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Representing algebraic integers as linear combinations of units

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Abstract

In this paper we consider representations of algebraic integers of a number field as linear combinations of units with coefficients coming from a fixed small set, and as sums of elements having small norms in absolute value. These theorems can be viewed as results concerning a generalization of the so-called unit sum number problem, as well. Beside these, extending previous related results we give an upper bound for the length of arithmetic progressions of \(t\)-term sums of algebraic integers having small norms in absolute value.

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Notes

  1. Here and in the sequel under a unit of \(K\) we mean a unit in \(O_K\).

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Acknowledgments

The work of D. Dombek was supported by the Czech Science Foundation, Grant GAČR 201/09/0584, by the grants MSM6840770039 and LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic, and by the Grant of the Grant Agency of the Czech Technical University in Prague, Grant No. SGS11/162/OHK4/3T/14. The research of L. Hajdu was supported in part by the OTKA grants K75566 and NK101680. The research of L. Hajdu and A. Pethő was supported in part by the OTKA Grant K100339 and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, cofinanced by the European Social Fund and the European Regional Development Fund. The paper was finished, when the author was working at the University of Niigata with a long term research fellowship of JSPS. The authors are grateful to the referee for his helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 , Prague 2, Czech Republic

    D. Dombek

  2. Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen, 4010, Hungary

    L. Hajdu

  3. Department of Computer Science, University of Debrecen, P.O. Box 12, Debrecen, 4010, Hungary

    A. Pethő

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  1. D. Dombek
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  2. L. Hajdu
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  3. A. Pethő
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Correspondence to D. Dombek.

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Dombek, D., Hajdu, L. & Pethő, A. Representing algebraic integers as linear combinations of units. Period Math Hung 68, 135–142 (2014). https://doi.org/10.1007/s10998-014-0020-9

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  • DOI: https://doi.org/10.1007/s10998-014-0020-9

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