On the fractional parts of a set of points

Heilbronn [5] proved that for any e > 0 there exists C (e) such that for any real θ and N ≥ 1 there is an integer x satisfying where ‖α‖ denotes the difference between α and the nearest integer, taken positively. The result is uniform in θ and so analogous to Dirichlet's inequality for the fractional parts of n θ. The result has been generalized to simultaneous approximations by Danicic [1] and Ming-chit Liu [6]. Here we shall extend the result to any finite number of simultaneous approximations when x 2 is replaced by a k -th power.