Geodesic
On fractional parts of rapidly growing functions
Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 727-748.

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We study the behaviour of fractional parts of functions $\alpha\exp([\log^c x]\log x)$, where $\alpha$ is a real algebraic number of degree $n\geqslant 2$ and $c$ is an arbitrary positive number less than one. 
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A. A. Karatsuba. On fractional parts of rapidly growing functions. Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 727-748. http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a5/

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