Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets
Abstract
Varjú and Yu have given a complete description of non-singleton self-similar sets of uniqueness on the real line ℝ. In this paper, we first prove an arithmetic product theorem for self-similar U-sets E: if F⊂ℝ is compact, then F· E is an M0-set if and only if F is uncountable. We next show that this conclusion remains valid for bounded integer-based homogeneous Moran sets. As an application, for every s∈(0,1], there exists a bounded integer-based homogeneous Moran set of uniqueness K such that dimH K=s and the self-product K· K is an M0-set. In sharp contrast, for every s∈[0,1/2] we can construct an unbounded integer-based homogeneous Moran U-set E with dimH E=s such that both E and E· E are U-sets, and 1/2 is the optimal threshold for U-sets to be closed under the arithmetic self-product. Moreover, we show that if E⊂[0,1] is a compact set whose self-product E· E is contained in an H-set, then underlinedimB E≤1/2.
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BibTeX
@misc{fu2026arithmetic,
title = {{Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets}},
author = {Yan-Song Fu and Xu-Guang Wang and Meng Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04956},
eprint = {2611.04956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04956},
}
RIS
TY - UNPB
TI - Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets
AU - Yan-Song Fu
AU - Xu-Guang Wang
AU - Meng Zhu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04956
UR - https://arxiv.org/abs/2611.04956
AB - Varjú and Yu have given a complete description of non-singleton self-similar sets of uniqueness on the real line ℝ. In this paper, we first prove an arithmetic product theorem for self-similar U-sets E: if F⊂ℝ is compact, then F· E is an M_0-set if and only if F is uncountable. We next show that this conclusion remains valid for bounded integer-based homogeneous Moran sets. As an application, for every s∈(0,1], there exists a bounded integer-based homogeneous Moran set of uniqueness K such that dim_H K=s and the self-product K· K is an M_0-set. In sharp contrast, for every s∈[0,1/2] we can construct an unbounded integer-based homogeneous Moran U-set E with dim_H E=s such that both E and E· E are U-sets, and 1/2 is the optimal threshold for U-sets to be closed under the arithmetic self-product. Moreover, we show that if E⊂[0,1] is a compact set whose self-product E· E is contained in an H-set, then underlinedim_B E≤1/2.
ER -