Continued fractions of Laurent series with partial quotients from a given set
1. Introduction. Van der Poorten and Shallit's paper [10] begins: " It is notorious that it is damnably difficult to explicitly compute the continued fraction of a quantity presented in some other form ". The quantity is usually presented either as a power series or as the root of a specific equation. There has been some success in the former case for continued fractions of real numbers, such as Euler's famous continued fraction for e [11] and more recent work [10] on " folded " continued fractions; however, other than the well-known results for quadratic real numbers, the only success with the latter has been for continued fractions of Laurent series rather than real numbers. In this paper we continue this line of investigation. We consider families of continued fractions of Laurent series whose partial quotients all lie in a given set. Following ideas of Baum and Sweet [2], we show that one may describe the zeros of certain collections of equations in terms of such families. The paragraphs that follow introduce the notation and definitions necessary to give a fuller description of our results. Let F q be the finite field with q elements and L q denote the field of formal Laurent series in x
On the metric theory of numbers in non-Archimedean settings
This thesis is a contribution to some fields of the metrical theory of numbers in nonArchimedean settings. This is a branch of number theory that studies and characterizes sets of numbers, which occur in a locally compact topological field endowed with a nonArchimedean absolute value. This is done from a probabilistic or measure-theoretic point of view. In particular, we develop new formulations of ergodicity and unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences. We use subsequence ergodic theory to establish a generalised metrical theory of continued fractions in both the settings of the p-adic numbers and the formal Laurent series over a finite field. We introduce the a-adic van der Corput sequence which significantly generalises the classical van der Corput sequence. We show that it provides a wealth of examples of low-discrepancy sequences which are very useful in the quasi-Monte Carlo method. We use our subsequential characterization of unique ergodicity to solve the generalised version of an open problem asked by O. Strauch on the distribution of the sequence of consecutive van der Corput sequences. In addition to these problems in ergodic methods and number theory, we employ some geometric measure theory to settle the positive characteristic analogue of an open problem asked by R.D. Mauldin on the complexity of the Liouville numbers in the field of formal Laurent series over a finite field by giving a complete characterization of all Hausdorff measures of the set of Liouville numbers.
Reid on Hypotheses and the Ether; A Reassessment
In recent years historians of science have come to see that Thomas Reid’s Essays on the Intellectual Powers of Man had a profound impact on the development of British methodological thought in the late-eighteenth and early-nineteenth centuries. Larry Laudan has argued that Reid was “the first major British philosopher to take Newton’s opinions on induction, causality, and hypotheses seriously”; and that Reid was thereby instrumental in changing the character of British methodological discourse in the Enlightenment.(1) Geoffrey Cantor has examined in detail the influence of Reid’s critique of ether theories and hypotheses in the Essays on leading protagonists in the nineteenth-century debates surrounding the wave theory of light,(2) and Richard Olson has traced the origins of the scientific style of British physics in the Victorian period to the writings of Reid and other Common Sense philosophers. (3) As well as delineating Reid’s influence, Laudan, Cantor, and Olson have attempted to identify the problems to which he addressed himself in articulating his methodology, and they have analysed with some care the internal structure of his methodological views. Thanks to the work of these historians, therefore, we have a fairly comprehensive picture of Reid’s methodology and its fortunes in the nineteenth century.