Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
Wouter van Doorn, Vjekoslav Kovač
TL;DR
The paper resolves Bleicher and Erdős's conjecture by constructing lacunary sequences of positive integers whose reciprocal sums attain all rational numbers in a nontrivial interval, and it develops a general framework determining when a lacunary reciprocal sum set can fill an interval. It introduces the achievement set framework $P((1/n_i))$, analyzes necessary and sufficient conditions, and defines the function $R(\lambda)$ to quantify maximal interval length fillable by $\lambda$-lacunary sequences. The authors prove three main results: (i) existence of $\lambda$-lacunary sequences representing all rationals in $[0,2]$ but not with $\lambda=2$; (ii) a precise upper bound $R(\lambda)$ via a recursively defined sequence $a_i$ and a divisor-augmentation construction; (iii) a construction allowing arbitrarily large jumps while still filling a positive-length interval, with optimal lacunarity thresholds discussed. Together, these results advance understanding of Egyptian fraction representations in lacunary settings and connect to prior work by Graham and Eppstein.
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.