Divergence of decreasing rearranged Fourier series on every infinite compact abelian group
Abstract
On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense Gδ in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines Körner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.
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BibTeX
@misc{nielsen2026divergence,
title = {{Divergence of decreasing rearranged Fourier series on every infinite compact abelian group}},
author = {Morten Nielsen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03506},
eprint = {2609.03506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Divergence of decreasing rearranged Fourier series on every infinite compact abelian group
AU - Morten Nielsen
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03506
UR - https://arxiv.org/abs/2609.03506
AB - On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense G_δ in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines Körner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.
ER -