Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
Number Theory
2025-12-04 v3 Classical Analysis and ODEs
Combinatorics
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.
Cite
@article{arxiv.2509.24971,
title = {Lacunary sequences whose reciprocal sums represent all rational numbers in an interval},
author = {Wouter van Doorn and Vjekoslav Kovač},
journal= {arXiv preprint arXiv:2509.24971},
year = {2025}
}
Comments
17 pages, 1 figure. v3: minor changes to the exposition