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On simultaneous diagonal inequalities, II

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

Scott T. Parsell
Scott T. Parsell
Affiliation:
Department of Mathematics, Texas A…M University, College Station, TX 77843-3368, U.S.A. Current address: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A. E-mail: parsell@alum.mit.edu
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In this paper we continue the investigation begun in [11]. Let λ1…., λs and μ1, …, μs be real numbers, and define the forms

Further, let τ be a positive real number. Our goal is to determine conditions under which the system of inequalities

has a non-trivial integral solution. As has frequently been the case in work on systems of diophantine inequalities (see, for example, Brüdern and Cook [6] and Cook [7]), we were forced in [11] to impose a condition requiring certain coefficient ratios to be algebraic. A recent paper of Bentkus and Gotze [4] introduced a method for avoiding such a restriction in the study of positivede finite quadratic forms, and these ideas are in fact flexible enough to be applied to other problems. In particular, Freeman [10] was able to adapt the method to obtain an asymptotic lower bound for the number of solutions of a single diophantine inequality, thus finally providing the expected strengthening of a classical theorem of Davenport and Heilbronn [9]. The purpose of the present note is to apply these new ideas to the system of inequalities (1.1).

MSC classification

Secondary: 11D75: Diophantine inequalities

Information

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 191 - 202
Copyright
Copyright © University College London 2001

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