Lacunary sequences whose reciprocal sums represent all rationals in an interval
Wouter van Doorn, Vjekoslav Kovač
Published: 2025/9/29
Abstract
Answering a question of Michael Bleicher and Paul 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 rationals 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.