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A counterexample to the Chung-Graham-Spiro gap-set conjecture
Abstract
Chung, Graham, and Spiro introduced slow Fibonacci walks and used them to partition the integers n≥2 into two sequences, the down-integers and the up-integers. They studied the local spacing of these two sequences and conjectured that their ℓ-step gap sets agree for every ℓ≥1. We show that the conjecture fails at ℓ=4 by proving 9∈ U4∖ D4 .
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BibTeX
@misc{aliabadi2026counterexample,
title = {{A counterexample to the Chung-Graham-Spiro gap-set conjecture}},
author = {Mohsen Aliabadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04473},
eprint = {2609.04473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04473},
}
RIS
TY - UNPB
TI - A counterexample to the Chung-Graham-Spiro gap-set conjecture
AU - Mohsen Aliabadi
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04473
UR - https://arxiv.org/abs/2609.04473
AB - Chung, Graham, and Spiro introduced slow Fibonacci walks and used them to partition the integers n≥2 into two sequences, the down-integers and the up-integers. They studied the local spacing of these two sequences and conjectured that their ℓ-step gap sets agree for every ℓ≥1. We show that the conjecture fails at ℓ=4 by proving 9∈ U_4∖ D_4 .
ER -