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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multiplicative approximation by the Weil height


Authors: Robert Grizzard and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 373 (2020), 3235-3259
MSC (2010): Primary 11J25, 11R04, 46B04
DOI: https://doi.org/10.1090/tran/7941
Published electronically: February 19, 2020
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Abstract: Let $ K/\mathbb{Q}$ be an algebraic extension of fields, and let $ \alpha \not = 0$ be contained in an algebraic closure of $ K$. If $ \alpha $ can be approximated by roots of numbers in $ K^{\times }$ with respect to the Weil height, we prove that some nonzero integer power of $ \alpha $ must belong to $ K^{\times }$. More generally, let $ K_1, K_2, \dots , K_N$, be algebraic extensions of $ \mathbb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $ \alpha \not = 0$ can be approximated by a product of roots of numbers from each $ K_n$ with respect to the Weil height, we prove that some nonzero integer power of $ \alpha $ must belong to the multiplicative group $ K_1^{\times } K_2^{\times } \cdots K_N^{\times }$. Our proof of the more general result uses methods from functional analysis.


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Additional Information

Robert Grizzard
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: grizzard@math.wisc.edu

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: vaaler@math.utexas.edu

DOI: https://doi.org/10.1090/tran/7941
Keywords: Weil height, projection operators
Received by editor(s): October 23, 2017
Received by editor(s) in revised form: July 5, 2019
Published electronically: February 19, 2020
Additional Notes: Research of the second author was supported by a grant from the National Security Agency, H92380-12-1-0254.
Article copyright: © Copyright 2020 American Mathematical Society