Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
Volume 223 / 2026
Acta Arithmetica 223 (2026), 275-295
MSC: Primary 11D68; Secondary 11B05, 11B13, 40A05
DOI: 10.4064/aa251001-13-1
Published online: 15 April 2026
Abstract
Disproving a conjecture of Bleicher and Erdős, 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.
