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The Quarterly Journal of Mathematics Advance Access published online on January 11, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham052
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© 2008. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org

ON THE ERROR TERMS AND EXCEPTIONAL SETS IN CONJECTURAL SECOND MAIN THEOREMS

Aaron Levin {dagger}

Centro di Ricerca Matematica Ennio De Giorgi I-56100 Pisa, Italy

David McKinnon

Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2T 2M2 Canada

Jörg Winkelmann

Institut für Mathematik, Universität Bayreuth, 95440, Bayreuth, Germany

Received 23 March 2007; Accepted for publication 30 November 2007.


  Abstract

We study the error terms and exceptional sets in conjectural Second Main Theorems in Nevanlinna theory and Diophantine approximation (Vojta's conjecture). In particular, we give a geometric description of the exceptional set in the case of surfaces and the trivial divisor. Examples are given which show that, in general, the exceptional sets in conjectural Second Main Theorems must depend on the parameter {epsilon} in these conjectures. As a consequence, we obtain counterexamples to a conjecture of S. Lang on the forms of the error terms in conjectural Second Main Theorems.

{dagger} E-mail: aaron.levin{at}sns.it


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