Optimization and Control Theory Department: 2014-05-20 Nonlinear Seminar

Adam Nawrocki
Asymptotic behaviour of a certain almost periodic function with regard to the Lebesgue measure

The function

$\begin{displaymath} f(x)=\frac{1}{2+\cos x +\cos{(x\sqrt 2)}} \end{displaymath}$ (1)
is a classical example of an unbounded and continuous $\mu$ -almost periodic function. For this function we have
$\begin{displaymath} \forall_{\varepsilon>0} \quad \lim_{x\to \infty}\frac{x^{-2-\varepsilon}}{2+\cos x +\cos{(x\sqrt 2)}}=0. \end{displaymath}$ (2)
During this lecture we will discuss the idea of the proof of the equality above, which uses diophantic approximations. We shall furthermore show, that the limit
$\begin{displaymath} \lim_{x\to \infty}\frac{x^{-2}}{2+\cos x +\cos{(x\sqrt 2)}} \end{displaymath}$ (3)
does not exist.