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A classification of
$\mathbb {Q}$-linear maps from 
$\overline {\mathbb {Q}}^\times /\overline {\mathbb {Q}}^\times _{\mathrm {tors}}$ to 
$\mathbb {R}$
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 21 October 2025, pp. 1-19
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- Article
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- Open access
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A 2009 article of Allcock and Vaaler explored the
$\mathbb {Q}$-vector space 
$\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$, showing how to represent it as part of a function space on the places of 
$\overline {\mathbb {Q}}$. We establish a representation theorem for the 
$\mathbb {R}$-vector space of 
$\mathbb {Q}$-linear maps from 
$\mathcal {G}$ to 
$\mathbb {R}$, enabling us to classify extensions to 
$\mathcal {G}$ of completely additive arithmetic functions. We further outline a strategy to construct 
$\mathbb {Q}$-linear maps from 
$\mathcal {G}$ to 
$\mathbb {Q}$, i.e., elements of the algebraic dual of 
$\mathcal {G}$. Our results make heavy use of Dirichlet’s S-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.
