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Here we prove that every real quadratic irrational α can be expressed as a periodic non-simple continued fraction having period length one. Moreover, we show that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of α. We close with an application relating the structure of a quadratic α to its conjugate.

In this paper we generalize classical results in Diophantine approximation to the setting of an arhitrary numher field in the context of the ring of 5-integers. Specifically, we present theorems pertaining to simultaneous approximations of linear forms and develop the notion of badly approximable ^-systems. In addition, we expand the subject of the geometry of numbers over the adele ring of a number field by developing the concept of the adelic polar body. This theory is then used to produce transference theorems in this general situation.

Abstract. In 1962 Erdős proved that every real number may be decomposed into a sum of Liouville numbers. Here we consider more general functions which decompose elements from an arbitrary local field into Liouville numbers. Several examples and applications are given. As an illustration, we prove that for any real numbers α 1 , α 2 , . . . , α N , not equal to 0 or 1, there exist uncountably many Liouville numbers σ such that α σ 1 , α σ 2 , . . . , α σ N are all Liouville numbers.