UBC Theses and Dissertations

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UBC Theses and Dissertations

Fractional parts of powers and related topics Bennett, Michael A.


In this thesis, we employ a variety of explicit approximations to tackle some problems in Diophantine analysis. We are generally concerned with constructing rational function approximations to certain polynomials and to the exponential and other related functions. From these systems, we deduce arithmetic information about the original problem. Chapter 0 is an introduction to the field and contains background information and known results from the literature. The situation for algebraic numbers in general is briefly surveyed. In Chapter 1, we utilize Pade approximation in the manner of F. Beukers to generalize and sharpen results of Beukers, D. Easton and G. Xu about lower bounds for fractional parts of powers of rationals. Some theorems of a "semi-effective" nature are discussed and density results for certain constructed sets are proved. In Chapter 2, we utilize Euler-Maclaurin summation to describe the content of Pade-type approximants to the binomial function. These results form the basis for the afore-mentioned improved bounds. Through similar techniques to those employed in Chapter1 we derive lower bounds upon fractional parts of values of the exponential at integers. These are given in Chapter 4. Chapter 3 contains an application of the above theory to additive number theory, in particular to Waring's problem and related questions about additive bases. The Hardy-Littlewood-Vinogradrov circle method is used with a theorem from Chapter 1 to prove a version of the Ideal Waring problem with restricted summands.

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