Van der Corput sets in Z^d

In this partly expository paper we study van der Corput sets in $\Z^d$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for $d=1$ in \cite{K-MF} and \cite{R}, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions

On the behavior at infinity of an integrable function

International audienceWe prove that, in a weak sense, any integrable function on the real line tends to zero at infinity : if f is an integrable function on R, then for almost all real number x, the sequence (f(nx)) tends to zero when n goes to infinity. Using Khinchin's metric theorem on Diophantine approximation, we establish that this convergence to zero can be arbitrarily slow

Random walks in Weyl chambers and crystals

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined by Biane, Bougerol and O'Connell for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice.Comment: The second version presents minor modifications to the previous on

On the sequence of integer parts of a good sequence for the ergodic theorem

summary:If $(u_n)$ is a sequence of real numbers which is good for the ergodic theorem, is the sequence of the integer parts $([u_n])$ good for the ergodic theorem\,? The answer is negative for the mean ergodic theorem and affirmative for the pointwise ergodic theorem

Intersective polynomials and polynomial Szemeredi theorem

Let P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}] be a family of polynomials such that p_{i}(\Z^{m})\sle\Z, i=1,\ld,r. We say that the family $P$ has {\it PSZ property} if for any set E\sle\Z with d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0 there exist infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of the form \hbox{\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}}. We prove that a polynomial family P=\{p_{1},\ld,p_{r}\} has PSZ property if and only if the polynomials p_{1},\ld,p_{r} are {\it jointly intersective}, meaning that for any $k\in\N$ there exists $n\in\Z^{m}$ such that the integers p_{1}(n),\ld,p_{r}(n) are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemer\'{e}di theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p_{1},\ld,p_{r}\in\Q[n] are jointly intersective integral polynomials, then for any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist i\in\{1,\ld,k\} and $a,n\in E_{i}$ such that \{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}

Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

We introduce and characterize central probability distributions on Littelmann paths. Next we establish a law of large numbers and a central limit theorem for the generalized Pitmann transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we show there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and explain how it can be computed.Comment: 27 pages, minor corrections and a simpler definition of the Pitman invers

Conditioned one-way simple random walk and representation theory

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].Comment: 32 page

Powers of sequences and recurrence

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mend\`es-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.Comment: 30 pages. Numerous small changes made. To appear in the Proceedings of the London Mathematical Societ