On the fractional parts of αn2 and βn

Abstract

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such thatThis extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].