On the Distribution Modulo 1 of the Sequence αn2 + βn

Abstract

Dirichlet's Theorem says that for any real α and for N ≧ 1, there exists a natural n ≦ N withwhere || || denotes the distance to the nearest integer. Heilbronn [2], improving estimates of Vinogradov [3], showed that for α, N as above and for ϵ ≧ 0, there exists an n ≦ N with