# Distribution modulo 1 of some oscillating sequences. III Academic Article

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### abstract

• For some oscillating functions, such as h( x ) = xp log3 cosh\left( x \right) = x^\pi \log ^3 \times \cos \times , we consider the distribution properties modulo 1 (density, uniform distribution) of the sequence h( n )h\left( n \right) , n \geqq 1{n \geqq 1} . We obtain positive and negative results covering the case when the factor xp log3 xx^{\pi } {log}^3 x is replaced by an arbitrary function ff of at most polynomial growth belonging to any Hardy field. (The latter condition may be viewed as a regularity growth condition on ff .) Similar results are obtained for the subsequence h( p )h\left( p \right) , taken over the primes p = 2,3,5,....p = 2,3,5,...\;.

### publication date

• January 1, 2002