Nil Bohr0-sets and polynomial recurrence

In Huang et al.  [17] it was proved that for any Nil_d  Bohr₀-set A$A$, there are a minimal system (X,T)$(X,T)$ and a non-empty open subset U$U$ of X$X$ with A⊃{n∈Z:U∩T⁻ⁿU∩?∩T^−dnU≠0?}$A\supset \{n\in \mathbb{Z}:U\cap T^{-n}U\cap ?\cap T^{-dn}U\ne 0?\}$, and for any minimal system (X,T)$(X,T)$ and any open non-empty U⊂X$U\subset X$, the set {n∈Z:U∩T⁻ⁿU∩?∩T^−dnU≠0?}$\{n\in \mathbb{Z}:U\cap T^{-n}U\cap ?\cap T^{-dn}U\ne 0?\}$ is an almost Nil_d Bohr₀-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem.     

Intersective polynomials and the polynomial Szemerédi theorem

Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p₁(n),…,a+p_r(n)}. We prove that a polynomial family P={p₁,…,p_r} has the PSZ property if and only if the polynomials p₁,…,p_r are jointly intersective, meaning that for any there exists such that the integers p₁(n),…,p_r(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi 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 are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nE_i such that {a,a+p₁(n),…,a+p_r(n)}E_i.     

Factors of generalised polynomials and automatic sequences

The aim of this short note is to generalise the result of Rampersad–Shallit saying that an automatic sequence and a Sturmian sequence cannot have arbitrarily long common factors. We show that the same result holds if a Sturmian sequence is replaced by an arbitrary sequence whose terms are given by a generalised polynomial (i.e., an expression involving algebraic operations and the floor function) that is not periodic except for a set of density zero.     

Formality in generalized Kähler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kähler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kähler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.