Cohen-Egyptian fraction and engel series expansions

Abstract

It was shown in the book written by Perron in 1951 that each nonzero real number can be uniquely written as an Engel series expansion. This series expansion represents a nonzero rational number if and only if each digit in such expansion is identical from certain point onward. In 1973, Cohen devised an algorithm to uniquely represent each nonzero real number as a sum of Egyptian fractions, which we refer to as its Cohen-Egyptian fraction expansion. Cohen also characterized the real rational numbers as those with finite Cohen-Egyptian fraction expansions. In this thesis, we extend their work to three complete fields with respect to discrete non-archimedean valuations, namely, the p-adic number field and two kinds of function fields (the one completed with respect to the degree valuation and the one completed with respect to a prime-adic valuation). We present algorithms for constructing these two types of series representations of elements in these fields and establish rationality criteria through the use of these expansions. In the last part, we analyze the relationship between these two expansions in all felds involved