Consistent maps and their associated dual representation theorems

TL;DR

This paper develops dual representation theorems for spaces of locally constant functions on algebraic number fields, extending previous work on vector spaces related to algebraic numbers and their heights.

Contribution

It introduces new dual representation theorems for locally constant function spaces using consistent maps, building on prior results about algebraic number spaces.

Findings

Established dual representation theorems for locally constant functions.

Recovered a main theorem from previous research as a corollary.

Extended the framework of consistent maps in number theory contexts.

Abstract

A 2009 article of Allcock and Vaaler examined the vector space $\mathcal G := \overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ over $\mathbb Q$, describing its completion with respect to the Weil height as a certain $L^1$ space. By involving an object called a consistent map, the author began efforts to establish Riesz-type representation theorems for the duals of spaces related to $\mathcal G$. Specifically, we provided such results for the algebraic and continuous duals of $\overline{\mathbb Q}^\times/{\overline{\mathbb Z}}^\times$. In the present article, we use consistent maps to provide representation theorems for the duals of locally constant function spaces on the places of $\overline{\mathbb Q}$ that arise in the work of Allcock and Vaaler. We further apply our new results to recover, as a corollary, a main theorem of our previous work.