Irrationality Sequences

Abstract: Sometimes one can prove the irrationality of the sum of reciprocals of a sequence of positive integers using only information about the growth rate of the sequence. Erdős and Straus introduced the notion of an irrationality sequence in order to isolate nontrivial aspects of this relationship. Despite its elementary formulation, the theory of irrationality sequences still contains many open problems. For instance, the question of whether $2^{2^n}$ is a (type 2) irrationality sequence is a particularly interesting open problem. Recently, Kovač and Tao obtained several interesting results on the asymptotic behavior of irrationality sequences. We study sums of reciprocals of doubly exponential sequences and show, among other results, that there are at most countably many real numbers $a>1$ for which $a^{2^n}$ is a (type 2) irrationality sequence. We also explain how such questions are related to certain greedy Egyptian fraction expansions.

dynamical systemsnumber theory

Audience: researchers in the topic

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One World Numeration seminar

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