Underapproximation by Egyptian fractions

DOI10.1016/J.JNT.2022.07.005arXiv2202.00191WikidataQ114156428 ScholiaQ114156428MaRDI QIDQ6389794

Melvyn B. Nathanson

Publication date: 31 January 2022

Abstract: An increasing sequence $(x_i{)}_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $hetain(0,1]$ if $su{m}_{i=1}^{n}frac1x_i<heta$. A greedy algorithm constructs an $n$-term underapproximation of $heta$. For some but not all numbers $heta$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.

Mathematics Subject Classification ID

Diophantine inequalities (11D75) Diophantine equations in many variables (11D72) Rational numbers as sums of fractions (11D68) Other number representations (11A67) Representation problems (11D85)

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