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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.

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