Combinatorical and Algebraic Structures Seminar

Session details

Date: 13.10.2009
Speaker: Karel Klouda, FJFI, České vysoké učení technické v Praze
Title: Non-standard representations of $p$-adic numbers
Abstract: The field of $p$-adic numbers, $p$ prime, is defined as a completion of the set of rational numbers with respect to the $p$-adic absolute value. This absolute value has rather nonintuitive properties; however, the theory of $p$-adic numbers proved to be useful in many fields, especially in the number theory. Each $p$-adic number has a unique (standard) left infinite representation in the form of a power series in $p$. This representation is finite for positive integers, eventually periodic for rationals, and aperiodic otherwise. We will introduce a new way how to represent $p$-adic numbers based on the rational base number system for positive integers proposed by S.~Akiyama, Ch.~Frougny, and J.~Sakarovitch in 2008. The generalization of this system for $p$-adic numbers is very natural; it is also straightforward generalization of the standard system. Therefore, for certain setting, these two systems share most of their properties. However, for some rational bases properties of the new system are very strange. Here is a little foretaste: \begin{quote} Zero as an element of the set of $3$-adic or $2$-adic numbers has uncountably many representations in base~$30/11$; one of them reads: $$\cdots\ 7\ 2\ 3\ 7\ 4\ 7\ 8\ 7\ 7\ 5\ 5\ 6\ 6\ 5\ 6\ . $$ \end{quote}

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