Generalised polynomials and integer powers

We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) = 0$ for all $n \geq 0$ then there exists infinitely many $m \in \mathbb{N}$, not divisible by $k$, such that $g(mk^n) = 0$ for some $n \geq 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.Comment: 51 page

Algorithmic classification of noncorrelated binary pattern sequences

We show that it is possible to algorithmically verify if a given pattern sequence is noncorrelated. As an application, we compute that there are exactly $2272$ noncorrelated binary pattern sequences of length $\leq 4$. If we restrict our attention to patterns that do not end with $\mathtt{0}$, we put forward a sufficient condition for a pattern sequence to be noncorrelated. We conjecture that this condition is also necessary, and verify this conjecture for lengths $\leq 5$.Comment: 20 page

On multiplicative automatic sequences

We show that any automatic multiplicative sequence either coincides with a Dirichlet character or is identically zero when restricted to integers not divisible by small primes. This answers a question of Bell, Bruin and Coons. A similar result was obtained independently by Klurman and Kurlberg.Comment: 12 page

Automatic sequences as good weights for ergodic theorems

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.Comment: 31 page

On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations

The dispersive effect of the Coriolis force for the stationary Navier-Stokes equations is investigated. The effect is of a different nature than the one shown for the non-stationary case by J. Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier. Existence of a unique solution is shown for arbitrary large external force provided the Coriolis force is large enough. The analysis is carried out in a new framework of the Fourier-Besov spaces. In addition to the stationary case counterparts of several classical results for the non-stationary Navier-Stokes problem have been proven