For some oscillating functions, such as \documentclass[12pt]{minimal}
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$$h\left( x \right) = x^\pi \log ^3 \times \cos \times $$
\end{document}, we consider the distribution properties modulo 1 (density, uniform distribution) of the sequence \documentclass[12pt]{minimal}
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$$h\left( n \right)$$
\end{document}, \documentclass[12pt]{minimal}
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$${n \geqq 1}$$
\end{document}. We obtain positive and negative results covering the case when the factor \documentclass[12pt]{minimal}
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$$x^{\pi } {log}^3 x$$
\end{document} is replaced by an arbitrary function \documentclass[12pt]{minimal}
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$$f$$
\end{document} of at most polynomial growth belonging to any Hardy field. (The latter condition may be viewed as a regularity growth condition on \documentclass[12pt]{minimal}
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$$f$$
\end{document}.) Similar results are obtained for the subsequence \documentclass[12pt]{minimal}
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$$h\left( p \right)$$
\end{document}, taken over the primes \documentclass[12pt]{minimal}
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$$p = 2,3,5,...\;.$$
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