On the Distribution of the Sequence <i>n²θ</i> (mod 1)

Abstract

In a paper with the above title H. Heilbronn [ 4 ] proved that, given any real θ , there exist infinitely many positive integers n such that the distance ║ θn 2 ║ from θn 2 to its nearest integral neighbour satisfies the bound He actually proved a somewhat stronger statement which shows that the integers n occur with some regularity and he suggested that perhaps the exponent may be replaced by . This theorem has attracted considerable attention and spawned a number of generalizations (see [1], [7] and the references therein), yet no essential improvement has been given for the original problem (but see [6], [8] ).

Locations

• Canadian Journal of Mathematics - View - PDF