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Durham e-Theses
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Cutting Sequences and the p-adic Littlewood Conjecture

BLACKMAN, JOHN,EDWARD (2020) Cutting Sequences and the p-adic Littlewood Conjecture. Doctoral thesis, Durham University.



The main aim of this thesis is to use the geometric setting of cutting sequences to better understand the behaviour of continued fractions under integer multiplication. We will use cutting sequences to construct an algorithm that multiplies continued fractions by an integer $n$. The theoretical aspects of this algorithm allow us to explore the interesting properties of continued fractions under integer multiplication. In particular, we show that an eventually recurrent continued fractions remain eventually recurrent when multiplied by a rational number. Finally, and most importantly, we provide a reformulation the $p$-adic Littlewood Conjecture in terms of a condition on the semi-convergents of a real number $\alpha$.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Cutting Sequences, Continued Fractions, Diophantine Approximation, The p-adic Littlewood Conjecture
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2020
Copyright:Copyright of this thesis is held by the author
Deposited On:09 Oct 2020 15:03

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