95-507 S. De Bievre, G. Forni

On the growth of averaged Weyl sums for rigid rotations (37K, Latex) Nov 28, 95

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Abstract. Let $\omega\in\R\setminus\Q$ and $f\in L^2(\To)$. We study the asymptotic behaviour of the {\it Weyl sums } $S(m,\omega)f(x) =\sum^{m-1}_{k = 0} f (x+k \omega)$ and their averages ${\hat S}(m,\omega)f(x) ={1\over m}\sum^{m}_{j = 1} S(j,\omega)f(x)$, in the $L^2$-norm. In particular, for a suitable class of Liouville rotation numbers $\omega\in \R\setminus\Q$, we are able to construct examples of functions $f\in H^s(\To)$, $s>0$, such that, for all $\epsilon>0$, $||{\hat S}(m,\omega)f||_2 \geq C_{\epsilon} m^{{1\over {1+s}}-\epsilon}$ as $m\to \infty$. In addition, for all $f\in H^s(\To)$, $\liminf m^{-{1\over{1+s}}} (\log m)^{-1/2} \parallel {\hat S}(m,\omega)f \parallel_2<\infty$, for all $\omega\in\R\setminus\Q$.

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