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Strongly small-2 sets that are not Riesz sets on compact abelian groups
Abstract
We prove that every infinite compact abelian group G admits a countably infinite proper set E⊂Ĝ that is strongly small-2 but not Riesz. More precisely, |α|*|β| is absolutely continuous for all α,β∈ ME(G), whereas ME(G) contains a singular complex measure of total mass one with exact Fourier support E whose convolution square belongs to L2(G). When the image of the doubling map on Ĝ is infinite, E may moreover be chosen so that #(E∩(γ−E))<∞ for every γ and E∩(−E)={0}. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which E∩(E+γ) is finite at every nonzero shift.
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BibTeX
@misc{ohrysko2026strongly,
title = {{Strongly small-2 sets that are not Riesz sets on compact abelian groups}},
author = {Przemysław Ohrysko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12628},
eprint = {2609.12628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12628},
}
RIS
TY - UNPB
TI - Strongly small-2 sets that are not Riesz sets on compact abelian groups
AU - Przemysław Ohrysko
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.12628
UR - https://arxiv.org/abs/2609.12628
AB - We prove that every infinite compact abelian group G admits a countably infinite proper set E⊂Ĝ that is strongly small-2 but not Riesz. More precisely, |α|*|β| is absolutely continuous for all α,β∈ M_E(G), whereas M_E(G) contains a singular complex measure of total mass one with exact Fourier support E whose convolution square belongs to L^2(G). When the image of the doubling map on Ĝ is infinite, E may moreover be chosen so that #(E∩(γ−E))<∞ for every γ and E∩(−E)={0}. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which E∩(E+γ) is finite at every nonzero shift.
ER -