On pairs of additive forms modulo one

Abstract

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.