### abstract

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