The problem concerning the distribution of the fractional parts of the sequence ank (k an integer exceeding one) was first considered by Hardy and Littlewood  and Weyl  earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov , Heilbronn  and Danicic  (see ). Recently Heath-Brown  has improved the unlocalized versions of these results for k ≥ 6 (a slightly stronger result than Heath-Brown's for K = 8 is given on page 24 of . The method mentioned there can, after some numerical calculation, improve Heath-Brown's result for 8 ≤ k ≤ 20, but still stronger results have recently been obtained by Dr. T. D. Wooley). The cognate problem regarding the sequence apk, where p denotes a prime, has also received some attention. In this situation even the case k = 1 proves to be difficult (see  and ). The first results in this field were given by Vinogradov (see Chapter 11 of  for the case k = 1,  for k ≥ 2). For k = 2 the best result to date has been supplied by Ghosh , and for ≥, by Harman (Theorem 3 in , building on the work in  and ). In this paper we shall improve the known results for 2 ≤ k ≤ 12. For larger k, Theorem 3 in  is more efficient. The theorem we prove is as follows.