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arXiv · 2509.24971

Lacunary sequences whose reciprocal sums represent all rational numbers in an interval

Wouter van Doorn·Vjekoslav Kovač
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Abstract

Disproving a conjecture of Bleicher and Erd\H{o}s, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all rational numbers from a non-empty open interval. We also study several stronger variants of their original problem: determining the value of the optimal lacunarity parameter, representing rational numbers infinitely many times, finding such lacunary sequences with arbitrarily large jumps, and relating the maximal length of a filled interval to a prescribed lacunarity parameter.

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Wouter van Doorn, Vjekoslav Kovač. 2025-09-29. Lacunary sequences whose reciprocal sums represent all rational numbers in an interval. https://doi.org/10.4064/aa251001-13-1

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