated in detail. In particular, we obtain a complete description of the geodesics starting from the distinguished point called the root. We give applications to various properties of large random planar maps.

International audience; This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic (\cite{ShaWilCriKan02CDL}, chapter \ref{KPCA1chap}). Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. We prove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence this allows to infer stability results for these estimated spaces.

The problem of manipulation by low-complexity robot hands is a key issue since many years. The performance of simplified hardware manipulators relies on the exploitation of nonholonomic effects that occur in rolling. Beside this issue, more recently, the attention of the scientific community has been devoted to the problems of finite capacity communication channels and of constraints on the complexity of computation. Quantization of controls proved to be efficient for dealing with such kinds of limitations. With this in mind, we consider the rolling of a pair of smooth convex objects, one on top of the other, under quantized control. The analysis of the reachable set is performed by exploiting the geometric nature of the system which helps in reducing to the case of a group acting on a manifold. The cases of a plane, a sphere and a body of revolution rolling on an arbitrary surface are treated in detail. (c) 2005 Elsevier B.V. All rights reserved.

Rolling a ball on a plane is a standard example of nonholonomy reported in many textbooks, and the problem is also well understood for any smooth deformation of the surfaces. For nonsmoothly deformed surfaces, however, much less is known. Although it may seem intuitive that nonholonomy is conserved (think e.g. to polyhedral approximations of smooth surfaces), current definitions of "nonholonomy" are inherently referred to systems described by ordinary differential equations, and are thus inapplicable to such systems. In this paper, we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces. We provide a description of such reachable set, discuss conditions under which the set is dense, or discrete, or has a compound structure, and provide a method for steering the system to a desired reachable configuration, robustly with respect to model uncertainties. Based on ideas and concepts encountered in this case study, and in some other examples we provide, we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of (kinematic) nonholonomy to nonsmooth, discrete, and hybrid dynamical systems. To capture the essence of phenomena commonly regarded as "nonholonomic," at least two irreducible concepts are to be defined, of "internal" and "external" nonholonomy, which may coexist in the same system. These definitions are instantiated by examples.

In this paper, we consider linear switched systems x(t) = A/sub u(t)/x(t), x /spl epsiv/ /spl Ropf//sup n/, u /spl epsiv/ U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.

Consider the control system (/spl Sigma/) given by x/spl dot/ = x(f + ug) where x /spl isin/ SO(3), |u| /spl les/ 1 and f, g /spl isin/ so(3) define two perpendicular left-invariant vector fields normalized so that ||f|| = cos(/spl alpha/) and ||g|| = sin(/spl alpha/), /spl alpha/ /spl isin/ (0, /spl pi//4). In this paper, we provide an upper bound and a lower bound for N(/spl alpha/), the maximum number of switchings for time-optimal trajectories of (/spl Sigma/). More precisely, we show that N/sub S/(/spl alpha/) /spl les/ N(/spl alpha/) /spl les/ N/sub S/(/spl alpha/) /spl les/+4, where N/sub S/(a) is a suitable integer function of /spl alpha/ such that N/sub S/(/spl alpha/) /sub /spl alpha//spl rarr/0//sup /spl sim//. The result is obtained by studying the time optimal synthesis of a projected control problem on /spl Ropf/P/sup 2/, where the projection is defined by an appropriate Hopf fibration.

International audience; In this paper we deal with the problem of stabilization of uncertain systems in the presence of input constraint. First algebraic conditions are derived for input-to-state stability of linear system with saturated linear feedback of low dimension. Then a recursive design procedure is derived for robust stabilization of block upper triangular nonlinear systems with feedforward structure.

In this paper, we address the issue of motion planning for the control system L R that results from the rolling without slipping nor spinning of a two dimensional Riemannian manifold M 1 onto another one M 2 . We present two procedures to tackle the motion planning problem when M 1 is a plane and M 2 a convex surface. The first approach rests on the Liouvillian character of L R . More precisely, if just one of the manifolds has a symmetry of revolution, then L R is shown to be a Liouvillian system. If, in addition, that manifold is convex and the other one is a plane, then a maximal linearizing output is explicitely computed. The second approach consists of the use of a continuation method. Even though L R admits nontrivial abnormal extremals, we are still able to successfully apply the continuation method if M 2 admits a stable periodic geodesic.

It has been established by Bao, Lin, and Sontag (2000) that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain l/sub p/ stabilization can be achieved by linear output feedback, for every p/spl isin/(1,/spl infin/] except p=1. An explicit construction of the corresponding feedback laws was given. The feedback laws constructed also resulted in a closed-loop system that is globally asymptotically stable. This note complements the results of Bao, Lin, and Sontag (2000) by showing that they also hold for the case of p=1. In establishing our results for p=1, we also allow the presence of certain additional external disturbance.

In this paper, we address the issue of solving the motion planning problems (MPP for short) for a class of left-invariant control systems Σ whose state spaces are semisimple compact Lie groups. The sub-Riemannian metrics induced by the dynamics of Σ admit nontrivial abnormal extremals. This fact a priori represents an obstruction for the procedure we use to tackle the MPP, which consists of a homotopy (or continuation) method. We are however able to provide complete answers for the MPP. © 2002 Elsevier Science B.V. All rights reserved.

In this paper, we propose high-gain numerical differentiators for estimating the higher derivatives of a given signal. We consider time varying high-gain vectors converging exponentially to the high-gain vectors introduced by F. Esfandiari and K.H. Khalil (1992) in an earlier paper. The dynamics of these time-varying high-gain vectors can be chosen in order to achieve specific objectives, such as peaking attenuation and low sensitivity with respect to noise disturbance. In particular, we show that the numerical differentiator introduced in an earlier paper avoids the peaking phenomenon in the sense of H.J. Sussmann and P.V. Kokotovic (1991), i.e., there is no unbounded overshoot of the error estimate during the initial times. We also propose another numerical differentiator which filters the reference signal with respect to a very simple quadratic cost.

In this paper, we address the issues of controllability and motion planning for the control system S/sub R/ that results from the rolling without slipping nor spinning of a two dimensional Riemmanian manifold M/sub 1/ onto another one M/sub 2/. In the first part of the paper, we describe precisely the control system under consideration together with its Lie algebraic structure. This leads to a recovery of the result of Agrachev and Sachkov (1999) who provided a necessary and sufficient condition on the manifolds for complete controllability of S/sub R/. In the main part of the paper, we present two procedures to tackle the motion planning problem when M/sub 1/ is a plane and M/sub 2/ a convex surface. The first approach is based on differential algebra. We show that S/sub R/ is a Liouvillian system and if M/sub 2/ has a symmetry of revolution, we compute a maximal linearizing output. The second technic consists of the use of a continuation method to attack the motion planning problem. Even though S/sub R/ admits nontrivial abnormal extremals, we are still able to successfully apply the continuation method if M/sub 2/ admits a stable periodic geodesic.

International audience; The aim of this work is to investigate the structure of time-optimal trajectories for a control system modelizing a satellite with two rotors attached along its two fixed axes. Our results extend to the general case those obtained by Sussmann and Tang in an unpublished paper where they treat a particular case described below. We end up finding a sufficient family of four parameters trajectory types. The main tools used are the Pontryagin Maximum Principle, switching functions and envelope theory. © 2001 EUCA.

We consider a finite-dimensional control system (Σ) x(t) = f(x(t), u(t)), such that there exists a feedback stabilizer k that renders x = f(x; k(x)) globally asymptotically stable. Moreover, for (H, p, q) with H an output map and 1 ≤ p ≤ q ∞, we assume that there exists a K∞-function α such that ‖H(xu)‖q ≤α(‖u‖p), where xu is the maximal solution of (Σ)k x(t) = f(x(t), k(x(t))+u(t)), corresponding to u and to the initial condition x(0) = 0. Then, the gain function G(H, p, q) of (H, p, q) given by is well-defined. We call profile of k for (H, p, q) any K∞-function which is of the same order of magnitude as G(H, p, q). For the double integrator subject to input saturation and stabilized by kL(x) = −(1 1)T x, we determine the profiles corresponding to the main output maps. In particular, if 0 is used to denote the standard saturation function, we show that the L2-gain from the output of the saturation nonlinearity to u of the system x = σ0(−x−x+u) with x(0) = x(0) = 0, is finite. We also provide a class of feedback stabilizers kF that have a linear profile for (x, p, p), 1 ≤ p ≤ ∞. For instance, we show that the L2-gains from x and x_ to u of the system x =σ0(−x − x − (x)3 + u) with x(0) = x (0) = 0, are finite. © 2001 EDP Sciences, SMAI.

In this paper we consider the problem of controllability for a discrete linear control system xk+1 = Axk + Buk, uk ∈ U, where (A, B) is controllable and U is a finite set. We prove the existence of a finite set U ensuring density for the reachable set from the origin under the necessary assumptions that the pair (A, B) is controllable and A has eigenvalues with modulus greater than or equal to 1. In the case of A only invertible we obtain density on compact sets. We also provide uniformity results with respect to the matrix A and the initial condition. In the one-dimensional case the matrix A reduces to a scalar λ and for λ andgt; 1 the reachable set R(0, U) from the origin is R(0, U)(λ) = {∑j=0n ujλj uj ∈ U, n ∈ N} When 0 andlt; λ andlt; 1 and U = {0, 1, 3}, the closure of this set is the subject of investigation of the well-known {0, 1, 3}-problem. It turns out that the nondensity of R(0, Ũ(λ))(λ) for the finite set of integers Ũ(λ) = {0, ±1, ..., ±[λ]} is related to special classes of algebraic integers. In particular if λ is a Pisot number, then the set is nowhere dense in R for any finite control set U of rationals.

International audience