On pairs of additive forms modulo one

Throughout this paper ∈ denotes an arbitrary positive number. For real α, ‖α‖ denotes the distance from α to the nearest integer. For natural numbers k we write K = 2k−1. In 1948 Heilbronn [8] showed that for any real α and N > C1(∈)This theorem has since been generalized in many ways. In particular, results of the following type have been proved for natural numbers k ≥ 2, h = 1,2 and s.

Small fractional parts of quadratic and additive forms

We denote by ∥…∥ the distance to the nearest integer. Let ε be an arbitrary positive number. Danicic(6) showed that for N > c1(s, ε) and a quadratic form Q(x1, …, xs) there exist integers n1, …, ns withhaving

On the fractional parts of certain additive forms

In this paper, k is a positive integer, k ≧ 2; K denotes 2k−1, and ε is an arbitrary positive number. X = (x1, …, xs) and U = (u1, …, ur) are integer vectors; is the distance from the real number y to the nearest integer.

An unsolved problem on the powers of 3/2

Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than, i.e the integral part of, and rn is its fractional part and so satisfies the inequality