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I'm not entirely sure whether any books will directly tell you how to solve those particular problems, but if you're looking for a good number theory book. Hardy & Wright's An Introduction to the Theory of Numbers may be of good help (both in the continued fractions topic and in number theory in general.) Or for algebraic number theory, I would ...

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This thread may help. The subject is called "Diophantine analysis" (or "Diophantine approximation(s)") and I enjoyed much Edward Burger's introduction "Exploring the Number Jungle". Other references may be found in Steuding's fine online course.

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normal $\; \not\Rightarrow \;$ irrationality measure $2$ There exist numbers that are normal with irrationality measure $>2$. In fact, there exist normal numbers (meaning normal with respect to every base) that have irrationality measure $\infty.$ This is Theorem 2 in Bugeaud [2] (2002). For related results, see [1] and [3]. irrationality measure $2$ ...

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