Distinct unit generated number fields and finiteness in number systems

Abstract: A distinct unit generated field is a number field K such that every algebraic integer of the field is a sum of distinct units. In 2015, Dombek, Masáková, and Ziegler studied totally complex quartic fields, leaving 8 cases unresolved. Because in this case there is only one fundamental unit $u$, their method involved the study of finiteness in positional number systems with base u and digits arising from the roots of unity in $K$. First, we consider a more general problem of positional representations with base beta with an arbitrary digit alphabet $D$. We will show that it is decidable whether a given pair $(\beta, D)$ allows eventually periodic or finite representations of elements of $O_K$. We are then able to prove the conjecture that the 8 remaining cases indeed are distinct unit generated.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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