Zero Harmonic Density implies Zero Density
Abstract
A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.
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BibTeX
@misc{cantuba2026zero,
title = {{Zero Harmonic Density implies Zero Density}},
author = {Rafael Reno S. Cantuba},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38240},
eprint = {2609.38240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38240},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zero Harmonic Density implies Zero Density
AU - Rafael Reno S. Cantuba
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38240
UR - https://arxiv.org/abs/2609.38240
AB - A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.
ER -